Линейные формы и статистические критерии. ІІ
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| Datum: | 1953 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | Russisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1953
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/7733 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512726228402176 |
|---|---|
| author | Линник, Ю. В. Линник, Ю. В. |
| author_facet | Линник, Ю. В. Линник, Ю. В. |
| author_sort | Линник, Ю. В. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-08-15T10:12:02Z |
| description | - |
| first_indexed | 2026-03-24T03:33:22Z |
| format | Article |
| fulltext |
1953 YKPAI1HCKI1Y! MATEMATI14ECKI1YI /KYPHAJI
HHCTHTYT MATEMA THKH
JIHueiiHhie Q>opMbi H cTaTHCTwteCKHe KpHTepHH. II
( 0J<OH'l a~me)
10. B. JluHHUK
§ 33
T.V,N~3
Bo3spamaHCb K npe,n,c-rasJieHmo (20,9) H y<IHTbiBaH (21, 1), nepemuneM
lf>opMyJiy C y<IeTOM (32,3) TaK:
('1f12 (u)d =-u2 + 8 () + _1 ~,, tfi( )• E(z) dz
) u u 2 1 u 2ni ..:. 1 ';) llUo za (z) ·
0 m=-oo Llll
(33,1)
PHJJ. cxo,n_HTCH paBHOMepHo B [0, 1], H eCJIH K(a) HJl.pO yKa3aHHoro s § 32
THna, TO
L au L u u L
( ( (IP2(u) \ () llptz(az)dz=(dllu(K(~)u'z(~u) ·d~. J K a) da J ll du = ) K a da J J J ~ r· ~ ~
0 0 0 0 0 0
DpH 3TOM, npH Lu < 1 Mbi MO)I<eM HCnOJlb30BaTb (33, 1 ). AHaJiorH'!HbiM
o6pa30M, ecJIH K 1 (a1), . .. , K q (aq) H,n_pa Toro )l<e THna n L 1, L 2 , •.• , L q
COOTBeTCTBYIOW:He sepxHHe npe,n_eJibl HHTerpaJIOB, noJiy'IHM
Lq Lq-l L 1 a1a 'l . .. a 9 u
~ K q(aq)daq ~ Kq_1 ((Xq_1)daq_l · ·• ~ K 1(a1)da1 ~ lp2u(u) du=
() 0 0 0
u Lq '-q-1 L,
= ~ ~ ~ Kq(aq)da9 ~ Kg_1 (aq_ 1)dag_1 • • • ~ K 1 (ai) da1 llf2 (al. · · "'2u ), (33,2)
0 0 0 0
npwieM npH L1, L 2 , . • • , L ,,u < 1 MmKHO .ucnoJib30BaTb ¢opMyJiy (33, 1) .
DycTb 1Jo < 0,01 cjJHKCHposaHHOe noJIO)KHTeJibHoe 'IHCJIO. D oJIO)I<HM
'I' = 1! (r) = 4rl-t· •:. r = 1, 2, 3, . . . ) H sse,n_eM '!HCJia
1 1
L,= 1 + - = 1 + 4 t +•·
l' r 10
(33,3)
H H,LI_pa
a• v+ l
K,(a) = y:;-- = L•+l a~.
S a• da r
(33,4)
0
247
PaccMOTpHM orrepaUHIO (33,2) rrpH KaKo~t -Jm6o u rro,LI. ycJIOBHe~t
Lt , L 2, . .. , L qu < 1.
1 1
,llJIH BCHKOrO KOMITJieKCH0£'0 Z 113 rrp mviOyrOJibHHKa Ot - 2 < X .:::;;: lit + 2 ,
m < y < m + 1 flOJIYliHM
L•-1
r
z- 1 ·
1+ v+ 1
IloJIO)I<HB Z = X+ iy, HaXO).I,H~I
z-1 x - 1 iy
1 + v + 1 = 1 + I'+ 1 + ~-+ 1 ,
OTI<y,LI.a
I z-1 ~2 ( x-1)' y'J m 2
1 + v+ 1 = 1 + ;,+ 1 + (1•+ 1)~ > 1 + 2 v2
3 1
BBH,LI.Y TOrO, '-ITO X - 1 > IJ1 - 2 > 2 ·
.li.aJiee, v = 4r1+•; •• EcJIH q ): 2 [ m [, TO ITOJI)''-IHM
2m lml nl z-1 ~ ( ct )2 t-2•;, 1+ - - > 1 + ~ > ea, l ml ,,+1 m-•,., '
r=m
r,LI.e ao, a l flOJIO)I<HTe.TJbHb!e KOHCTaHTbl.
LI.aJiee,
I £ •-1 I = I £><-I I ~ {1 + _ 1_)o·,-1 .
,. ,. """"' \ 4r1+•;.
(33,5)
(33,6)
BBH,LI.Y CXO,L!.HMOCTH pH,LI.a i: _ l _ H MeeM ,Ll.JIH JII06oro q > 1 H .'IIOOOfO Z
r=l r1+r..'
yKa3aHHOH flOJIOCbl
q
r=l
r,LI.e a2( '1}o) flOJIO}J<HTeJ1bHa51 J<OHCTarna, 3aBHC51lUa5I TOJl bKO OT ?j0,
.llaJiee,
q
r=l
BBH,LI.y (33, 1) , (33,2), (33,6), (33,7) "H (33,8) rrpH BC5IKOM
1
q > 1 H U < - (- ) HMeeM
218
aa r,J
u l.q Lq-1
~ ~u ~ K (aq) du9 ~ K (uq_1) daq_ 1 •
0 0 0
I.,
. . ~ K(r;.1 ) da1 'P, (a1 u2 •• • :x2 u) =
0
1 ~' p· E(z) dz =-A (qlu 2 +S(ql(u) + - . L Al9l (z)(uu0) • ___:_....!......,._
1 2nl m = -00 "' za (z) '
Lm
(33,7)
(33,8)
(33,9)
me rrpH JJJ060~! q, A (q) H Bee K03cpqmu.He HTbl s~q) (u), a T3K)Ke Bee cpyHK
U.HH A~l (z) orpai-IH'-IeHbi rro a6eoJJionroi'I Be.riH'-!HHe a6eoJJIOTHoif KOHeTaH
q
Tal-i A <a: l, H rrpHTOM ,nJJ H I m I < i, HMeeT MeeTo ou.eHKa
I A~l(z) 1-< exp (- a 1 1 m 11_ 2.''). (33,10)
I(pOl\!e TOfO, H3 (33,5) li!OfKeM 33KJJ JOtiHTb, '-ITO I1pH ,J.8HHOM r H y ~ + 00
Jm ec llr
Lr
I ~ K r(f3){3zdpl =B(r) J ~ J ' (33,11)
()
r,ne B (r), KaK u paHce, orpaHH'-ICH II OC 'IHCJJO, 38BHenmee OT r H He o,n:Ho
11 TO )Ke ,nJm Beex y.
0Te!O,LI.3 3 8KJII048e M: rrpH q > 2 p51,U (:33,9) 6y,neT a6eOji!OTHO eXO,UHTbeH,
1
11 rrpu u < - (- ) ero "O)!UIO ,nucj:>cpepeHu.upoBaTb rro u. BbiiiOJ1H51H 3TO,
a3 "'o
I10•1Y4H~ l H3 (33,9 )
Lq L q-1 L 1
~ K(ct.q)drxq ~ K(uq_ 1)duq_ 1 ••• ~K(r~1 )dv. /P2(o:1a2· · . aqu)=
0 ll 0
. 1 .::,, p _E(z)dz
=-2A(ql u 2 +TlqJ(u)+ - . ) Alq)(z)(uu0 ) - ,
1 2nz m~-:xl . m a(z)
Lm
r,ne
§ 34
06o3Ha4 rnr orr epaTop
Lq L,
~ K(r•q)daq .. . \ K(o:1)do:11J(C1.1rt 2 ••• a1u) 'Iepe3 Lq(p).
\) 0
06palll35leb K (32,4) H (32,5), ll3XOt"lH~l
Lq(/ 1) = 1 + L q(W2 )+ Lq(g},
r.Je L ,,(f1) HoBan x. cp . e.riy'-laihwi'I BeJJ U'-IHHbi , TaK)Ke l!enwH.
flpH yeJ10BHH
(33,12)
(33,13)
(34, 1)
1 0-< u < - ( - ) (34, 2)
as ''7o
Lq(W,) Bbi pa)KaeTen pa3JJmi<ei-IHeM (33,12).
1laJ1ee, rrpu yCJJoBHU (34,2), BBH.UY (32,4) H (32,5), Haxo,nHM e yqe
TOlll ( 28,3) ~ ( u) = B u4 ; rrpu JJIOOOM q > 2
Ha1a2 ... aqU) = B(at, .. , aqtt) 4 = Btt4 rrpu O< a; < L, OTKy,na
paBIIO\!ep!IO ITO q
(34,3)
249
PaccMoTpHM nocJie,lloBaTe.Ji bHOCTb cpyHKU.Hi'i L" (/1).
BBH.llY Toro, 'ITO .a 1a 2 • • • a" < a 3 {1Jo) np11 scex q, ~Ibi BH,Ll,Hl\1, l.JTO
L" (/I) paBHOCTeneHHO HenpepbiBHbi H orpaHH'ICHbi B ( - =, =). BBH.llY
::noro B03MO)l{HO Bbi6paTb no,Ll,nocJieJJ.oBaTeJibHOCTb L,,i (/1 ), paBHOMepHo
CXO,ll,Hlll.YIOCH B K3)!(,ll,OM cerMeriTe K Henpepb!BHOH Cj)yHKU.HH [} ( ll), KOTOpaH
6y,ll,eT xapaKTepHCTH'ICCKOH ( CM. [. J<paMep [8]).
Bo3bMeM noCJie,ll,oBaTeJibHOCTb L"i (!1) , cxo,Ll,Hlll.YIOCH K g (u). llJIH ~~:arli:
,li,Oro '!JICHa nOCJIC,Ll,OB3TC.TJbHOCTH HpH 0 < U <"...:: U1 6)',Ll,CT Hl\ICTb ~ I CCTO cpop
MyJia (33,12). CooTBeTCTBYIOIU.He cpyHKU.HH A~~i 1 (z) perymipilbl H orpa
HH'ICHbl Ha KOI-ITypax Llll paBHOMe pHO no m. no TeopeMe, ).l,OK33a!-lllOi·i
B KHHre n. MoHTCJl51 ([9], CTp. 28)' Mbl 1\IQ)J(C~I ,ll,Jl51 m = 0, 1, 2, .. . Bbl-
opaTb no,ll,nOCJIC,ll,OBaTeJ!bHOCTH, CXO,ll,5llll.HCC51 !{ peryJIHpH bl M A~"') (z)
O}lHY HJ .llpyroii. B peJyJibTaTe JJ.JIH npe,ll,eJibHOi·i cpyHKU.HH npu 0 <; u < u2 =
= u.) llOJIY'IHM cpopMy.rJy THJla (33, 12). DpH I u I < Uo npe,ll,eJibH351 cpyHKU.HH
6y.lleT no.rJO)KHTe.TJbHOti H In g (u) HenpepbiBHbiM u peaJibHbi M.
§ 35
060J H3'1HM noJiy'IHBWHCC5! npe,ll,CJibHble cpyHKU.HH L!e p e3 [} ( u), h ( U), C ( tt),
T3K liTO
g(u) = 1 + h(u) + e(u); I u I < Uo. (35.1)
BsH,ay BbiwecKa3aHHoro, npH 0 < u < U0 no.riy'IHM
h(u)= - 2E0u2+T<"'l (zz) + ~ ~~ rh A ( "') (z) (llUo)z E(z) dz (35,1')
' 2nl ..L.J J m a(z) '
m=-oo Lm
r,ll,c Eo, KoacpcpHU.HCHTbi Tl"'l H I A\,':"l (z) I orpaHH'ICHbi n o MO.llYJIIO a6co
JJIOTHOH KOHCT3HTOif.
llanee,
e(u) = Bu 1
I A~"'.> (z) I< exp ( - al l m l1- 2 '!u) HaL"'.
(35,2)
(35,3)
EcJiu P1, P~. . . , P"" HeoTpuu.aTe.TJbHbiC l!Hc.na, B cyl\Il\Ie paBIIbie 1,
N
a fi 1, fiz, ... , f3s HeoTpuuaTeJibllbie •l!HCJI<:J, TO ~ Pjg({Jjll) 6y,ll,eT X. c!J.,
j = l
npH'IeM np.H 0 < fJiu < Uo 6y,ll,eT scpHa cpopMy.ri a
N N ~
~ Pih({:J7u)=- 2E0 (~ Pij' ) u 2 + 7~ PjT~"'l([3ju) +
1 "' , .
+ -- ~ A; A<"'l(z) (uu )zp (z)E(z) dz
2ni m= - oo ;t' m o N a(z) '
(35,4)
r,ll,e
'" PN(z) = ~ P/j ·
j= i
DycTb 3a,Ll,aHo u eJioe l!HCJio M > 2. PaccMoTpH l\I see noJIIOCbi cpyHKUHH
_ z E (z)
WN(z)- (uuo) P :-,- (z) a(z) (35,5)
250
H nocTaBHM ce6e 3a,n:aqei1: nocTpOHTb TaKyiO cpyHKU.HIO PN (z), tiT06bi <DN(z)
He HMe.TJo noJJIOCOB B KOHTypax Lm npH I m I < M H tiT06bJ npH 3TOM
tiHCJJa fli He TIOJJytiaJJHCb CJJHUJKOM 60JJblliHMH.
Bo3bMeM KaKoi1-JJH6o noJJIOC ( aKTHBHbiH ny.'lb o (z)) a KOH1)'pe
Lm( I m! < M); nycTb sTo 6y,n:eT (}=a+ i7:; 3,n:ecb o, < a< o2 ; m- I<
< 1; < m + 2. Bae,n:eM tiHCJJO {Je = I + .d~ , r,n:e L1 e > 0 no,n: yc.TJOBHeM:
fJg= -(1;, T. e. {J~~ = (1 + LJQ)i< =- 1. QqeBHJI:HO, MOMHO Bbi6paTb ,diJ
no,n: ycJJOBHeM
(35,6)
UoJJaraH Tenepb A(, = {1~, ilOJJYtiHM, tiTO cpyHKU.HH A~ + {J~ HMeeT e npo
CTbiM nyJJeM. CocTaBHM npoH3Be,n:enHe ll(A~ + flP no seeM noJJIOcaM B Lm
rrpH I m I < M, CtiHTaH KpaTnocm.
CorJiacno JieMMe III tiHCJJo rroJIIOCoB B KOHType Lm He npeaocxo,n:HT
KOHCTaHTbi a5 npH acex m. BBH,n:y ~noro Hame npoH3Be,n:eHHe pa3aepHeTCH
N
B cyMMY BH,n:a ~ A; (3~, r,n:e tiHCJia f3i rroJiy<IaiOTCH KaK rrpoH3Be,n:eHHH <IHCe.TJ
j=l
fle B J<OJIHtieCTBe, He rrpeBOCXOJ1:5IIU.eM 2a5M. 0TCIO,n:a HMeeM .li:JIH BCHKOfO
j<N
r,n:e
M( a)2a,
1 < {Ji < C23 fl 1 + __! < Ma• .
Ill
m = l
lY
.ll:eJJ51 cocTaBJJeHHYKJ paHee cyMMY Ha ~ A;, rroJiytiHM cyMMY
1-l
JY
PN(z) = ~ P/Jj,
i=l
N
~ Pj= 1; 1 < fJi < M a•.
i=l
§ 36
(35,7)
(35,8)
(35,9)
06paTHMC51 K (35, I) H rro 3a.LI.aHHOMY M ;> 2 rrocTpoHM ¢YIIKU.Hio g M(u)
yKa3aHHoro Bbime THrra. UoJIYtiHM
N N
gM(u) = 1 + ~ Pih({Jiu) + ~ Pje({Jiu) (36,1)
]=1 j=l
(36,2)
Pa3yMeeTCH, Pi> f1J H N 3aBHCHT oT M. CorJJacHo cKa3aHHOMY Bbillle,
11pH
(36,3)
HMeeT MeCTO (35,4).
25 1
flyCTb )!,aHa KaKaH-JJH60 qeTHaH X. cp. f(u) l1 COOTBeTCTBYIOIUHi{ eii
00
3aKoH pacrrpe,ll,eJJ e iHIH V(x). Tor)J,a HMeeM f(u) = ~cos ux dV(x).
Terrepb 6y,ll,eM CllHTaTb, liTO _ oo
2 < 03 < 4.
HcrroJJb3yeM paseHCTBO
si rr 4 ux = } (3- 4 cos 2ux + cos 4ux),
113 KOTOporo BbiBO,ll,llM
00
3[(0) -4f(2u) -f-f(4u) = 8 ~ sirr 4 ux dV(x).
00
- 4hM(2u)+h111 (4u)-4eMG2u)+eM (4u)=8 ~ sin'1 uxdV(x).
- 00
Ilo.10)!<HM Terrepb
(36,4)
(36,5)
(36,6)
(36,7)
Tor)J,a rrp11 )l.OCTaTOllHO 6oJJbUJOM M (36,3) 6y,ll,eT HMeTb MecTo ,ll,JJH 4u,
TaK 'lTO 6y,ll,eT sepno 11 paseHCTBO (35,4). Pa3,ll,emiM paseHCTBO (36,6) npn
U, paBHOl\1 (36,7), ITOIJJJeHHO Ha U 4 H OIJ.NIHM CBepxy Bblp3)!{eHHe
00
~ sin 1 uxdV(x) (36,8)
- · 00
rrpH ITOMOLUH .TJeBOH IJaCTll.
I1MeeM rrpe)!{)J,e scero
.N
eM(u) = ~ Pie(f1p). (36,9)
, - l
Cor.TJaCHO (35,2) rroJJyiJHM
N
I eM(u) I= Bu~ ~Pi;, (36,10)
i=l
OTI{y,ll,a
(36,11)
.UaJJee, ou.eHHllt
(36,12)
Jl.'IH .3TOfO HCITOJib3yeM ( 35,4). flo ITOCTpOeHHIO p N ( Z), COOTBeTCTBYIOIUero
IJHC..JIY M, see KOHTYPHbie HHTerpa.TJbi HClle3aroT rrpH I m I < M. JlaJiee, H3
(35,8) Ha scex L 111
(36,1 3)
252
Bsn.ny 3Toro cor.nacHo (35.4) npn m > M H "}o < 0,01
I rfl A(""l(z)(zw )z P (z) I!__~) dz I=
;) m o N u(z)
(36,14)
Cy;-.IMa KOHTYPHbiX Hmerpa.los c ! m I> ~M HMeeT B BH;IY (36, 14) ouenKy
B (c) u",- ' exp (- ~ M 0•9) . (36, 15)
§ 37
Bo3sparii,a5Icb K cpopl\IY•lHposKe JieMMbi XI, i\ Ibi BH}IHM, tJTo eum .TJeMMa
HesepHa, TO s 2 > a 3 + 2 > 4, TaK tJTO cyMMa - 2E0u2 + T\"'> (u) = T(u)
Ill\!CeT BHJJ:
T(u) = Eu2 + Bul . (37, I)
0TCIOJJ:3 H3XOJJ:H"I
- 2T(2u) + T( 4u) = Bu 1•
BsH,n:y 3Toro s Bbipa,KeHHH nma (35,4) JL.TIH - 4hM(2u) + hM(4u) BHe
N
mnerpa.1bHbie lJJieHbi HMeioT ouermy Bu4 ~ Pjl~.
BBH}LY (36,15) HaXO}LI!M
l
-4hM(2u)+ hM(4zz) I B (c) u2-• exp ( - ~ Mo,n) N
--- --zz-, ---- = zzt + B ~ P)3~.
,= l
(37,2)
l13 (36,6), (37,2) H (36,1!) 33KJIIOlJaeM, lJTO
~ si::ux dV (x)< C2 ; ;~ PiJj (37,3)
- - 00
,Ll,.TI5I scex 6oJiblllHX iV!.
3 }recb l'(x) - 3ai<Oll pacnp e;:re .TT emm .nm i x. cp. gM(u). ]la,1ee, HMeeM npH
lux I<;-' I X ! < ;~' sin 1 l!X > (~ llXr' BBH,Ll,Y t:Iero
2u
TaK liTO (37 ,3) }L3CT
2u
\ x 4 dV (x) < C2 u .1~ PJl; ·
"'
(37,4)
2u
y
,Ua.TJee, H3 (36,2) gN(u) = ~ Pj g(f/p). ECJm o6o3HalJHM qepe3 W (x)
• -- l
33KOH pacnpe,LI,e.lei-IH5I ;:1,.'151 X. Q:>. g ( U), TO g ({Jill) 6y}LeT OTBe<J3Tb 33KOH pac-
253
npe)J,eJieHHH W (i.), a X. <P· gM(u) 6y}l.eT OTBC43Tb 3 3KOH pacnpe)J.e,JieHHH
J
0TCIO)J.3
BBHJJ.Y TOro, t.ITO flj < Ma', u = e-Vu, uaxOJJ.HM
n n y- 'VA1 - - ::;:;: - e M M-a, > e" = M
2uflj c:Y 2 1
JJ..'IH M > const. TorJJ.a (37,6) )J,aeT
:!u J.l/1
~ x ·ldV (x) :;?- ~
"' -111,
-2u
N
x!dW(x) ~ P .(3·1. L..J J J
j=i
N
CpaBHHBaH c (37,4) H coi<pau(aH Ha ~ Pi~,. no.TJy tiaeM
j = i
!If,
~ X~ dW(x) < G2G ·
- .\11
(37,5)
(37,6)
(37,7)
(37,8)
'V-
3 ,J,ecb M 1 = e·2 M l\lmKe T 6biTb CJJ.CJiaHo CKOJib yroJJ.HO 60JlbW IBI, a W (x)
ecTb 3 3KOH pacnpe,L( eJi e iiHH )1..!151 x. ¢. g (u), He 3 aBHCHIJ.Uiii OT 1VI1• BBHJJ.Y
00
~JTOI'O 33I<JII04<l ei\l 113 (37,8), liTO \ X 4 dW(x) < C~6 , T. e. pacnpe)J.eJICHHC
-oo
W(x) m1eer llemepTbiii MOl\I CHT a 1(W).
§ 38
<Pymm.IIH g ( u) n oJJYliHJJach c noMOU(biO o n peJJ.e.TieHHoro npeJJ.eJibHoro
nepexo~a H3 ¢YHKUHH
Lq L,
L 2 (/J) = ~ K q(o:q)do:2 ••• ~ K 1 (o:1) d o:J 1 (o:1 ••• o:2u),
0 0
r)l,e /t(u) = e-u'+'!•,(u) X. Cp., OTBe4aiO!l(351 n JIOTHOCTH B Cp051THOCTH y(x),
HenpepbiBIIol'i H orparmtiCIHIOl'i Ha ucei-i ocH. XapaKTepHCTHlleCKOi-i cpyr·IKUHH
L q(/I) 6yJJ.eT OTBel!aTb nJIOTHOCTb
L~q l~. , d K 1 (o:J .. . K 2 (o:g) ( x ) ( )
drt 2 ••• 0'.1 ,fj = ,fj x, q.
• . • a . .. . a 1 .al ... .);1 ..
(38,1)
254
3aMeTHM, 4TO B CHJIY (33, l)
Kr (ar) _ 1' + 1 a•·-l
~ - M;:+l ,. He npepbiBHbl rrpn 0 < ar < Lr·
Mbl )l.OKa)KeM Tenepb, 4To )l.JJH nJJ OTHOCTH y (x) )l.O.TJH\eH cymecTnonaTb
qeTBepTbiH MOMeHT
nycTb 3 TO He Tal<. Toma npu )l.OCTaT04HO 60JJbWOl\I M HHTerpaJJ
1~1
~ x 1 y(x)dx
-M
l\IO)!{eT 6b!Tb C)l,eJiaH CKO.TJb yrO,ll,HO 60.TJ b llli1M.
:Vb (38, 1) noJJyqael\r
M
(38,2)
(38,3)
},! Lq L 1 a1 ••• u q
~ x 4y(x, q)dx = ~ daq . .. ~ do:1 /{1 (o:1) •• • K2 (uq)o:~ ..• o:; ~ x 4y(x) dx >
-},f 0 0 - J.ll
ill
> r,:f' x'!J(x)dx
- M
~-::I:;,
Lr
a 1 •• • a q
Lq L,
~ da2 ••• ~ du.1 K(o:J)· . . K(o:2) o:; ... :~.~ . (38,4)
0
,Ua·Jiee, \ e~.: K(a,.) do:,.< L,", H no (33,8) Lh L,2 ••• , Lq ('_ a 8(170). 0TCIO,ll,a
0
M
M a
~x4y(x,q)dx > c27 ~ x 4y(x)dx,
- ,1! M
- --
"
X
06o3 J-Ja4HM Y (x, q) = ~ y (x, q) dx. Tor)l.a HiiiCeM npH )l.OCTaTO'IHO
M
6oJibWOM M mnerpa.n ~ x '1 dY (x , q) > A, r,ll,e A I<aKaH yro)l.HO npe;I.nucaH-
- M
HCI51 KOHCT3f1 T a. Ope,ll,CJibl-!i.IH X . <j.J. g(u) HMeeT COOTBeTCTB)'IOll.(HH eii 3 3KOH
pacnpe,ll,eJICHHH W (x) , 11, no H313ecn-Ibl~1 Teope~taiii Xe.'IJIH, nO.'l)"IHM
Jj[
~ x·1 dW (x) :>A npH mo6o 11 r A 1-1 no,TJ.XO)l.Hll.(eM M. 3Ha'IHT, <JeTnepTbiii :\IO-
- M
l\Iein a.1 (W) He cyme CTsyeT, 4TO nponmope4HT Bbi BO/ IJlM § 37.
255
YJTaK,
(38,5)
- oo
n.:wnwcm s epomi-IOCTH y (x) cor.TiaCHO § 32 OTBel!a.na npH u > 0 x. ¢.
II (u) = e'I~ (nl- u'. CorJiaCIIO H3 BCCTHbTM TeopCM<l~l [8] cyrn.ecTBOBaHHe a4
B.'1elle T cylll,eCTBOB8!-IHC 'ICTBCpTOH npOH3BO}lHOii fltl (u), Henpe pb!BHOii
11 orpaiiwremwii 1-w nce}i ocH. B tiaCTHOCTH, JlOJI)Kel! cyiJLCCTBoBaTh
lim f~ll (u).
u~+o
.Ua.Tiee, B cer~rciiTe [0,1 ], !I (u) =I= 0 no noCTpoe i-Imo l!HCJia tlo. BBI-I;ly
:noro B cerl\reHTe [0, I ] cyuLeCTBJ"CT lJ-'\41 (zz), a B cenretne [0, uo] cyw:eCTsyer
Henpepb!BII35! <p!4l(zz) H, B liilCTIIOCTH, lim <p(4) (u) ( 3TO CJJe}l)"CT 1!3 TOfO,
(
ll-+ + 0
'-ITO 1Jf ( u) = lift z~J ) .
H o cor"1acHo .·IeMi\tC IX B 3TOM c.nyqae }J,OJDKIIO 6biTb
3TO npoTHBopel!HT npe}lriO.TIO)J\eiimo 2 < o.1 < 4, H, cTaJIO 6biTb, .r r e~I\Ia XI
JlOKa3aiH1 npH Tai<Ol\I orpaH rrqe uun Ha oa. T enepb 6y;Le ~r npo}lOJI)K3Th ee
JlOK33ilTe:IbCTBO, CtiHT<IH, liTO 03 > 4.
§ 39
oy}leM Cl!HT3Tb O:; >- 4. Ec.TIH JIC\l~la XI l!CBCpH<i, TO f!OJIIOC 82, 0 KOTOpOM
ronopHTCH B ee cj:Jop i\I)'.'IHposKe, }lOmi<eH YJlOBJieTBOpHTb ycJIOBHIO
(39, I)
ll.lll B006m:e He C)'I.UCCTBOB<lTb.
Bse)J:e\1 qerHoe l!HCJIO 2p no}l ycJrOBHe ~r: 2p - IImnreiJbtuec l!CTHOC lJHc.no ,
KOTopoe 6oJJbille o3 , T. e.
2p - 2 < 0 3 < 2p.
Bo3BpaiLLa HCb K § 32 11 33, coorHourcHHC (33, I) H:JnHmei\r:
r.n:e
11
\ P2 (u) U 2 J - u du=- 2 + S1 (zz) + I1 (u),
\)
I (ll) - 1 ~· f'h (uzz )J!J'(:zl dz·
1 - 2"i L.l ;t' 0 za (z) '
ru --.:.-00
Lm
ll~
- ·2 + S1 (u) = Vo (zz) + V 1 (u); V 0 (u) = H BBHJlY (39,1)
r .n:e i,' > 0 !10.'10)IOHCJibH35! KOHCT<IHTa.
(39,2)
(39,3)
(39,4)
(:39,5}
JleB3 5I 'l3CTb (39,3) JlHcpcpepeHUHp)'eMa npH ll > 0, TO )1\C I<aCaeTCH
SI (u). 3HatJHT, II' (u) cymecrnyeT npH u > 0. Bee 3TH npoH3BOJlHble Henpe
pb!Bllbl npu u > 0.
256
noJraraH - u2 + uSt'(u) = S2(u), ult'( u) = l z( u), l!all/J,C~I
lJI2 (u) = S z (u) + l z (u). (39,6)
Mbi IIaWJIII B § 3 1, <tTo P '(u) H W"(u) cymecTsytoT H HenpepbiBHbi
B [0 ,1]; TO )1\C Kaca e TCH 'F/ (u) H P 2"(u). I1MeeM: 'ZV(O) = 0; lfl/'(u) = B
Dpii ll E [0,1]. 0TCIO,TI:a:
'Pz' (u) = Bu. (39,7)
T e rrcpi. .n.oKa)KC~I oaer-IKy:
(39,8)
I13 (39, 1) BbiBO.D.HM, liTO s2>4. Bsn.rr.y 3Toro S~'>(zz) cymecTBYIOT H orpa
HHlleHbi B [0, 1] rrp11 n <; 4; Sz'(O) = 0. Tor.rr.a H3 (39,6) rroJiyllaeTCH , liTO
f / (u) l:I / 2" (u) TO)KC cymeCTBYIOT Ii orpaHHliCHbl , 11
12' (u) = Bu; [2 (u) = Bu2 . (39,9)
I1MeeM .rr.aJree, HCllOJib3YH (21, 1),
ll
\ I . (zz) J ~du = /I(u) = B(E)zz"'- ' · (39,10)
0
EcJIH ~ liHC.ilo < 1, To H3 (39,10) HaXO.lJ.HM
(39,11)
nycTb s up !l z (.!l) I= ~ 13 npH ll E r l , §]; 00;;::,: fJ > 1 rrpH .lJ.OCT aTOlii-10 Ma
.T!OM ~- MriO)KCCTBo TOlleK u, .D.JI~ 2rwTopbiX l lz(u) l= ~;ii npn ue[~. ~ ]
3 3!\IKHYTO B CHJIY HenpepbiBHOCTH / 2 (u ). 2
EcJIH yKa3mmoe M I-rmKecTso rryCToe, TO / a(u) coxpaHHCT 3HaK (HJIH 0)
s Haw eM cerMeHTe, 11 Jl z (u) I> -~ ~~. OTK Y .rr.a
~ ~:' du ~ c,.o' ~ B(,W·-.
.;
2
O TJ(y.lJ.a
(39,12)
Ec.TJH OHO HCllYCTOC, TO B H3WCM cen1CHTC Har~i.D.CTCH TOliKa ~2, 6.'lH)K3HWaH
c .rresa K lWI Ke ~ ~ . r .n.e .n.ocTHra eTcH supremum (me / 2 (s) = + §~). Jin6o
TaKa 51 )KC TOliK3 ~a Crrpasa, ripHli C~I /2 (s~) = / 2 (Sa)=+~ S'\ H 3 H 3K / 2 (zz)
coxparrHeTCH Ha centeHTax [;2, §t] H [ ~1 , s3].
D yCTb .lJ.JIH OllpC.lJ.CJICHHOCTH 3TOT 3H3K ITOJIO)KliTCJJCH , H CY lli.CCTBYCT
cerMe HT [~ t , §a]. Tor.rr.a HMeeM :
l12:(§a)- /2 (St) l = I I2 '(.'t)l(§,-~r); .'l e[~r• ~n1·
237
(39, 13)
·. ..
I ~ I:?z) du i = ~ I 12~u) I du > f.~ ~r (;3- ~I) >} c29 ~!~-I .
·' ' .,;1 .;,
CpaBHHBaH c (39, II ) , Haxo,LJ,HM
g2,g-I = B(sH''•-'; (3 > a3 +21-s;
0TCIO,LJ,a: np1:1 U€ [ }, ; J
cr,+1+•'
! 2 (u) = B (t') u- 2 - (s < ;0(s') ). (39, 14)
Cp aBHHBa51 c (39,12), BH,LJ,HM, lJTO ou.eHKa (39, 14) sepua no ncex cJiyqaHx .
O,LJ,HaKo oua ne,LJ,ocTaT0411a .UJIH ,LJ,aJibHeilumx u.eJi eH. 4To6br ee YJIY"IlllHTb,
.D.OCTaT04HO yJiy4WHTb ou.em<y I / (u). l1Meel\r corJiacuo (39,14)
$ "+1-•
~ I2'(u) du = !2 (s)- !2 (~) = B (<"); ~.
1:
2
Il,aJ1ee, I z" (u) = B. 0 6o3Ha4aH sup I /.2' (u) I= ~r H paccy)K,LI.aH, KaK l\!bi
paccy)J(,LI.3JIH BbiWe OTI-!OCHTCJibHO I 2 ( u)' l10.1IYlJHM s3- sl > Cu~sfl, H
cr,+t-E
!_ c,s • C. -!:,3 = _!_ Cso§211 = B (e) ~-2-
2 !> 3(, .., 2
f.l t1n + 1-e' ( - c ( ' )
I' > - -4 - - ~ < !>o E ) •
0TCIO,IJ;a, I-13KOHe U.,
<11,+ 1-E
12 ' (u) = B(c)u- 4 -
D o,LI.CTaBHB 3TY OU.eHKY B (39, 13), H3XO,LI.HM
<T3+1-E
. t I 12 (l;g)- I! (~t) I -....... c ( ) / - -· -4 -
~S-!>1 = lf2'(.'J )I / 29 li -,
0TCIO,LI.3
corJiacHo (39, 11) .
258
<1.,+1-E
C (c) 2~- 1--·--4 - .
~§ = B(eH"·- ·
(39,15)
Eho .u.aeT
(i > _,cx8,_+_ 1_-_r: + _0"3_+_ 1_-_E
2 8
ll
2(u,+tJ- •
fz(u)=B(r:)ull .
.3TO ll eCTb )KeJJ3eM351 OU.CHKa.
Ho
§ 40
)l.aJJee .u.eiicTsyeM, KaK B § 32: paccMaTprmaeM
f 1 (u) = e '~';(u)=e8•(u)el, (u) npH UE (0,1).
YLIHTbiBaH oueHKY S z ( u) = B u2 H ( 39,8) , HaXO.lJ.HM
! 1 (u) = e8 •<uJ (1+/2 (u)+B /12(u) J!).
HaXO.ll.llM:
1
c = - , TIOJJYLIHM
8
1
I 12 (u) J2 = Bu2P+s
[2p LIHCJJO ll3 (39,2]. )l,aJJee, COrJJaCHO (39,5)
eS• (u) = e v, (u) + Bu2p+~; '."> 0,
p
TT ( ) - ~ r 2lll .,2 u - ~ . 11 ~,u ,
m=l
T3K LITO ( 40,1 ) nepeDHllleTCH TilK:
f 1 (u) = e v, (u) (1 + / 2 (u}) + Bzz%P+ ~.
3 aMeTHM eme, LITO
BBH.li.Y 3Toro, ( 40,3) MO)KHo n epenHcaTh
21>
f 1 (u) = 1 + v 2 (u) + ~ anu"/2 (u)+ Bu2P+~ .
n=l
Mbi HMeJJ.H B (39,4) Bbipa)KeHHe .U.JIH / :I (u) :
1 ~, · tfi z E(z)
11 (u) = 2ni ~ ;;r (uuo) za(z) dz,
m = - oo 1
Lm
(39,16}
(40,1}
(40,2)
(40,3)
(40,4)
(40,5)
(39,4)
npH 3TOM 12(u) = ul{ Cu). O .u.HaKo pH.U. (39,4), soo6me rosopn, HeJJb3 51 .U.H<P
<PepeHUHpoBaTb DOLIJJ CHHO, TaK KaK DOJJYLIHTC51 p aCXO.lJ.Hl.ll.HHCH p5I.lJ. .
259
Mbr )J.OKa)KeM, O,'J.Hai<o, 'ITO npH u.e.ri biX no.:JO)KHTeJibHbiX n HMeeM _
u -
\ _u_"l--=--2--'----(u--'--) du= _I ~· A; u=+nu z _ E_(_z) __ dz
) ll 2 Tci m f::_ oo L:t' 0 (z + n) 0" (z) •
0 m
( 40,6)
B caMoM .ne.Tie,
U II l l r u"l (u) \ \ ~ --z~ du = ) u"/1' (u) du = u"ldu)- ) nu"- 111 (u) du.
Cl 0 0
PHi l. (39,4) MmKHO )'MHO)KHTb Ha nu"- 1 11 HI-ITerpHposarb nO'IJreHHO. O T,neJib
:Hbrii 'IJieH ero .nacT
m u z+r• ( 1 - ___!!_____)- uz E (z) dz = ..-h uz+nuz E (z) dz '
";j" n+z • za(z) ~ • (z+ n)a(z)
Lm
'ITO H npHBO)J.HT K ( 40,6). 3aMeTHM, 'ITO BBH)J.Y TOfO, 'ITO CTeHKH KOHTypoB
1
Lm JieMaT npasee -03 - e ;> 4 - e, - - p e ryJIHpHo BHYTPH HX H Ha HHX.
z+n
06o3Ha'IaH, .naJree,
H
2p
13 (u) = ~ rr ,.u" !, (u),
n=l
II
~ 13 (u) 1 ~-;, p -·- du= -2 . uzH(z)dz,
ll ILl
0 nt = -oo L,n
ft(u) = 1 + V2(u) + l 3 (u) + Hu),
np114eM pH ,a, ( 40,8) CXO,G.I1TCH paBHOMepHO no U E [0,1 ], H
~(u) = Bu2P+C.
§ 41
(4~.7)
(40,8)
( 40,9)
(40,10)
T enepb B ( 40,9) npHMeHHeM paccy)K)J.eHH e § 33 11 34. M MeeM s o6o-
3 Ha4eHHHX § 34
npu yCJIOBHH
1
O <; u < - .- ).
a3 V'7o
( 41 '1)
(41,2)
CosepillHB npe,neJibi-IbiH n ep exo.n, yKa3a HHbiH s § 34, noJiy'IaeM rwsyro x. ¢.
g(u), npH4eM
g(u) = 1 + h(u) + e(u); I u I<; Uo.
Dp11 0 < u-<;: U0
h(u) = V~"' l(u)+ 2~i i:' g'> Al,:)(z)uzH(z)dz, ( 41,3)
m=-oo Lm
•160
r.II,e V\""l (u) '-leTHbiH noJIHHOM cTeneHH 2p, H A~:' I (z) y,ll,OBJieTBOpHIOT
YCJIOBHIQ:
IA~l(z)l < exp(-atl m II-2'lo),
T3K '-ITO pH.II, B ( 41 ,3) CXO,ll,HTCH a6COJIIOTHO li paBHOMepHO. LlaJiee,
e (u) = Bu2P+' .
(41,4)
( 41,5)
llaJiee, no 3a,ll,aHHOMY '-IHCJIY M ;:;;;, 2, ,ll,eHCTBYH MeTo,ll,a~m § 35, CTpoHM
ij:JyHKUHIO
N K N
gM(u) = ~ Pig(f3p)= 1 + ~ Pih(t1p) + ~ Pje(f3p) =
j = l i =I J=l
= 1 + hM(u) + eM(u).
3,ll,ech
I \'
eM(u) = Bu2P+~ ~ PJ3]P+~,
j=l
npH 3TOM
N
~Pj=1, 1 < t1i < M'".
j = l
ua KOinypax L"' ( 41 ,8) 6y,ll,e T HMeTb MecTo npH
1
O< u < - (- )Ma . a, 'llo •
( 41 ,6)
(41,7)
(41,8)
(35,9)
(35,3)
(36,3)
Tenepb ,ll,eiiCTsyeM, KaK s § 36, onepHpyH c '-IHCJIOM 2p BMecTo '-IHCJia 4.
HcnoJih3yeM paseHCTBO
(-1)P P cos (2p-2k)ux ~
sin~P(ux)= 22p-I ~(-l)kC~P [.!::.] = k.JCkpcos(2p-2k)ux
k=O 2 p k=O
{CM. [10], CTp. 268). p
IloJiaraH x = 0, Haxo,ll,HM : ~ Ckp = 0.
k= U
IlycTh x. <P- gM(u) OTBe'-!aeT 3aKoHy pacnpe,ll,eJieHHH V(x), TOr,ll,a Hallie
paBeHCTBO ,ll,3CT
p p
~ CkpgM[(2p-2k) u] = ~ CJcphM((2p-2k) u) +
k=U le - U
00
+ k~ CkPeM((2p-2k)u) = ~ sin2PuxdV(x). (41,9)
-oo
IloJiaraeM
(36,7)
2 . ~lK p auncKun MaTeMaT. IKypnaJI, T. V, Nt:! 3. 261
DpH .D.OCTaTOliHO 6oJiblllOM M (36,3) 6y.n:eT BepHo .D.JIH 2pu H 6y.n:eT HMeTb
MeCTO ( 41 ,8). )J,oKaMeM Terrepb, liTO eCJIH m < p IJ.eJIOe liHCJIO, TO
p
L Ckp(2p-2k)2".'= 0 (41,10)
k=O
(npH m = 0 rroJiyllaeM, B llaCTHOCTH, k~ Ckp= o). Bo3bMeM paBeHCTBO
p
sin2Pz = L Ckp cos (2p-2k)z.
k=O
)J,HcpcpepeHU.HPYH ero 2m pa3 ITO Z H ITOJiaraH Z = 0, ITOJIYliHM 0 rrp 11
2m < 2p, H6o BeJIHl!HHa CJieBa pa3JiaraeTCH B pH.D: MaKJiopeHa, H.aliHI-IaiO
mmkH c z2P . 0TciO,n:a
p
(- 1)"' L: Ckp (2p-2k)2"'= 0,
k=O
H ITOJiyllaeM (41,10). TaK KaK V~oo) (u) lleTHbiH ITOJIHHOM CTerreHH 2p, TO
p L Ckpy~oo) [(2p-2k)u ]= Bu2P, (41,11)
k=O
)J,aJiee, ou.eHHBaH P N(z) corJiacHo § 36, Haif,n:eM, liTO cyMMa KOHTYPHhiX
HHTerpaJIOB B ( 41 ,8) 6y.ZI,eT
B(e)u<>a-• exp (- i M0,9). (36,15)
BBH.LJ:Y 3Toro n ( 4 1,11) rroJiyllaeM
I L;C,.,hM [(2p-2k) u] I B(e) u">-• exp ( - }.J..zo,9) N.,
= +B\P~
u 2P u 2P :"-" J ;
j=l
YliHTbiBaH, liTO u = e-VM", na:H,n:eM ,n:JIH (41 ,12) OtteHKY
N
B ~P)J]P.
j=l
Jl.aJiee, H3 ( 41,7)
I L;CkpeM[~:- 2k)u] I= N
Bu~ (sup f15) L Pi]P.
J=l
BBH.LJ:Y (35,9)
H no.ilyllaeTcH ou.eiiKa
N
B ~ P.~2P.
~ J 1
j = l
262
(41,12)
(41,13).
(41,14)
1:5BHJlY 3TOfO, ( 41 .~) )l3CT
C sin2;ux dV(x) = B ~ P.(PP. j uP .l..l J J
- · 00 j=l
(41,15)
Paccy)K)laH, )laJiee, KaK B § 37 H 38, r)le neJJHCb paccy)K)leHHH c GHCJJOM
2p = 4, TIOJiyGaeM, tiTO B [0,u0) Cyi1J,e CTByeT Henpe pb!BHaH npOH3BO)lHaH
lJ'2P (zz) H, B qacn-IOCTH, lim 'P2P ( u) . ITo Ji eMMe IX, B ::>TOM cJJyqae JlO.mKHO
u--+0
6biTb 0 3 ;> 2p. 3ro npOTHBope'IHT (39,2). Tervr caMbiM JJeMMa XI, B qacTH
)'TBep)!<)leHH5J (28, 1), OTHOCHilJ,CfOCH K TIOJIIOCY 82, H3KOHe U, )l0K33aHa.
§ 42
OcTae TCH )lOKa3aTb nropy1o tiaCTb JJ e MMbi XI - HMeHHO, coorHowe
H He (28,2).
EcJIH S 2. OTCYTCTBye T, TO, KaK HBCTByeT H3 npe)lbiJlYill.HX pacCy)!()leHHH,
cyi1J,e CTBOBaHHe 0 3 nponmopetiHBO, H TIOTOMY KOMTIJieKCHb!X HyJJeH HeT.
B BHJlY 3Toro, npH 0 ::::;; u · ;; tto HMee M
'P (u) = P (1tt2),
r)le P ( u2 ) noJJHI-!OM ti enwi1 CTeneHH, He 6oJibwe ii o2• BBHJlY tieTHOCTH <p (u),
l!aXO)lHM
np11 l•t I ::( tto. (42,1)
EcJIH 6bi paneHCTBO ( 42,1) co6JIIO)laJJ o Cb Ha ncei1 ocH, TO corJJacHo y)!<e
npHMCHHBWeHcH HaMH T e ope Me [2] 6hiJIO 6hi lJJ(u) = Au2 H A ::::;; 0. Ho Mbi
pacnoJJaraeM paneHCTBOM ( 42,1) TOJtbK O npH I u I ::::;; uo. .lloKa)KeM Tenepb
JJeMMy, npe)lCT3BJJ5JIOill.YIO H HeKOTOpbiH CaMOCTOHTeJJbi-IblH HHTepec.
J1 eM M a XI I. nycTb xapaKTepHCTHtieCKa H cpyHKUHH HeKOTOpOH CJIY
' Ia i-'niOi-i BeJIHGHllbl B He i<OTOpOH OKpeCTI-IOCTH I-IYJIH, - 0 ::( U < 0 HMeeT BH.l!.
f (u) = eP(u), (42,2)
r)le P(u) noJJHHOM. Tor)la
P(u) = Au2 + Ciu,
r)le A ::::;; 0 H C p e aJJbHbie l<OHCTai-ITb i, H pane HCTBO ( 42,2) B e pi-IO Ha nce i1 ocH.
)1, o K a 3 a T e Jt b c T B o . f ( u) HMeeT nee npoH3 BOJlHbie npH u = 0. TaK
K<JK f(u) eCTb X. cp.
00
f (u) = ~ ei"T dF(x),
r)le F (x) COOTBeTCTBYIOill.Hii 3 aKOH pacnpe)leJJeHHH, TO OTCIO)la, KaK H3-
BeCT!-IO, cJJe)lyeT cylll.eCTBOBaHHe ncex MOMeHTOB a, )lJJH F (x). f1pH 3TOM
d"
I a" I = If<"> (0) 1- OueHHM poeT I an I BMecTe c n.l1M eeM: f<"> (0) =c: du" eP<">
npH u = 0. eP iu) ueJiaH cpyHKUHH HI P(u)l < K I u IP npH I u I > 1. BnH)lY
3Toro Hl\IeeM
11! l'fi eP (u)
J ln ) (0) = 2ni ;J' u"+l du,
CR
r)le CR oi<py)KHOCTb I u I= R.
263
CTaJJo 6biTb, np!i JJI06oM R > 0
I j(n) (0) I<~~ . ~n eKRP.
1
Bbi6epeM 3.LI.CCb R = n P, TOr.LI.a np!i n ;;> n0 , noJiyiiHM
( n I ~) n! exp K11 - 2 n n
I t<"l <O) I < " < n~ .
R2 n2P
BB!i.li.Y 3Toro noJiy'lHM np!i m > no
Cl.2rr· < (2m2)_ml • li Y2m-1 < CX2m>
(2m) 2P
(42,3)
(42,4)
r.LI.e Y2m-I a6cOJIIOTHbiH (2m- I) -ii MOMCHT. BB!i.li.Y 3Toro np!i mo6oM r > 0.
00 "'
~ e'xdF(x)= ~~; ~ xndF(x) = B(r) + B ~ ( ~ ) < C£ .
_ 00 n= O _ .:J; n=n0 n2p
0TCIO.LI.a HBCTBYCT, 'ITO f(u) IJ.CJiaH cpyHKU.IiH , 11, llOCKOJlbKY OHa COBlla.LI.aCT
np!i - o ~ U ·< o C IJ.C•10H cpyHKU.HCH e~' ( u), TO f (u) = e''(ul BCIO.li.Y
11, B IJaCTHOCTH, Ha BCCH p eaJibHOi:i OC!i.
no TeopeMe Mapu.HHKCB!i'Ia [2], f(u) B 3TOM CJJyqae CCTb X. cp. HOp
MaJibHOro 3aKoHa, TaK 'ITO P(u) = Au2 + Ciu; A ~ 0, C peaJJbHO. ECJJH
f(u) IICTHa, TO C = 0 . . 9To .LI.aeT .LI.OKa3aTCJibCTBo BTopoi1 IJaCTH JJCMMbi XI:
ecm1 82 OTCYTCTByeT, TO 11MeeT MCCTO (28,2). T eM caMbiM JJeMMa XI .LI.OKa -
3aHa llOJIHOCTbiO.
§ 43
Mbi 6y.LI.eM Tenepb 113yiiaTb x. cp. BH.LI.a
I (u) = exp ~ A 1 1 u 11 P ,- (ln I u! ), (43,1)
nre )' npo6eraeT KOHelfHYIO CI1CTeMy p a3JJ J1l.IHbrx peaJibHbi X 'IHCCJJ, A 1 =I= 0
peaJJbHbre KOHCTaHTbr, P 7(1n I u I) no.rJHHOMbr cTeneHeif d 1 co CTapwHMH
K03cpcpHIJ.HCHTaMH = I.
3aKOH, OTBeiJaiOW.HH x. cp. ( 43, I ), o6o3~ra'IHM F (x). BBH.li.Y Toro, 'ITO
f ( u) HenpepbiBHa 11 f (0) = I, .LI.OJDKHO 6biTb y > 0. )lm<a:t~\CM Tenepb CJJe
.LI.YIOW.YIO JICMMy.
JI e M M a XII I. .UJIH BCHKOH x. cp. BH.LI.a ( 43, I) MO)KHO nocTpOHTb .nse
JJHHeHHbiC cpop Mbl
L. (x) = a1xr +
L 2(x) = b1x1 +
+ a,x,
+ b,x,
c HCOTPHU.aTeJJbHbi MI1 Ko3cpcp11U.HeHTaMH, Taime, 'ITO
o(z) = a 1z + ... + a,z- b1z- · · · - b;¢ 0 ( 43,2)
H 'ITO L 1 (x) ~ L 2 (x), Kor.na HC3aBHCHMbtc Ha6JIIO.LI.eHHH X; pacnpe,neJJeHbi
no 3aKoHy F(x).
264
.il. 0 K a 3 a T e Jl b C T B 0. Do 3a.D,a H HOH KOHC'IHOH CH CTCMC <JHCCJI y H d1,
( In 3).
nOCTpOHM HOBbie 'IHCJia e1 = exp y H COCTaBHM QJYHKll,HIO
u(z) = fl(e~ -3)dr+ 1 • (43,3)
r
3Ta QJYHKU.HH, O'ICBHAHO, 6y.n,eT H MCTb 'IHCJia y HYJIHMH I<paTHOCTH d1 + I
(ocTaJibHbie ee HYJIH 6y.n,yr KOM n JieKCHhiMH ). P acKphrBaH CK06KH B Bbrpa)f{e
HHH ( 43,2) H p a36HBa5I 4 JICHbl C ll,CJI04HCJICHHbiMH K03QJQJHll,HCHTaMH Ha
cy~1Mbl O.D,HHaKOBhiX 4JICHOB, llOJIY4HM Bblp a)f{C~IHC THna ( 43,2).
0oCJIC :noro MO)KCM COCTaBHTb JIHHeHHbi C QJOp Mbl
LJ ( x ) = aJxJ + a2x2 + · · · + a,x,; L 2 (x) = bJXJ + · · · + b,x,
C HeOTpHll,aTCJibHbiMH K03QJQJHll,HCHTaMH.
X. cp. 3 THx cpopM 6y;wT
r
n f (ap) H n f (b;u). 0HH =/= 0.
i=l i= l
.il.JIH AOIW3aTeJihCTBa TOro, liTO LJ ( x) ~ L 2 ( x ) , .n,ocTaTOlJHO noKa3aTb, liTO
4aCTHOe HalUHX npOH3Be.n,eHHH paBHO l. 3TO lJaCT HOe 6y.n,eT HMCTb BHA
(43,4)
d 'l
.UaJiee, P.1(InaJu l) = ~ C,. (u)(lna)'". 4HcJio y eCTh HYJih o ( z ) KpaT-
m= O
HOCTH d.1 + I . BBHAY 3Toro npH m ~ d-l HMeeM
r r
<T1n (y) = ~aT (In a)'"- ~ bi (In b)m = 0.
i=l i=l
BBHAY 3Toro (43,4 ) pasHo I , H LI(x ) ~ L 2(x) 4 . T . .n,.
§ 44
B CHJIY .TJeMMhi XII I H 3aMe4 aHHH o e.n,HHCTBCHHOCTH pa3JIO)!<CHHH B [0, I]
B pHJJ:bi BH.n,a , yKa3a HHoro B § 20, K QJYHKU.HHM (43,1) 6yJJ:yT npHJIO)!<HMbJ
see npeJJ:biJJ:YW.He pe3yJihT3Tbl. B q acTHOCTH, ecJm OJ H a2 HaHMCHbWee
H HaH60JihWee H3 4HCCJI y, TO Aa, (--l )da,+l > 0 (snpo4eM, 3TO OlJCBHAHO He
nocpe.n,cTBerniO). vb JieMMbl X 3aKJII043CM, 41'0
(27,1)
Mbi AOJI)!{Hbi Tenepb HCCJie.n,osaTh, npH K aKHx ycJIOBHHX cpyHKU,HH
exp ~ Ar \ u 11 P1 (In I u I) (44,1)
('l)
MO)f{eT 6hiTh x apaKTe pHCTH'ICCKOH QJYHKU,Hel'i . B cJiy4ae OJ < 2 Mbi AJIH npo
CTOThi ( H TOJlbKO H3 3TH X C006pa)f{CHHH) p a36epeM TOJibKO TaKHC Bblpa)f{e
HHH ( 44, I), n 1.e P1 (In I u I)= 1, cJiy4aH: )f{e OJ = 2 npH.n,eTCH p a3o6paTb
flOJJH OCTbiO.
26'>
OepeHyMepyeM liHCJJa ( B DOpH,li.Ke B03p3CT3HHH
01 = YI < Y2 < ... < l;t = 0 2· ( 44,2)
Mbr ,li.O IWR<eM Tenepb T eop eMy IV. 0Ha pamroc HJibHa YTBep):<JJ:emno: nycTb
a 1 < 2, H 3a,ll.aHbl npOH3BOJ!bHbie peaJibHbi e ·liHCJia )'2, ... , f(l -l)' 0 2 DO,ll.
"CJIOBHeM (44,2), npaH3BOJibHbre peaJibHbre liHCJr a A .. , . •. , A r H npo-
J ~ ~ l - 1
H3BOJibHOe OTpmr:aTe.1JbHOe lJHCJIO A "• ·< 0; TOr,ll.a MO }KHO DO;I:biCKaTb I<OH-
CTarny A 0 > 0, TaKyro, liTO nprr BCHKOM A a, < - A 0
6y,11.eT xapar{TepHCTHlieCIWH QJYHKII,He H CJiylJaHHOH BeJIHlii-!Hbl.
D. o I< a 3 aTe JIb c T B o. 06a3 HalJ.HM Bbrpa)Ker-nre ( 44,3)
H cocTaBHM npeo6pa3onmme <Pypbe
00 ~
y (x) = 2_ ~ e-iux f (u) du = 2_ ~· ei"x f (u) du
2rr 2n .
B CH.flY lJ CTIIOCTH f(u). J1HTerpaJI a6COJIIOTHO CXO,li.HTCH BBll,li.Y
A a, < O.
(44,3)
liepe3 f(u)
(44,4)
HaM HY)!<HO ,li.OKa3 aTb, liTO y (x) ;> 0 npH a cex x, ecJIH I A ,, I ,li.OCTaTatmo
seJI Him.
B B H,li.Y l!CTHOCTH f(u) HHTerpaJI (44,4) MO)KHO 3anHcaTb Tare
00
y (x) = ~ ~ eiux f (u) du. (44,5)
0
BBH,ll.y TOrO, liTO II ;> 0, B ( 44,5) MO)!<tiO HanHCaTb
f (u) = exp ( - Au"' +% A-1p"1j - Cu"•) , (44,6)
n1.e BBe,ll.eHbr o6o3 HalJeimH: - A = A", < 0; - C = A,,< 0. ClJHTaH f ( u)
f!OJIO)J<HTeJibHOii npH U ;> 0, 3a~ reqaeM , <ITO ec l\W)l<llO npOJ.J,O.l)KHTb H a KO\I
D.1Jei<CHYlO f!JIOCKOCTb <mceJJ 'W = (jCi'P C pa3 p e30M B,li.OJib BCe i-i f!OJIOrKHTeJib
H OH no.flyacH. 0Ha 6y,11.eT p e r yJI5rpHoii BHYTPH yKa 3 ar·IJIOj:J o6JlaCTH H n e npe
pbJmrori Bf!JIOTb ,11.0 KOIITypa o6.1J aCTH.
BBHJJ.Y ::nora ~Ibl Mmr<eM ,ll.ar<a3aTb, <ITO npu 1v = u +- iv
00
~ eiux f (u) du = ~ eiw.< f (w) dw + ~ eiwx f (w) dw, (44,7)
0 L, L,
r ,11.e Lr e CTb OTp e3aK MHHMOH ocn tt = 0, 0 < v < A; A mo6oe noJIO)!<H
TeJibHae l!HCJIO, a L 2 ·- noJiynpHMaH v = A, 0 < u < =·
ll.JIH )IOK33aTeJibCTBa DpHMeHHM TeopeMy I\OIIlH I< DpHI\10YfOJ!bHOl\!Y
KOHTypy, HMeiOII.J,e My ,li.Be CTOpOHbl Ha OCHX KOOp,li.HHaT H Bb!pe3
7(
w=Qei'P, Q- 0, 0 <. g; <. 2-.
.LJ.JIH .li.OK333Te.TJbCTBa (44,7) .li.OCT3TOlJHO 06Hapy2KHTb, 'ITO IIpH U ~ oo,
A
~ ei(u+ivJ f (u+iv) i dv--0. (44,8)
v==O
TioJiaraH u + iv = Rei'P, BH.LI.HM, 'ITO npH 0 < v <A H v ~ oo, q; --0.
BBH.li.Y ::noro (u + iv)"• = R "• ei'P · "• HMeeT 6ecKoHelJHO MaJibiH apryMeHT,
H R { (u + iv)"•} > }R"•. 0TCIO.ll.3 HBCTByeT, 'ITO f(u + iv) ·~ 0 paBHOMepHO
no v u ( 44,8) nMeeT oueHKY
~ e-v if (u+iv) I dv-- 0.
v=O
§ 45
Mbr .n.oKa3aJin (44,7) H 6y.n.eM npo.n.oJI2K3Tb .n.oKa3aTe.TihCTBO Teo-
peMbi IV. )J,Jr5J 3Toro c noMOlllbiO Bhr6opa noli,XO,il,mnero A= A(x) Mbi
cnepBa H3H.LI.eM 3CHMIITOTHlJeCKOe Bblpa:tKeHHe .li.JIH y (X) H .li.OKa2KeM, 'ITO
y(x) > 0 npH x > x 0 (A) . 3aTeM Mbi rroKa2KeM, 'ITO npn .n.ocTaTO'IHO 6oJib
UIHX A y(x) > 0 T3IOI<e npH 0 <:;X< Xo(A) . BBH.li.Y lJeTHOCTH y(x) TeOpeMa
6y.n.eT .n.oKa3aHa.
0yCTb
_!_ + •lo
x>K0A"' =K0A•,+•Io; A >- 1, (45,1)
1
r.n.e qepe3 1'1 o6o3HatJeHo - , a K 0 noJI02KHTeJibHaH KOHCTaHTa, 3HalJeHHe
at
I<OTOpOii 6y.n.eT <flHKCHpOBaHO B )1.3JlbHeHUieM. 0yCTb
A(x) =x-I ln2 x. (45,2)
00
/2 = ~ eiwxf(w)dw= ~ ei(u+iA (x)) xj(u+iA(x))du,
L;! u=O
OTKy.LI.a
00
112 1-<:e-xA(x) ~ if(u+iA(x))!du. (45,3)
0
naJiee, npH 0 < u < KlA( x) HMeeM I/ (u + iA(x)) l = exp BA(A(x))"•,
a npH u > K 1A(x), noJiaraH u + iA(x) = Rei'P , BH.LI.HM, 'ITO <p CKOJib yro.n.Ho
MaJio npu .n.ocTaTOlJHO 6oJiblllOM K1, H
If ( u + iA(x)) I = exp (- AR"• cos rp (h + i: A;j RYj cos qy • 'Yj- CR"• cos rp • CJ2) •
7=2
DpH )l.OCT3TOlJHO OOJiblllOM KI IIOJIY'IHTCH
H
1 1-1
2 CR"• cos qJCJ2 > L A r;Rrj cos qJ • Y;•
j=2
261
BBH.IlY 3THX o u.eHoK HaXO.IlHM: ..
~ I/ (u+iA(x) ) idu=BA(x) exp (BA(A(x) )a•).
0
Teneph coc<JHTaeM
I=~ ei"''"/(w)dw.
L,
DpH
w=iv =ve 2
A (x) 1f i . 1_ 1 cri :ti
/ 1 = ~ e-vrexp (- Ae2 "'v"•+ ~ A1fe2\'~J- Ce-2 a') idv.
v=O J= 2
B CHJIY yCJIOBHH (45,2) .llJ151 A(x) HMeeM
]n!a, x )n!(K A•,+'1o)
Av"• < A (A(:r) )"'•= A --" - < --K" Ao ., ,,- .
X 1 01 lO I
( 45,4)
( 45,5)
(45,6)
(45,7)
3TO lJHCJIO CKOJlb yrO,li.HO MaJJO npH .ll.OCTaTOlJHO 60JibWOM (He3aBHCHMO OT
A) K 0•
5 y.lleM ClJHTaTn K o CTOJih 6oJJbWHM, 'ITO cyMMa a6comoni biX BE'j nilJHH
<JJieHoB a 3KcnoHe HTe (45,6) MeHhwe I. T or,ll.a noJJy<JHM
ni /-1 n; 1ti
( 2 "• ~ 9 Yj .. 2 "• ) exp - Ae v''• + ~ A1l~ v'f- Ce v"• =
f=2
:ri . 1_ 1 ni :ri
-~ . -~ - ~
= 1-Ae2 v"• + ~ A 11e 2 v1i- Ce 2 v7•+ BAtv~o.,
j=2
A(x) n;
11 =i ~ e -v"(- A)e2 o'v"• dv+
0
1_ 1 '" A (x) :T i . A (x) A ( x)
+ ~A1ie211 ~ e-vx . v'l dv-Cie2 "' ~ e-vxv"• dv+i ~ e-vr dv.
J-2
0 u
DpH v > A(x)
f - _!In' x
) e - "" vc dv =Be 2
, t (r)
eCJIH C > 0 KOHCTaHTa.
BBH.IlY 3Toro noJJy<JHM
:ti oe l-1 rti .. oo
11 =-iAe2 "' ~ e-vxv"•dv+ ~iA1l1.- ,i~e-vrv1i dv-
o l-2 0
~a, f f f -_! In' x
- i· Ce J e-vxv"• dv+BA 2 ) e-vxv %<1, dv+i) e-vx dv+ BA'e 2 •
0 0 0
268.
(45,8)
(45,9)
(45,10)
HMeeM, .D.a.Jlee, npH c > 0
• 00
~ e-vxvcdv= x;+1 r(c+ 1).
0
+ c . n r(a%+ 1) r(2(JI + 1) - -} ln'x
sm 2 (J: x"'+ 1 + BA% -x2a,+-r- + BA 2e
OcTaTOtiHbiH '-IJJeH B ( 45,2) B CHJIY ( 45,1) 6y.D.eT HMeTb BH.D.
- -~ ln'x
Be 4
(45,11 }
(45,1 2}
(45,13}
eCJJH Ko 6biJJO qmKcHposaHo .D.OCTaTO'-IHO 6oJJblllHM (He3aBHCHMO OT A) .
.[laJJee, H3 ( 45,3) H ( 45,4) H3XO)I.HM
II: I< e-ln' x . BA(x)eBA(. l(>)) ct, = Be-ln' x. (45,14}
Tor)l.a (45,13) , (44,5) H (44,7) )l.aioT
A . n P(cr1 +1)
y(x)> 4;c sm2a1 x "•+J (45,15)
npH x > K 0A•·,+•;• ; A ;> I; K 0 KOHCTaHTa (pa3yMeeTCH, He3aBHCHM3 H OT A).
§ 46
T enepb paccMoTpHM cJJy'-laif , KOr)l.a 0 :s;;; x < K 0A''•+·•·. floJJO)!<HM
~ = xA-•·,, 11 B (44,5) noJJmi<HM U = tA-''•.
T or)l.a (44,5) npeBpaTHTCH B
00
y (x) = ____!,. 9l (' exp (i~ t) · ·exp (- F• +
rrA' )
0
1-1 (')"(• (' ' ") + ~2 A,j A"' ) - c A~.) I dt.
floJJO)J{HM l 1 = ln2''• A; Tor.n:.a
t,
y (x) = ~ 9l \ exp (i~t) · exp ( -!"•) X
rrA' )
0
(
/-I t y. ( t )" ) X exp ~Arj(----,;;)J-C -----,;; ' dt+Be-ln'A.
1= 2 A .A
(46,1}
(46,2)
oy)l.eM Cl.JHT3Tb A CTOJib 60JJblUHM, '-ITO npH t :s;;; tl cyMMa MO)I.yJJeH l.JJJeHOB
TpeTbero 3Kcn oHeJ-ITa B (46,2) MeHbllle 1. Tor)l.a, p a3JJa raH TpeTHif 3KCnoH e HT
B pH.U M aKJi opeHa, Ji en<O H axo.n.HM
t,
Y(x) = - 1- 9! \ e'~te-t"•dt+B · -1- A-~.r •.
nA•' J A•'
(46,3)
0
269
l ·L1H BBH.ll.Y onpe)I.eJieHIUI f1 rrpH )I.OCT3TO'IHO 60JlbWOM 1-
oo
Y (x ) = _ l _ 9l \ ei;t e-1" ' dt + B · __!_ A -•·,r,,
rrA''' .) A¥•
(46,4)
0
§ 47
OcTaeTCH nccJie.II.OBaTb HHTerp aJI
"' 1 \ ·" (1 z m = ;; 91 .) e z;t e-t l dt. (47,1)
0
Ha oCHOBaHH'H § 44 H 45, rroJiaraH A = 1, BH.II.HM, liTO rrp n ~ > K o
6y.II.eT
(47,2)
H eoTpHI.J,aTeJibHOCTb mnerpaJia ( 4 7 ,1) rrpH Bcex ~ H3BeCTI-Ia H3 TeopHH
ycTOH'lHBb!X 33KOHOB (E. B. r He)I.eHI<O II A. H . I\OJIMOropoB [11]) , a TaioKe
M O)!{eT 6biTb H3Hl.Il,HO 060CHOBai·Ia C ITOMOlll,b!O TCOpHH ITOJIQ)KHTeJibHO orrpe
J{eJieHHbiX cpyHKUHH Ha rroJiyMeTpH<IeCKHX rrpocTpaHCTBax ( Kn <I> an [12]),
HO J{JIH ITOJIHOTbl H3JIQ)£<eHH5J Mbl rrpHBC)I.Cl'vl HOBOe .ll.OI<a3aTeJi bCTBO ITOJIOii<H
T eJibHOCTH Z a) rrpH ; > 0, a 3 H3'IHT, H rrpH BCe X p eaJJbHbi X ~ -
CrrepBa 3 aMeTHM, 'ITO H3 HeoTpnuaTeJJbHOCTH mnerpaJia z ( ~) npH Bcex
peaJibHbiX ~ CJie.LJ:yeT ero fiOJJO)KHTeJi bHOCTb H3 BCe !'i peaJi b HOH OCH. .[lJIH
.ll.OI<333TeJibCTBa CITCpBa paCCMOTpH M CJiyqai.J: 1 < o 1 < 2.
B 3TOM cJiyllae z U) nepo?.THOCTIIaH n JIOTHOCTb c x. cp.
e- 1 t 1"• = q; (f).
"'
z (g) = ~ z!{;- 17)z1 (?J) d 1,·;
-00
Z 1 ('7) > 0 BCIO.ll.y.
BBH.ll.Y He rrpep biBHOCTH 2 1 ('7) 3aKJIIOll aeM, liTO )I.JIH Bcex '7 2 1 a- "J) Z 1 ('7) = 0,
-eC.' lH z U) = 0 B KaKo}i -JIHOO TOliKe ~ -
.llaJiee,
i ?) I < 170·
1
O'leBH.ll.IIO Z 1 ("/) > '2 npH I '7 I< 17o, TaK 'ITO Z 1 (~- '7) = 0 rrpH
Bnn,uy Tor o, lJTO o 1 ::::.., I, H3
H Z1 a) IJ,eJI'bie cpyHKIJ,HH, H H3
(47,1)
TaKoro paBe HCTBa
00
5IBCTByeT,
C.'Je,UOB3.TIO
'ITO'
6bi,
zW
'ITO
Zi( ~) = 0, liTO HeB03~JO)KHO, Ta l< K3K ~ Z 1 (~) d~= 1. 9TO H ,UOK33 biB3eT Hawe
- oo
YTBep )K,UeHHe .ll.J!H cJiy<IaH I < o 1 < 2.
B CJiy qae 0 < 0 1 < 1 Mbl MQ)KeM rrpHMeHHTb TeopeMy r . IlOJJHa [13]
v BbiiTYKJibiX cpyHKUHHX. <I>yHKUHH f (u) = e -1" I"• rrpu 0 < o1 <;; 1 HBJIHeTCH
•JeTHOH H npll 1l > 0 BbiiTYKJIOH KHH3y.
2 70
B caMOM .neJie, HMeeM rrpH u > 0
I" (u) = e-ua, [( a1 ua·-1)~+a1 (1-al) u"·- 2] > 0,
liTO ,llOK33biB3eT Tpe6yeMoe.
§ 48
HeOTpm~an'JibHOCTb ( 4 7,1) rrp11 0 < or .:;:;; 2 BbiBO}~HTCH H3 c.rre,n:yromeli
JICMMbl , HMCIOW,eii 11 IlCKOTOpbii:i: C3MOCTOHTeJibHblH HHTepcc.
Jl e !II M a XIV. Dpu BCH KOM a rro.n ycJrOBHHMH
O< a < 2 ( 48,1)
cpyHKUHH
1
I (u) = 1 +I u Ia (48,2)
6y,n:eT xaparnepHCTHliCCI<OH cpyr-IKIJ,HeH HeKOTOporo pacrrpe,n:e.nez.IHH.
Ll o K a 3 aTe Jl b c T B o. Opu a = 2 rroJryll aeTCH H3BeCTHaH x. ¢ . .II.JIH
pacrrpe.neJieHHH nma x:. Op11 0 <a <( 1 Mo:nmo rrpHMCI-IliTb UHTHposaHHYIO
Bblllle TeopeMy f . 00JIIfa [13), H60 rrpH U > 0, K3K JICfKO rrpoBepHTb,
f'(u) > 0. OcTaeTcH CJiyllaii 1 <a< 2. L(.TIH .na1-moro cJiyllaH rrp11 JIIo6oM
peaJibHOM E cpyHKUHH
00
vm = ,},c ~ e;~ ., I (u) du
co
v (s) = ~ 9\ ~ ei'iu I (u) du = ~ 9l(M(g) ), (48,3)
0
r.ne
co
M(s) = ~ 1 :~~a du. (48,4)
0
P acCi\WTpHM ITJIOCKOCTb KOMITJieKcHoro rrepeMeHHoro u c pa3pe30M B,llOJib
Bcei:i rr oJrmKHTeJibHOi'i ocH. BI-IyTpH TaKoi-i o6J1 3CTH cpymuLHH u" perymrpHa
1
11 nerrpepbmHa BITJIOTb .no KOI-ITypa. Jla.rree, cpyr-ll:UHH
1 + ll o
He HMeCT
oco6emwcTeli B rrepBOM KBa.npmne o6Jiacnr, n6o H3 Zla = - l cJie.nyeT
(2k+l) n
I u I = 1; arc u = , 11 BBH.li.Y Toro, liTO 1 <a < 2, see 3Ha'ICHHH
' a
a rc u JIC)!<aT BI-Ie Ksa.npmna.
BBH.UY TOfO, liTO a> 1, JICrKO 33K.TI IOli3CM, liTO KOI-ITyp HHTerpHpOBaHHSJ
B ( 48,2) !I!O)t<Ho rroBepHyTb Ha 90° rrp oTHB tzacoBoi'i CTpemm TaK, liTO ou
n orra,UCT Ha MHHMYIO OCb.
Tor.na rroJiyliHM
27!
OTKy,ua
3THM JICMMa )I.OK333H3.
§ 49
J-13 JJCMMbl J!CfKO CJI C.li.YCT HCOTpHU.3TCJ! b HOCTb (47,1). J1MCIIIW, HapH,U
1
c f (u) = 1 +I u I" , npH mo6oM N > 0 cj:Jym<U.HH
f ( ~) = 1 + I :N 1 1"
TaK)!{e 6y )l.eT X. cj:J.
DycTb N" npo6eraeT HaTypaJJhJJbJe '-IHCJH1 n. Tor,ua cj:JyHKU.HH
J f (~-\ }" = (- 1 _ _ )" l Ni 1 + u"n-1
6y,ueT TaK)!{e x. cj:J. DpH 3TOM oHa 6y,ueT CTpe~IHTbCH np11 scex u K HenpepbiB
Hoii cj:JyHKlJ.HH e- I u !<> , KOTOpaH TO>I<C ,UOJI/Kl-13 6biTb xapaKTe pHCTH'-! eCKOfl (8] .
.7rHM ,uoKa3 aHo, 'ITO z ( ~ ) > 0 so scex liCCJi e,weMhiX c.ny'-! aHx. B CHJIY
0'-ICBH,UHOH Henpepb!BIIOCTH z ( n JHICCM z ( ~) > au ( Ko) > 0 npH I ; l < Ko-
Tor,ua (47,2) ,ua eT z (~ ) > (;+~),,+1 npH s c ex § > 0.
Bo3Bpaw.aHCb K ( 46,3), Haxo,uHM
a, 1 1 , ,
y(x) > A¥• (~+ 1 )"d:i +B· A,., A- .1.. (49,1)
Do,ucTaBJIHH 3Ha'-!eHHe ~ = xA-¥•, BH,UHM, 'ITO npH A > Ao
ecJIH 0 < x < A ¥t , H
( ) > 1 A + B A-•·r
y x 2a2 x "'•+l A''• ' •,
A¥• <;: x <: K 0 A ••,t.; •.
Ec.1u 1)o 6biJio ,uocTaTO'-IHO 1\taJio, '-ITO Mbi 6y,ueM npe,unoJiaraTh, TO
272
B co'-!eTaHHH c ( 45, 15) 3TO ,uoKa3biBaeT TeopeMy.
§ 50
Tenepb Mbi MmKe M npHl~TYilHTb K ,uoKa 3 aTeJibCTBY TcopeMbi L
DycTb ,uaHbi ,use JlHHeiiiibie cj:JopMbi
Li(x) = a1x1 +
L 2(x) = b1x1 + ·
. + a,x,}
· + b,x,
(49,2)
(1,1}
c onpe.n;emuoUJ,eii cpyHKUHeii
a(z>=lat [z+ ···+I a, lz·-1 b1l %-···-lb. 1%.
<:ocTaBHM scnOMoraTeJibHbie cpopMbi
(50,1)
(50,2)
L{(x) = I a1 I Xt + · · · +I a, I x,)
L{ (x) = l bt I Xt + · · · + I b, I x, 1·
M'bt 3HaeM, 'ITO BCflKHii CHMMeTpH4eCKHii saKOH (l)(x)
. .LI.JIH KOTOpOrQ
pacnpe.n;e.rreHHH Xt.
Lt(X) ~ L2(x),
-6y .n;eT TaK.:m:e y .n.osJJeTsopHTb yCJioBHIO
L{ (x) ~ L2' (x).
(50,3)
{50,4)
Pas6epeM cnepsa TpHBHaJJbHbte cJJytJaH .,Bbtpo:m:,D,eHHH" onpe.n;ensuoUJ,eH:
¢YHKUHH.
TiycTb ;o (z) = 0. Tor.n;a cpopMbi L{ (x) H L 2' (x) cosna.n;aiOT .D:JIH scex
.3aKoHOB (l)(x) . ECJIH Tenepb (l)(x) JII06oH: CHMMeTpHtJecKHH saKoH, To
L1 (x) ;:;;,;; Lt' (x) == L/ (x) ~ L 2 (x)
H, TaKHM o6pasoM, .D:JIH JJI06oro TaKoro 0(x) HMeeM (50,3) .
.ll,pyroH: CJiytJaH: Bbtpmi<.n;eHHH 6y.n;eT, Kor.n;a
o(z) = cz; c > 0. (50,5)
B aTOM CJiytJae, eCJIH )J.JJH saKoHa F (x) ItMeeM ( 50,3), TO .LI.JIH cHMMeTPH'le
-cKoro 3atwHa <D(x)=F(x)*(l-F(-.x)) HMeeM L{(x)~L21"(x) .
Ho B CHJJY (50,5) )I;Jl:H X. cp. f (u) 3aKOHa (/) (x) 6y.n;eM HMeTb B OKpecT
IIOCTH 0
f ( u) = 1 (! u I <: Uo) .
Ho Tor.n;a oHa .LI.OJJH<Ha 6biTb nepHO.D:HtJeCKOH c nepHo.n;OM u0 , T. e. paBHOH 1
)I;Jifl BCeX U. JiTaK, (l)(x) = f{x) - BbipO:m:)l.eHHbiH 3aKOH H TaKOB :m:e .p (x).
floCJie HCKJJIOlJeHHH TpHBHaJJbHbiX CJJytJaeB Haif.n;eM, 'lTO o (z) npHMeT BH)I;
a(z)=m1e1•z+m2e1•z+ · · · +m,.e1,.z, (50,6)
r.n;e mi UeJibte 'IHCJJa =1= 0, noJJo:m:HTeJibHbte H oTpHuaTeJJbHbte, H lt < .. h <
< . . . < An peaJibHbie 'lHCJia.
flo H3BeCTHbJM Teop,eMaM TeOpHH cpyHKUHH I<OMnJJeKCHOrO nepeMeHHOrO
o (z) 6y.n.eT UeJIOH <iJYHKUHeH H 6y.n.e-r HMeTb 6eCKOHe'IH0 MHOrO KOMnJieKC
HbiX HyJJeH, pacnpe.n;eJie.HHe KOTOpb!X H3y'leHO B § 8 H 9. LITO KacaeTCH
peaJibHbiX HYJieft o (z), TO HeKOTOpbie CBe)J.eHHH 0 HHX .n.aeT H3BeCTHaH TeO
peMa Jlareppa ([14], CTp. 3), npe.n;cTaBJIHIOI.UaH o6o6UJ,.eime npaBHJJa sHaKos
. .LJ:eKapTa. ECJIH o6osHa'IHTb <Iepes W 'IHCJIO nepeMeH 3HaKa B nocJie)J.t!>Ba
TeJibHOCTH mt, m2, ... ' mn H qepe3 N 'IHCJIO peaJJbHbiX HyJJeH 0 (z)' TO
W - N = 2k > 0, r)J.e 2k 'leTHOe 'lHCJIO. ECJm )J.JJH KaKoro-JJH6o saKoHa
F(x) HMeeM (50,3), TO JLJIH (l)(x) =F(X)*(l-F(-x)) HMeeM (50,4).
ECJIH f(u) x. cp. c!J(x), TO f(u) 'leTHa H npH O<;u-<.uof(u)=I=O, H
·lJS(u) =In f(u) y)J.OBJieTBOp5IeT COOTHOilleHHHM (20,3) H (20,4) § 20.
TaK 'ITO
u
~ IJ1(uu0 ) ~ 1 ·p· % E(z) du= ~ 2~. (uu0) ~( )dz+R,.(u)
U 1 m I < n nl L ZU Z
m
(20,3)
B(E) u"·-·
R,.(u) = n JnTUT; O< u-<. L (20,4)
273
CorJiacHo JieMMe VI a6cu.Hcca 01 )J.OJI)!{Ha 6biTb aKTHBHhiM peaJibHhi l\I
HyJieM a(z). CTaJio 6biTh, eCJIH peaJihHhiX HyJieti: y .a (z) soo6me HeT, TO HeT
ll
HHKaKHX aKTHBHhiX HyJieH, H ~ 1J1 (~uo) du = 0 npH 0 ~ U <. 1.
0
0TCIO)J.a lJJ(u) =0npH [u[ ~ Uo, f(u) = 1, TaK )!{e, KaK$(x) H F(x).-
HeC06CTBeHHbie 3aKOHbl.
BBH.li.Y ::noro 6y.n.eM .n.onycKaTh, 'ITO a (z) HMeeT XOTh O.llHH peaJihHhiU
HYJih; Tor.n.a 0 1 6y.n.eT TaKOBhiM. ITo JieMMe X
0 < G! ~ 2. (27,1)
.ll.JIH )J.OKa3aTeJihCTBa TeopeMbi I HY)!<HO Tlll,aTeJibHO uccJie.n.osaTh CJlY·
qaii a1 = 2. Mbi .n.oKa)!{eM HCCKOJ1hKO JieMl\1, aHaJIOrulJHhiX TeopeMe IV.
§ 51
JI e M M a XV. ITycTb 3a.n.aHhi .n.sa IJHCJia no.n. yCJioBHeM
)'2 >)'I> 2
H llOJIHHOM C peaJibHb!MH K03cpcpHI].HeHTaMH CTeneHII n > 0
P(z) = aozn + a1zn-! + · · · +an; ao < 0,
npuiJeM co6mo.n.eHo cJie.n.yiomee yCJioBHe: ecJIH l' I ecTh u.eJioe IJeTHoe IJHCJio,
TO n ;:;. 1. Tor.n.a npu .n.ocTaTOIJHO 6oJibWOM A > 0
1) JI.H60 MQ)KHO llOCTpOHTb xapaJ\TepHCTHIJeCKYIO cpyHKU.HIO BH)J.a
/1 (u) = exp (- Au2 - A1 I u [1• Jnm I u [), (51,1)
r.n.e 0 ~ m <. n u.eJioe IJHCJio u A1 > 0 noJIO)!<HTeJihHoe IJHCJio,
2) Jm6o
f2(u) = exp (- Au2 - I u 1r, P(In I u I) - ur•), (51,2)
6y.n.eT xapaKTepHCTHIJeCKOH cpyHKU.HeH .
.ll o K a 3 aTe JI h c T B o. CocTaBHM cnepsa cpyHKU.HIO / 1 (u), ciJHTaH A,
H m IJHCJiaMH JII06biMH, y.n.osJieTBopmOlll.HMH yKa3aHHhiM BbiWe ycJIOBHHl\1,.
H BHaiJaJie 6y)J.eM )J.eifcTBOBaTh, KaK B § 44.
IToJIO)!<HM
00
y 1 (x)= 2~ ~ e iuxf 1 (u)du;
-00
SICHO BH)J.HO, 'ITO HHTerpaJI a6COJIIOTHO CXO)J.UTCH H 'ITO
00
Y1 (x) = ~':R ~ eiux f 1 (u) du. (51,3)
-oo
3arrHCbiBaH rrpH U > 0
/ 1 (u)=exp(-Au1 -A 1 ur• In"'u) (51,4)
33MeiJaeM, 'ITO /1 (u) peryJIHpHa B ll.HOCKOCTH 'W = U + iV C pa3pe30M B,LI.OJ!b
sceii rroJIO)!<HTeJibHoii ocH H Henpephmna snJIOTh .n.o pa3pe3a. YIJHThiB&H, 'ITO
In w =In I w [ + i arc?.:.., .n.oKa)KeM, KaK B § 44, •no npu mo6oM A= A(x) > 0
u
00
~ eiux f 1 ( u) du = ~ eiwx f 1 ( w) dw + ~ eiLQx f 1 ( w) dw,
0 L, L,
r,LI.e £1 H £ 2 Te x;e KOHTYPhi, 'ITO H B § 44.
274
1
1 2 +'Io In2 X
DoJiaraH a1 = 2; 1' J = 2 ; x > K 0A ; A(x)= ~ , Haii.n.eM cooT-
BeTCTBNIHO ( 45,3) 11 ( 45, I4)
I 121 = Be-In' ".
Ey.n.eM, .n.aJiee, Bbi4HCJIHTb
! 1 = -~ eiwx f 1 (w) dw =
L,
A(x)
" ( ~7, ( nf)"' ) = ~ e-vx exp Av2 - A 1e vr• In v+ 2 dv.
v=O
Dp11 K 0 .n.ocTaT04HO 6o.nhmoM, I-Ie3aBHCHMO oT A, 6y.n.eT
A 2 < A In4 x ln4(K0 A ++'~•) _1_
v x2 < K~Az•;. < 4
(51,5) .
(51,6)
•
.), I ·;. 1 1 I v·;, (tn v+ ~ < v2 < :; < 4 npH HamHx 3Ha4eHHHX x. BBH.n.y 3Toro DO)J,-
HHTerpaJibHYJO ¢YHI<IJ)110 B (51 ,6) MOJI<HO n ep enHCUTb
ie-"" eAv' ( 1- A1e:;j;•, vr, (In v + ~if + Bvz·;, In"11 v).
P eaJibHaH qacTb :noif rpyHI<IJ.HH 6y.n.eT
91e-vx i (-Ate~.,, vr• (In v + ~ir
- A~Ae;~;\r.+2(In v+ ';i;"' + BA2v2;·,+2In2mv) .
11MeeM, .n.aJiee, np11 c > 0, /-( > 0 u.eJioM
I e-vxvc lnu v dv = ~ e-t (~) c (In t-In x)t' ~ =
0 0
1 oo~ B (In x)l'- 1
= - - ( - In x)." • e-ttcdt+ = x c+ l xc+l
0
(51,7)
(51,8)
Do.n.cTaBJIHH .3TO B (51,7) H Y4HTbmaH, 4TO HHTerpHposaHHe (51,7) OT A (x)
.n.o oo .n.aeT 4JieH Be-In'", Haii.n.eM
91 (/1) = 91 (- iA1 e;j;-, f e-•xvr, (In v+ ~T' dv +
0
(InxY" (lnx)"") + BA xr,+3 + BA z x2r,+ 3 . (51,9)
275
IlpH yKa3aHHbiX 3HaqeHHHX x nocJie.llHHe .usa qJieHa .uaioT
A (In x)'"
B X~ xr,+3 '
A
r.ue 2<1.
X
3'aifMeMCH nepBblM qJICHOl\l (51 ,9)
( '¥·,, ':' ( :rri)!ll )
9l - iA1e · \ e-r•:r • v7• • ln 1J + 2 dv = T (x).
0
(51,10)
(51 ,11)
IlycTb HMeeM cnepsa: )'I He ecTb ueJioe qeniOe qHcJio H [y; 1 qeTHO. Tor.ua
.:rr
B Bbipa}KeHHH (51, I) .llJl5l !I ( u) rro.rJO}KHM m = 0 11 noJiyqm\1 s m 2" y1 > 0.
- • 7f r(yl + 1)
T(x) -A1 sill 2- y1 • - -.,-+1 ,
x·•
H B CHJIY BbiUieH3JlO}KeHHblX O!l,CHOK
A 1 • n 1
Yt (x) > 2 Sill 2 Yt x·t.+I (51,12)
npH
(51,13)
)laJiee Mbi cMmKeM noKa3aTb, KaK 3To 6biJio c.ueJiaHo B § 46, qTo YI (x ) > 0
_1_ + r 0
H npH 0 < x < K 0A 2 ', eCJIH A .uocTaTOLillO BeJIHKO.
IlycTb Tenepb Yl He ecTb ueJioe q eTHoe qHcJio H I 'd J HeqeTHO. EcJIH rrpn
3TOM cTeneHb noJIHHOMa P(z ) n 6biJia > I, TO noJiara eM m = I, AI = I,
TOr,lla (5J,JJ) ,llaeT BBH.llY (51,8)
_ • :rr ( rcr1 + 1) ) n
T(x)- Sill 2 Yt - x 7•+I ln x + x ·r.+I,
• 7f I · 7f I -Sill 2 y1 = Sill 2 /'1 > 0.
B yCJioBH5IX (51,13) noJJyqHM !/I (x) > 0. EcJm rrpH Tex }Ke ycJIOBlHIX cTeneHb
n = 0, TO P(z) = a0 < 0, H B 3TOM cJiyllae Mbl 6y.ueM cocTaB.lJHTb cpyHK
u.mo ( 51,2), KOTOp35! ITpHMeT BH,ll
/ 2 (u) = exp(-Au'-a0 !u ir.- ur•).
Paccy}K.lleHHHMH, aHaJiorHqHhlMH rrp e.llbl.llYmHM, noKa}l<eM, liTO np11
..!_+'lo
x > K 0A 2
1 C . \ ao sin~ /'1 ' 1
y, (x) = ;; 9l ~ e•IJX f 2 (u) du > 2 xr.+1·
0
•276
(51,13)
(51,14)
§ 52
flepeH'.n,eM Tenepb K CJJytiaiO, KOr.n,a Yt ue.rroe tieTHOe tiHCJJO. flo YCJJOBHIO
.ll,OJDKHO 6biTb n ;> 1.
By.n,eM pa3JIHtiaTb cJiytiaH: y1 = 0 (mod 4) H y1 = 2 (mod 4). B CJJyqae
/'1 =0 (mod 4), o6paiJJ.a.SICb K (51,9) H (51,10), llOJIYtiHM
A,= 1 H c ytieToM (51,8)
n:t
-r.
e 2 = 1 npH
00
T(x) = 9l ( -i~ e-vxvr• (In V + ;)m dv=
0
= nm P(r 1 + 1) (-In x)m- 1 + B(In x)"'-•
2 xr.+l X'l,+l
flpH m HetieTHOM T (x) 6y.n,eT noJIOiKHTe.TihHhiM H 6y.n,eT
( ) > :rem r (rl + 1) (In x)"'-1
U2 x 4 x r.+I
Mbi MOiKeM B3HTh m = I. n:i
flycTb, HaKoHei.J., /'I= 2(mod 4). Tor.n,a e21' = -1 H
00
T(x) = 9l ( i\ e-vxvr• (In v + ~T" dv) =
0
= _ nm T(/'1 + 1) (-In x)"'_1 + B(In x)m-•
2 xr,+I xr.+l
ECJJH ueJioe tiHCJJo m = 2, TO
ECJJH iKe n = I H HeJih3H noJIOiKHTh m = 2, TO
P(ln I u I) = ao In I u I+ a,; ao < 0.
B .9TOM CJJyqae cocTaBHM ¢YHKI.J.HIO (51 ,2) B BH.n,e
f 2 (u) = exp (-Au2 -I u 11• (ao ln I u I+ a,) -I u 11'),
BBHJl.Y (52,3) noJiytiHM
(52,1)
(52,2)
(52,3)
(52,4)
(52,5)
L(JIH .n,oKa3aTe.TihCTBa JieMMhi XV Tenepb ocTaeTCH noKa3aTb, tiTO BO acex
TeX CJiytia.SIX, KOr.n,a Mbl llOKa3aJIH, tiTO Yi (x) > 0 {j = 1, 2) npH 0 < X<
++•;o < K 0A- , 6y.n,eM HMeTh Yi(x) > 0 npH
L(JIH :;noro paCCMOTpHM
_!_ + 'lo
0 <X< KoA 2
00
gi(x) = ~ 9l \ eiuxjj(u)du
0
3. YKpa nncKHn MaTeMaT. mypnaJI, T. V, J\1? 3.
(52,6)
(52,7)
277
B yCJioBHHX (52,6). DpH 3TOM fi(u) Bbi6paHo cooTBeTCTBeHHO npe,nhi,nyrn.eMy,
TaK 'ITO
/i(u)=exp( -Au2 +I u I'• Q(ln I ul )-b lzzl1'') ,
r.ne Q (ln lul) - noJIHHOM cTerreHH <; n H b = 0 HJIH 1. PaccMoTpHM
H llOJIO)!{HM
Tor.na
..
yj(x)= ~ 9l ~ eiux f (u) du
0
(52,8)
B CHJIY npe,nbi,nyrn.ero TaKaH no,ncTaHOBKa .naeT HaM (poJib A HrpaeT 1)
(52,9)
HJIH
1 oo~ B -~
y.(x)= v-9l ei;te- 1'dt+ v- A 2 InnA=
; n A A
0
~· 'lo 1 - -2 B - -+ A ~ Inn A.
= V2nA e VA (52,10)
TipH ~ < Ko H .nocTaTO'IHo . 6oJiblliOM A Yi (x) > 0. 3THM JieMMa XV noJI
HOCTbiO .UOKa3aHa.
§ 53
By,neM npo,nOJI)!{aTb ,noKasaTeJibCTBO TeopeMbi I. EcJIH Ha6JIIo,neHHH X;
npHHa,nJie)!{aT K HOpMaJibHOMY THny, TO BBH.UY yCJIOBHH
1) a, + a2 + · · ·+ a,= b1 + b2 + · · · + b,
2) a(2) = 0
HCHo, 'ITO L 1 (x) H L 2 (x) HOpMaJibHbl H L, (x) ~ L2 (x). 06pan10, nycTb
BbinOJIHeHbi yCJioBHH 1) - 5). OycTb ,nJIH KaKoro-JI·H6o co6cTBeHHOro saKOHct
F(x) pacnpe,neJieHHH X; HMeeM Li(x) ~ L 2(x). BBH.UY yCJioBHH
sup (I at I,··., Ia. I) =;b sup Cl btl• · .. , I b. I>
noJiy'IHM: eCJIH / 1(u) x. q,. pacnpe,neJieHHH F(x), H
/(u)=f,(u)f!(-u) = lf!(u)l2 x. q, . .UJIH F(x) *( 1-F(-x) ) .
278
H eeJIH IJI(u) = In f(u), TO .LI.JIH JII06hiX u > 0
C 'P(u) du=S (u) + ~ ~' A\ uzE(z) dz
J u 1 2m ..:::.. ';) za(z) '
0 m=-oo~
(53,1)
r.rr.e S 1 ( u) eyMMa BhilJeTOB B peaJihHhiX aKTHBHbiX Hymix a (z), H pH.rr. exo
.LI.HTeH paBHOMepHo B JIIo6oM eerMeHTe oeH [0, oo). EeJIH 1) - 5) BbiiiOJI-
HeHbl, TO M bl .LI.OJDKHbl IIOKa33Th, lJTO
Pa3 6epeM enepsa CJiytJaif, Kor.rr.a Bee peaJibHhie noJIJOebi
E(z)
o(z)
Tor.rr.a JieMMa XI npoeThie IIOJIIOCbl B U.eJib!X lJeTHbiX TOlJKaX 2k > 0.
.rr.aeT: aKTHBHbiX KOMIIJieKeHbiX HyJieH HeT, H
'P(u)=Au2 ; A < O.
cyTh
cpa 3 y
CTaJIO 6biTh, f ( u) x. cp. HOpMaJibHoro 3aKoHa H no H3Beenroif TeopeMe
r. K:paMepa [8] F (x) TO)KC HOpMaJleH. HMeeM OI = 2; no JieMMe VII .LI.OJI)KHO
eym.eeTBOBaTb 02 >OJ = 2, H B eyMMe sl (u) B (53,1) lJJi eH u"• P(In I u I)
.LI.OJDKe H HMeTb OTpHU.aTeJibHblll eTaplliHH K03<!><!>HU.HCHT.
Ll.oKa)KeM , lJTO B cpopMyJie (53, 1) OTeyTeTBYIOT aKTHBHbie KOMnJieKCHble
HYJIH, T. e. HelJC3aiOT Bee HHTerpaJibHbiC lJJICHbl. BBH.Ll.Y .LI.OKa3aHHOrO Bblllle,
MQ)KHO elJHTaTb, lJTO eeTb nOJIIOC, KOTOpblll He 6y.rr.e T OJJ:HOBpeMeHHO npOCTbiM
H u.eJibiM tJeTHhiM tJHeJIOM. nyeTh enepsa see peaJihHbie noJIJOChi npoCTbie.
Tor,na 3TOT nOJIIOC He eCTb u.eJioe lJeTHOe lJHCJIO. no ycJIOBHHM 1) - 5) 3TO
eCTb r = 0 2 H [~] HClJeTHO. Tor,na IIOJIYlJHM H3 (53, 1)
ll k
f IJI(u) "\"" •m 1 1 "\""' rfi E(z)dz
J -u- du = m-7;1 amu - aou + 2ni ~ 'iJ' u• za (z) •
0 ~
(53,2)
.lJ.oKa)KeM, lJTO HHTerpaJibHbiC lJJieHbl HClJC3aiOT. nycTh 2r KaKOe-JIH60
qenwe I.!HCJIO > y. 113 .rr.oKa3aTeJihCTsa JieMMhi XV HBCTsyeT, lJTO cpynKU.HH
BH.LI.a exp (- Au2 -A 1 I u lr); At > 0 ne MO)KeT 6hiTb x. cp. H I.!TO <PYHKU.HH
<p(u) = exp (- Au2 + a0y I u 11-- u") (53,3)
npH ,nocTaTOI.!HO 6oJihlllOM A, H ao > 0 (tiTo eo6JIJO.LI.eHo) 6y,neT xapaKTepH
eTHI.!eCKOH. BBHJJ:Y 3Toro exp(P(u)+Jn <p(u)) TO)I<e X. cp., H
u k
~ IJI(u)+;n rp(u) du=- Au'+ 1: amu•m-a0 u1+aou1-u"+
m=l
0
1 1:' ~ E(z) + -. · u•-- dz=- Au'+
2nz za(z)
m. Lm
k 1 g'> E(z) + "\"" o: u"11 - u•• + -. "\""' - - dz .
..:::.. no 27ll ..:::.. ZCT (z)
m= l m Lm
no JieMMe XI TOTlJaC )I{C 38KJIJOI.!aeM: HHTerpaJibHbie lJJieHbl JJ:OJI)KHbl Helle-
38Th, H ln <p(ll) +W(u) = -Aou2, T8K lJTO no TeopeMe r. KpaMepa [8]
F (x) I-IOpMaJJeH.
279
§ 54
D.orrycTI1M, 'ITO He Bee peaJibHbie rroJI!{)Cbi rrpoCTbie. Tor.na B CI1JIY yCJio
BI1H 1) - 5) ITOJIIOC, 0 KOTOpOM IIJJia pe'lb BbJIIJe, eCTb 'leTHOe 'lHCJIO
-2(mod 4) 11 11MeeT KpaTHOCTb 2. OH pacrroJJO:tKeH B TO'lKe a 2 11 B St(u)
BHOCI1T 'lJieH - u4"+2 (a0 Irr u + a 1); a0 > 0. l13 ,n.oKa3 aTeJibCTBa JieMMbi XV
RBCTByeT, 'ITO QJYHKUI15I
exp(-Aull-u4k+2(bo In lui +bl))
np11 bo > 0 He MO)KeT 6b!Tb X. QJ. 11 'ITO QJYHKU.I1H
cp (u) = exp ( -Au'+ u4k+ 2 (ao In I u I + a 0 + (4k+2))- u4"+4)
6y.neT x. cp. rrp11 6oJibiiJOM A. Tor.na 11 exp (lft(u) +In <p(u)) 6y.neT x. cp.,
u
\ llf(u) +In cp (u)
11 B pa3JIQ)KeHI111 J U du OTCYTCTByeT 'lJieH C u4k+2, T3K 'ITO
0
u
~ llf(u)+~n cp(u) du= ~ a,.u""-u4k+4+ ~· g>
() m= l 111 Lm
E(z)
uz - ( )dz. za z
(54,1)
no JieMMe XI mnerpaJibHbJe 'lJieHbl 11C'le33!{)T
In cp(u) + P(u) =- Au2
11 ITO T e op e Me r. KpaMepa [8] F (x) HOpMaJibHblll 33KOH . TaKI1M o6pa30M,
Mbi .noKa3aJII1, 'ITO ycJIOBI1R 1) - 5) .nocTaTO'lHbi, 'lT06bi YTBep)J(,ll.eHI1H (A)
11 (B) 6biJII1 3KBI1B3JieHTHh!MI1.
Jl.OK3)Ke M, 'ITO 01111 11 He06XO,li.11Mbl, T. e. lJTO 11p11 HapyrneHI111 XOTH 6hl
O,li.HOrO 113 HI1X (A) 11 (B) CT3HOBHTCH He3KBI1BaJieHTHb!MH.
EcJII1 HapyrneHo O.li.HO 113 ycJIOB'I1H, 1) 11JII1 2), To, oqeB11.LI.HO, 113 (A) He
CJie,nyeT (B). l1TaK, 1) 11 2) He06XO.LI.11Mbl .LI.JIR 3KBI1BaJi e HTHOCTH (A) H (B),
T3K 'ITO 2 - KOpeHb a(z).
YcJIOBI1e 3) TaioKe Heo6xo,li.I1MO. B caMOM .neJie, rrycTb 3) HapyrneHo.
Tor.na ecTb .li.ByKpaTHbii1 KopeHb o (z) B11,ll.a s = 4k; k ueJioe. .Il.JIR Hero
11MeeM
a(s) =lazts + · · · +la, l5 - l bz r- · · · - l b, I' =O
a' (s) = I a, Is In I az I+ · · · +I a, !<In I a, 1- i bz I<In I bz I- ·
-I b, lstn I b, I = o.
113 ,li.OKa3 aTeJibCTBa JieMI\Ibi XV RBCTByeT, 'ITO np11 A > Ao > 0 QJYHKUI1R
rp (u) = exp(- Au2 -u4k In lui)
6y.neT x . cp. HeKoToporo CI1M~IeTp11'lecKoro 3aKoHa <P(x). D .JIH 3TOro 3aKoHa
pacrrpe.neJieHI1R X; HMeeM
L1 (x) ~ L/ (x) ~ L2' (x) ';;f L 2 (x).
3Ha'li1T, 113 (B) He cJie.nyeT (A) . DycTb Terrepb HapyrnaeTcH 4). Tor.na JIH6o
o (z) 11MeeT Tp exKpaTHbiH KopeHb s = 4k + 2 ?;> 2 (k u.e.rwe), JIH6o eCTb
,li.ByKpaTHb!H KOpeHb S = 4k + 2, KOTOpblH He HBJIHeTCH M3KCI1M3JibHb!M 113
ITOJIO:tKHTeJibHb!X KOpHeH, TaK 'ITO eCTb KopeHb
a> s = 4/l + 2.
280
PaccMoTpHM nepBbiH CJiyllaii. KaK HBCTByeT H3 ,ll.OKa3aTeJibCTBa JieMMbi XV,
<i>YHKU.HH
<p(u) =exp(-Au2 -lul 4k+2In2 lul) npH A > Ao
6y.LieT X. 4> . .CHMMeTpHlleCKOrO 3aKOHa (]) (x), .llJIH KOTOporo 6y.LieM HMeTb
L 1 (x) ~ L, (x), TaK liTO H3 (B) He CJie.LiyeT (A). Bo3bMeM Tenepb BTopoii cJiy
llaii: S = 4k + 2 ,!I.ByKpaTHbiH KOpeHb, H a > s;a (a)= 0. l·f3 ,llOKa3aTeJibCTBa
JieMMbi XV HBCTByeT, liTO <l>YHKU.HH exp (- Au2 - u4k+2 In I u I) He 1\WJKeT
6biTb X. c,p. H liTO <i>YHKU.HH
gJ(u) = exp (- Au2 + u•k+2 In I u 1- u " )
6y,!I.eT X. 4>. ,ll.JIH CHMMeTpHlleCKOrO 3aKOHa (]J(x), ,ll.JIH KOTOporo
L 1 (x) ~ L 2 (x). 3HaliHT, 4) Heo6xo.LIHMO.
TiycTb HapyrneHo 5). HMeiOTCH lleTbipe B03MORUIOCTH:
I) a(z) 11MeeT ITOJIOJKI1TeJibHbiH KopeHb y, He paBHbiH I.J.eJIOMY lleTHOMY
liHCJiy, H [ ~ 1 lleTHO. I-b ,ll.OKa3aTeJibCTBa JieMMhi XV ycMaTp11BaeM, liTO
<i>YHKU.HH
·gJ(U) = exp (- Au2 -I u lr)
np11 A > A 0 6y,!I.eT x. cp. CHMMeTp11ll ecKoro 3aKOHa W(x), ,ll.JI51 KOToporo
LI(x) ~ L 2(x), TaK liTO H3 (B) He CJie,ll.yeT (A).
II) { ~} =t= 0 ( {} 3 HaK ,!I.po6HOH llaCT11), [ ~] IIelleTHa, HO y He e.LIHH-
CTBeHHbiH TaKoll: I<OpeHb, Tal< liTO eCTb /'I > y C I ZJ ) =f= 0· rl t! J HelleTHO.
) 2 J ' 2
T or.Lia H3 ;'J.OKa3aTeJ1 bCTBa JieMMbi XV ycMaTp11BaeM, liTO exp (- Au2 -
-A1 1 u lr) np11 A1 > 0 He 1\WR<eT 6biTb x. ¢>. 11 liTO <p (u) = exp (-Au2 +
+ I u 1r- I u 1r,) npH A > Ao 6y.LieT x. ¢. CI1MMeTpHllecKoro 3aKoHa (lj (x),
,ZI.JHI KOToporo L 1 ( x ) ~ L 2 (x) 11 113 (B) He CJie,ll.yeT (A) .
III) KopeHb r co CBOHCTBaMI1, yKa3aiUibiMI1 B II), e,llHHCTBeHHbiH, HO
He npoCTOlL
H3 ,ll.OKa3aTeJibCTBa JieMMbi XV BI1)I;HM, liTO
<p (u) = exp ( - Au2 - I u l i' ]n I u!)
np11 A > A 0 6y,n:eT x. ¢>. CI1MMeTp11'I ecKoro 3aKOHa (/) (x), ,ll.JI51 KOToporo
L 1(x) ~Lz(x), 11 H3 (B) ue CJi e,ll.yeT (A ).
IV) YKa3aHHbi H I<opeHb /' e,n:HHCTBeHHbii1, npocToii, 110 He MaKC11MaJib
I-Ibii1 113 noJimimTeJibi-IbiX I<Opi-Ieii, Tai< liTO ecTb KopeHb a > y. B 3TOM cJiyllae
,n:el'fcTByeM, I<aK B II): ·113 ,n:oKa3aTeJibCTBa JieMMbi XV CJi e,n:yeT, liTO
~p(u)=exp(-Au~+ lu l' -lu l")
np11 A > A 0 eCTb x. ¢> . CI1MMeTp11llecKoro 3a Koi-Ia (/J (x), ,n:JJ51 KOToporo
L 1 (x) ~ L 2 (x), H 113 (B) H e CJJe,n:yeT (A). 3THM ,n:oKa3ai-Ia u eo6xo,n:HMOCTb
ycJJOBHH 1) - 5) 11 TeM C<lMbiM TeopeMa I.
§ 55
3ai-iMeivrcH 3,n:ecb ,ll.OITOJIHeHI1eM K Teop eMe II' § 2. TiycTb 3a,n:aHo u.eJJoe
'-IHCJIO 2m> 0. TioKa)KeM, LITO MQ)IUIO ITOCTp011Tb Taime JII1HeHI-Ibie CTaTI1 -
CTI1KI1 L1 (x) H L 2 (x), ,n:.rm KOTOpbiX onpe,!I.CJlmoma 51 ¢YHI<U.HH a (z ) 'F 0
11 cymecTByeT CHMMeTp11lleCI<Hll 3aKOI-I W(x), 11M e iOU~I1if 2m MOMellTOI3, HO
He HMeiOll1.11H (2m+ 4) -ro MOMeHTa, ,!I.J1 51 KOToporo L1 (x) ~ L 2 (x). Pa3y-
281
MeeTCH, f!>(X) He COBna.n.aeT C HOpM3JibHbiM 33KQHOM (KOTQpblH HMee.T BCe
MOMeHThl) .
.il.nH TaKoro noCTpoeHHH, cooTBeTCTBeHIJO .n.oKa3aTeJ!bCTBY JleMMbl XV,
§. 51-52, 33Me'laeM, 'ITO ¢YHKU.HH
qJ(u) =exp (-Au2 -rtt1 -u6_:_ ... -u2"'- j u j1) . (55,1)
6y.n.eT xapaKTepHCTH'IeCKOH, eCJIH y > 2m He eCTb U.eJIOe 'ICTHOe 'IHCJIO, [ ~ J
'Iento H A > 0 .LI.OCHl'l'O'IHO BeJIHKO. B KatteCTBe TaKoro 'IHCJia r Bbi6epeM
1
y = 2m + - , eCJIH m 'leTHO,
2
1
y = 2m + 2 + 2 , eCJIH m He'leTHO.
Tor.n.a, npH A> Ao, ~(u) 6y.n.eT x. ¢ . .LJ.JJH HeKoTOporo CHMMeTpH'IecKoro
33KOHa f!>{~·)
!l.aJiee, O'leBH.LI.fiO, cymecTByeT (/)(2m> (0), H He cymetTByeT q.><2mH> (0),
,..
HOO r <2m+ 4. BBH.LI.Y ;:noro FJMeeM U2m = ~ x2mdfP(x) < oo; U2m+4 = oo.
-00
PasyMeeTCH, aJ(x) He HopMaJibHbiH saKOH ('ITO cpasy BH.LJ.HO H3 (55,1)) ..
no JieMMe XIII § 43, .LJ.JIH Harneif x. ¢. (55,1) MO}!{HO nocTpOHTb .D.Be
JIHHeHHble CT3THCTHKH
· L1 (x) = a1x1 + · · · + a,x,; L2 (x) = b1x1 + · · · + b,x,
C HeOTpHU.3TCJ!bHbiMH K03¢qmu.neHT3MH H 0!1pe.LI.eJIBK>IU.eif ¢YHKU.Heif (} (z) '¢= 0,
TaKHe, 'Ito L1 (x) ~ L:z(x), eCJIH na6JIIo.LJ.eHH.sr pacnpe.n.eJieHhr no saKoay aJ(x) .
.3To n .n.ocTaBJI.sreT )KeJJaeMoe .nonoJIHeRne K TeopeMe. II'.
§ 56
3 'aliMeMcH s)l.ecb .n.oKasaTeJihCTBOM TeopeMbi III § 2.
TiyCTb H3M 33.LJ.3Hbl 'IHCJia Yh ·y2, 0 0 0 rl· no.n. YCJIOBfl.eM
o < ·r'l < · · · < r1 ~ 2. (56,1)
Mbr 6y.n.eM CTpOHTb 'leTBCpKy JIHHeHRhiX CTaTHCTHK £1 (x), £2 (x), La (x),
L 4 (x), T3KHX, 'I.TO ecJIH .n.mr CHMMeTpH'IeCKoro saKoHa ()) (x) pacnpe.n.eJieHHH
HaoJIIO.LJ.eHHH X; LI(x) ·~ L2(x) H L 3 (x) ~ L.1(x), TO x. ¢. saKoHa lP(x)
HMeCT BH.ll.
f (u) = exp (- ~ Ar1 i u j7i). (56,2)
j = l
Mbr 6y.n.eM cTpOHTh .rmueihrhre cTaTHCTHKH no o6pasu.y .n.oKasareJibCTBa
JieMMbl XIII § 43.
BoshMCM .n.sa u.eJJbiX B3aHMHO npocToJX
H Y 2 ~ 3. U COCT3BHM .LI.Be U.CJ!b!C <j.J.}'Hl(U.HH
a.(z) = ~[exp en,~. z)- g,. ];
noJioJKHTeJihHhix 'IHCJia Y1
v= 1, 2. (56,3)
l.J:HCJia r 1, y2 , ••• , ?'1 6y.n.yT, O'l.eBH.LJ.HO, o6ru.HMH. npoCTbiMH HYJJHMII 01 (z)
n a2 ( z). HHbiX o6ru..Hx HyJieli He cymecrsyer. B caM OM .n.eJi e, ocraJibHbie
282
2nim
HYJIH a~ (z ) HMeiOT Bli.U e = 'YI + ln u~ Yjt r.ne m .=~= ~ ~eJIOe. ECJIH 6bi .n.Ba
TaKHe HYJIH y a1 (z) H a2 (z) coBrraJIH, TO rroJiyliHJIOCh 6br
2nim 2nin
rj+ ln Ut yi= rk + lng2 Y~c·
OT}(y.zJ.a
':{TO HeB03MO)I{HO, TaK K3K U1 H g 2 B3aHMHO IIpOCTbie. fhaK, COBMeCTHbiMH Hy
JIHMH a1 (z) H -a2 (z) HBJIHIOTCH JIHlllb peaJibHbre HYJIH y1, '}'2, .. . , 'Yr
Do.no6Ho TOMY KaK B .noKa3aTeJibCTBe JieMMhr XII I § 43 3aMmiaeM, liTO,
pa3BOpaliHBaH CK06Im B (56,3) H pa36HBaH li.!IeJI.bi C ~eJIOtiliCJieHHhiMH K03<P
¢H~HeHTaMH Ha CYMMbi O.UHHaiWBbiX liJie HOB, IIOJIYtiHM
a1 (z) = a1z + · + a,Z- b1 z - · · - b,Z
a2(z) = c1z + · · · + c,Z- d1 z - · · · - d,Z,
r.ne liHCJia a1, b1, c1, d; IIOJIO::>KHTe.!IbHbi.
YCJIOBHMCH, liTO B 3THX paBeHCTBax Y*e npoH3Be.neHbi Bee· B03MO)I{Hbre
IIOIIapHbie COKpameHHH liHCeJI a; H b;; C; H d;. ilOJIHOFO COKpameHHH 6biTb
He MO)I{eT, H6o a~ (z) ;¢= 0. Tor}J.a IIOJIYliHM
sup (a1, ... , a,) =1= sup (b1 ,
sup (c1, . .. , c,) =1= sup (d~,
BBeJJ.e M JIHHeHHbre cTaTHCTHKH:
L1 (x) = a1x1 +
L 2 (X) · = b1X1 +
L 3 (x) = c1xl +
L4(x) = d1x1 +
+ a,x,,
+ b,x,,
+ c,x,
+ d,x,.
' b,)'
' d,).
(56,4)
(56,5)
Mbr BHJJ.HM, CJieJJ.yH JJ.OKa3aTeJibCTBY JieMMbi XIII § 43, '!TO JJ.JIH .BCHKoro CHM
MeTpaqecKoro 3aKOHa dJ(x ) C X. 4>. BH}J.a (56,2) H, B llaCTHOCTH, }J.JIH KOM
ll03Hll,HH COOTBeTCTBYIOmHX 'Yi CHMMeTpHlle:CKHX YCTQHliHBbiX 3aKOHOB,
llOJIYliHM
L1(x) ~ L 2(x ); L3fx) ~ L4(x).
06paTHO, IIYCTb }J.JIH K3KOFO-JIH60 CHMMeTpHtieCKOFO 33KOHa C X. 4>. f(u)
HMeeM
L1 (x) ~ [,2 (x); L3 (x) ~ L 4 (x).
DpHMeHHH TeopeMy · V, HMeeM, rroJiaraH IJf(u) =In I f(u) 12 = 2ln f(u),
npH u > 0
u 1 "' r IP(zz)d - ~ A<•·) r+ ~ s':~l( ) J U U- ~ Yj U I ~ m U ,
0 j = l m=O
(56,6)
r}J.e Y = 1, 2, H BTOpble cyMMbi UJJ.YT no KOM IJJieKCHbiM HYJIHM a~ (z) C IIOJIO
.JKHTeJibHOH MHHMOH qacThW. Ho TaKHe HYJIH y .a~ ( z ) He coBrraJJ.aiOT, H rro
TOMy, B CHJIY e.UHHCTBeHHOCTH pa3JIO::>KeHHH ( 56,6) ( CM. § 20) , BtOpbie
c yMMbi rrycTbre, H
u= 1, 2, ... , z).
283
I
0TCIO.LJ.a lJI (u) = ~riA~~) u1i npH U ;> 0, H BBH.Ll.Y tieTHOCTH
j = l
1 'F ( )
f(u) = e'i u
I
I (u) = exp (- 1~ Ari I u l1i) , r.u.e
1 (1} -A-;-=- r-Ar·.
J 2 J ]
3To H .u.oKa3bJBaeT TeopeMy III.
IlpaKTHtieCKOe ITOCTpoeHHe tieTBepOK JIHHeHHbiX CTaTHCTHK .Ll.Jifl npHMe
HeHHfl K KpHTepHHM corJiaCHfl (eM. pa3.LJ.. II) Bbiro.u.Hee npoH3BO.LI.HTb HHbiM
cnoco6oM, TaK KaK npH mmcaHHOM cnoco6e tiHCJIO r, HMeiOmee nopH.LJ.OK 21,
CJIHlllKOM 6biCTpO pacTeT C tiHCJIOM napaMeTpOB /.
§ 57
06paTHMCH K TeopeMe V'. Ilpe.u.cTaBJieHHe (2,2) npH 0 <; u -<;: uo o6o
CHOBbiBaeTCH B § 1-20. IlpHBe.LJ.eM npHMep, ITOKa3b!BaiOlll.Hlf, '!TO ITpH U > Uo
H u < u0 , B yCJIOBHHX TeopeMbi V', x. ¢. f(u) Mo:>KeT 6biTb JII06oii HeBomy
TOR KHH3Y <PYHKI.I,Heii.
IlycTb r tiHCJIO c yCJioBHeM
O<r< L (57, 1)
CorJiacHo .u.oKa3aTeJibCTBY TeopeMbi III (§ 55) MOJKHO nocTpoHTb .u.Be JIH
HeHHbie CTaTHCTHKH
Lt (x) =a tXt + · · · + a,x, H L 2 (x) = b1x1 + · · · + b,x,
c yCJioBHeM ai ;> 0, bi ;> 0 H a(z) = a1z + · · · + a,Z- b1z- · · · - b,Z
HMeeT O.LJ.IIH, II TOJibKO O.LJ.HH, peaJibHblli I<OpeHb y .
.[{JIH yCTORtiiiBOfO CIIMMeTpHtieCKOfO 3aKOHa <l>(x) C X. <!J. f(u ) = e-lul'l'
6y.u.eM HMeTb L 1 (x) :::;;;; L 2 (x). B u0 = 1, IIMeeM f (u0 ) = f ( 1) = e-1 = v0;
f' ( uo) = - ye-l = Vo'.
IlpoBe.LJ.eM KaCaTeJibHYIO K KpiiBOR y = f (u) B TOTJKe (tto, Vo): y - Vo =
= Vo' ( U - 1) . BBH.Ll.Y TOro, liTO Vo' < 0, OHa n epecetieT OCb a6CI.I,IICC
B TOtiKe U 1 . .[{aJiee, npii 0 <; U .· :. l HMeeM B CIIJIY YCJIOBIIfl (57,1) f"(u) > 0.
CocTaBHM Tenepb •IeTHyio <PYHKI.I,IIIO ft (u), KOTopaH npii u ;> 0 onpe.LI.eJIHeTCH
TalC
!1 (u) = f (u) npii uo <; u -<;: u1o
{I (u) =vo+ vo'(u-- 1) npH uo <; u .<;: u1,
f1(u) = 0 npH u ;> u1.
Mbl DOJIY'!IIM KYC04HO rJia,LJ,KYIO CIIMMeTpiitiHYIO KpiiBYIO, He BOfHYTYIO KHII3Y
H C Op.LJ.HI·!aTOH B 0, paBI-JOH 1. Ilo Teop eMe f. IlOJIHa [13], OHa 6y,LJ,eT X. <!J .
.LI.JIH HeKOToporo pacnpe,LJ,eJi e HH51 Wt (x).
Mbi BBeJiu ¢opMbi L 1 (x) H L 2 (x). IloJIOJKHM Tenepb s up (at, ... , a,.
b 1, . • . , b r) = a, II
L 1 (x) L: (x)
L 8 (x) = -a- + Xr+I; L~. (x) = -- + Xr+l· u1 au1
IloKa)KeM, 4TO .LJ.JIH 3aKoHa (J)I(x) c x. ¢. ft(u) 6y,LJ,eT L 3 (x) :::;;;; L 4(x), tieM
6y,LJ,eT 3aKOHti eH npHMep, HY)!<HbiH .LJ.JI51 TeopeMbi V' ( 3 al\IeTHM, 4TO ,LJ,.TI 51 cpopM
L 3 (x) H L 4 (x) T04Hbie Bepxmie rpmm K03cpcpHI.I,He HTOB coBna.u.aioT).
284
11MeeM: x. cp. L3(x) nMeeT BH.ll:
/1 (u) fl 11 (~~u) = rp(u)
j=l 1
u x. cp. ,n:JIH L4 (x) HMeeT Bu,n::
Dpu U;:;;;, U1 HMeeM ·t/J(u) = rp(u) = 0. Dpu 0 <; U < U1 HMeeM
b·u
_ J_ < I= Uo, H
au1
f () ( bl"l+ ... +b.' )
= 1 u • exp - aYu,r u1 = 1/J(u).
0TciO,n:a L 3(x) ;:;;;;r. L4(x) npu x;, pacnpe,n:eJieHHhiX no 3aKoHy (JJ1 (x).
Dpu 3TOM BMeCTO KaCaTeJihHOH K I<pHBOH y = f(u) MOMHO 6biJIO B351Th·
KYCOK JII060H BhiiTYKJIOH KHH3Y KpHBOH ,no ee nepece l.feHH5I C OCh!O a6CU,HCC.
II
CTaTHCTHLfecKue KpuTepnu
§ 58
T eopeMhi I - III pa3,n:eJra I MoryT cJiy)I<HTh ,LJ,JIH nocTpoeHHH HOBhiX CTa
THCTHI.fCCKHX KpHTepHeB. 1\\.bi 1-Ial.fHeM C BOnpoca 0 KpHTepHH HOpMaJibHOCTH'
Tuna.
nycrh ,n:aHa llOBTOpHaH Bhr6opKa o6neMa N
(58,1)
Mhr xoTH~r HCnh!TaTh nmoTe3y Ho o TO ~r . l.fTO 3TO - Bhr6opr<a H3 Hop~raJib
HOH r e HepaJibHOH COBOKynHOCTH C napaMeTpaMH a, a, KOTOpyro Mhl 6y,n:eM
o6o3Hal.faTh N(a, o).
ECJIH napaMe Tphr a H a H3 Be CTHbi, TaK l.fTO runoTe3a H0 npocTaH, TO
MO>KHO npHMe HHTb H3 BeCTHbiH KpHTepHii A. H. KoJIIvroroposa HJIH ,r. 2 - KpH
Tepuil ,LJ,JI5I BbiHCHeHH5I CO!JI3CH5I Ho c Bbi6opKoi:'r (58,1). npu 3TOM Ha,LJ,o
HMeTb B BH,LJ,y, 'ITO o6a 9 TH KpHTepHH HMe iOT aCHMllTOTH'!eCKHH xapaKTep ·
H LfTO npHMeHeHHe ._% 2 BCe r,n:a BHOCHT HeJ<OTopyro He onp e,II,e.TJeHHOCTh B CB513H
C B03 ~JO)J(J10CTbJO pa3JIH'IHh!X rpynnHpOBOK H36JIIO,LJ,CHHi{.
EcJIH >Ke, KaK u 6bmae T B no,II,aBJimouJeM 6o.TJbWHHCTBe CJiyqa e s, n apa
MeTpbr a H 0 HCH3 BCCTHbl H HX 33MCH5iiOT CT3THCTHK3MH C llOMOJJ..J.hiO BhJ-
6opKH (58,1) (06hiLfHO aHa 33Mei-IHIOT CT3THCTHK3MH X H s), TO ,n:eHCTBHC .7.~
u acHMnToTwre cKHH xapaKTe p pacl.feTa cooTBeTCTByromero pacnpe,II,eJiemm,.
soo6me rosopH, yxy,LJ,waroTCH.
HeKOTOpbre asiopbl (B. 11. PoMaHOBCKHH [15], A. M . .Ll.JIHII, A. K Ky
Tai:f [16] u ,II,p .) npuMeHmOT ,LJ,JIH HCnhrTaHHH Ho cJie,n:yro m ee BH,II,OH3MeHeHue
I<pHTepuH A. H. KoJIMoroposa.
285·
PaccMaTpHBaeTC.H
l = VNsup iS,.(x) -<r(x, x, s)l, (58,2)
%
x (x-;)'
- 1 ~ -~ a(x, x, s) = v- e . dx
2rrs
-oo
. H sll (x) KYMYJI.HTHBHa.H KpHBa.H Bbi6opKH (58, 1), H KpHTH<IeCKa.H 30Ha )l.JI.H
Ho Bbi6HpaeTC.H Hcxon:H H3 rrpe.nrroJIO)I<eHH.H, <ITO rrpH 6oJibll!HX n )l.JIH J.
. HMeeT MeCTo pacrrpe.neJieHHe A. H. KoJIMoroposa
P().<z) = ~ (-1)ke-2k'z' npH z > O.
k=-oo
: Eho rrpe.nrroJioMeHHe rrpH 6JIHMai1weM paccMoTpeHHH oKa3hiBaeTc.H Heo6-
0CHOBaHHbiM, 11 3KcrrepHMeHTbi c Ta6JIHU.eii CJiyqaifHhiX <IHCeJI yKa3biBaJOT Ha
B03MOMHOCTb 3H3<IHTeJibHblX paCXO)I<)l.eHHH [17].
AcHMITTOTH<IecKoe pacrrpe.neJieHHe craTHCTHKH THrra (58,2) 33BHCHT oT
THITa 33KOHa pacrrpe.ne.leHHH reHepa.lbHOii COBOKYITHOCTH H )l.JIH CJiy<IaH
(58,2) )l.OJIMHO 6biTb CITeU.H3.'!bHO paCC<IHT3HO.
T eopeMbi I, II H III pa3.neJia I MoryT 6hiTb yrrOTpe6JieHbi )l.JIH rrocTpoe
HH.H KpHTepH.H COfJI3CH.H H o C Bb!60pKOH (58, 1), KOTOpblH He HOCHT 3CHMITTO
. TH<IeCKOfO xapaKTepa H r)l.e ,,06beMbi" KpHTH<IeCKHX 30H MOfYT 6b!Tb TQqHQ
paCC<I!'fT3Hbl.
ilOCTpOeHHe T3l<Of0 I<pHTepH.H HOpMaJibHOCTH THIT3 OITHC3HO B MOeH
33MeTKe [19].
CorJiaCHO reopeMe I, rro.n6HpaeM .nse q>opMbi L 1 (x) 11 L 2 (x) or r rrepe
·MeHHhiX Xt, x2, ... , x, raKHe, qTo yCJIOBHe Lt (x) ,..._, L 2 (x) 3KBHBaJieHTHO
HOpMaJibHOCTH THITa H36JIJO)l.eHHH X; .
I13 Bbi60pKH (58,1) 6epeM nepBb!e r 3JieMeHTOB Xt, X 2, ... , X, H COCTa
B.H.HeM ¥3 HHX q>opM)' £1 (x), 33TeM H3 CJie)l.yiOllJ.HX r 3 J! eMeHTOB COCT3BJI.HeM
q>opMy L 2 (x), H3 nocJJe.nyJolll.HX r 3JieMeHTOB - q>opMy Lt (x) 11 T . .n.. Or.ne
_JIHeM rroJiy<IeHHbie ¢opMbi L 1 (x) OT paBHoro HM I<OJIH<Iecrsa ¢opM L2 (x).
B CHJIY reopeiVIbi I <IHc.'!a L t (x) 11 <IHC.ria L2 (x) cocraBJI.HIOT o.n.Hopo.n.HyJo
Bbi60pi<y TOf)l.a, H TOJlbi<O TOr)l.a, eCJJH THIT Ha6JIIO)l.eHHH HOp~!aJibllhJii.
0JI.HOpO,!!,HOCTb Hall!HX .nsyx Bbi6opoK, cocTO.Hlll.HX H3 <IHCeJI Lt (x)
H L 2 (x) H HMeiOllJ.HX OJ1.HHaKOBbli1 o6 h eM , MO)J<I-IO rrposepHTb c rroMOW.biO
HOBoro I<pHTepH.H cor.riaC'H.H 13. B. f He.neHKO H B. C. KopoJIIOKa [18], He
HOCHIU.ero aCHMITTOTHl!eCKOrO xapaKTepa .
B MOeH 3 aMeTI<e [ 19] )'I<a3aHO ITOCTpOeHHe mmeHHb!X cpop !V! OT <I eTbipex
rrepeMeHI-!blX L; (X) ( i = l , 2, 3, 4), OTDelJaiOI-ll.HX B COBOI<YITIIOCTH )'CJIOBH.HM
TeopeMbi I. TeopeTH<Iecim raKHe q>op Mbi orse<IaiOT .no.'!)I<Hbll\I Tpe6onaHHH~r,
HO rrpaKTH<Ie CI<H MOf)'T CJiaOO pearHpOB3Tb Ha 3 aMeHy HOpM3J1bHOf0 3ai<OHa
I<ai<HM-JIH60 ,6JIH3.TJe;.K3UlHM" H H~! CJOUJ,Hl\1 )l.HCITepcmo (HaiTpH Mep, paBHO
MepHb!M B [a- 2a, a + 2oj), T3K IG:IK Teopei\Ja J J51f!)'HOB3, KaK ITOK3 3biBaeT
3KCrrepHl\!eHTa.TJbHLiii MaTe pHa.TI, HHOf,!I,a )l.eHCTB)'eT DeCbMa TO<I!!O )')I<e rrpH
q eTbipex He3aBHCHMbiX CJiaraeMbiX, H rrocTpOCHHLi e ¢opMb! Lt (x) H L2 (x)
l\IOfYT B 3TOM CJly<Ia e OI<a33TbC.H 6 JIH3 KIHIH K HOplli3JibHOMY pacrrpe,!!,e.'l eHHIO.
'286
TipaKTH'ICCKH rop33.LJ:O JIY4We ,neiicTBYIOT CJIC.LJ:YIOIJJ.He <PopMbl, H3H.LJ:CHHb!C
A. n. Xycy:
Lt (x) = Xt 1
1 1 1 1 .
L 2(x) =-x2 +-x3 + -x,1 --X5 J
2 2 2 2
(58,3)
3.necb onpe,neJimou~aH <PYHKII.HH
o(z) = 1-4. 2-z
HMCCT e,nm-ICTBCHHblH peaJibHblH npOCTOH KOpCHb 2 H Y.LJ:OBJICTBOpHeT ycJIO
BHHM TeopeMbl l.
CooTHoweHHe L1 (x) ::;;;; L2 (x) 3KBHBaJieHTHO rwpMaJibHOCTH THna pac
npe.neJieHHH Ha6JIIO.LJ:CHHH.
Jl.aJree, OK33b!B3CTCH, 'ITO .LJ:JIH 33KOH3, paBHOMepHoro B [- 2, 2], 0 ,6JIH3-
KOro" K 'HHTerpaJibHOMY HopMaJibHOMY 33KOHY c napaMeTpaMH 0; 1, L 1 (x)
H £ 2 (x) HMCIOT p33 HOCTH 'ICTBepTb!X MOMCHTOB 6oJibWe, 'ICM .LJ:JIH ,npyrHX
n ap cpopM C 'ICTblpbMH nepeMCHHbiMH H YCJIOBHHMH TeopeMbl l.
.9KcnepHMCHT3JlbHbiH M3TepH3Jl T3K)!\C rOBOpHT B TIOJ1b3Y npHMCHCHHH
¢opM (58,3).
Pa36op npaKTHliCCKH y.no6HbiX KpHTepneB HOpMaJibHOCTH, OCHOBaHHbiX
H3 JlHHCHHbiX cpopMaX, 6y,neT C.LJ:CJl3H B ,npyroM MCCTC.
§ 59
TeopeMa III .naeT B03MOlliHOCTb nocTpOHTb KpHTepm1 corJiaCHH Bbi6opKH
Xp X 2, ... , XN
C rHITOTC30H Ho 0 TOM, 'ITO THTI H30Jl!O.LJ:CHHH HBJIHCTCH KOMTI03HII.HCH CHMMC
TpH4CCKHX yCTOH'IHBblX 33KOI!OB C X. <il·
/(u)=exp(- tAi\u\1j)•
J=l
(59,1)
r,ne 4HCJI3 f' i C YCJIOBHCM
0 < 1'i < 2
.LJ:OJI)KHbi 6biTb 3a,naHbi, a 'IHCJia Ai Mory T 6biTb HCH3BeCTHbiMH . .UJIH rro
CTpoeHHH KptnepHH CTpOHM .LJ:BC napbl JlHHCHHb!X CT3THCTHIC
L 1 (x), L 2(x); L 3 (x), L .i (x),
T3KHX, 'ITO CCJIH .LJ:JIH CHMMCTpH4CCKOrO 33KOH3 (/) (x) HMCCM
Lt(x) ::;;;; L2(x); L 3 (x) ::;;;; L .1 (x),
TO X. <il· 33KO!I3 (/)(X) HMCCT BH.LJ: (59, 1) H o6paTHO.
(59,2)
(59,3)
3.necb Ha,no, o.nHaKo, onre THTb, 'ITO cor.nacHo TeopeMe IV, K CO)!\aJie
Hmo, T3K 6y,ny T ,nonym,eHbl H X. cp. BH.LJ:3 (59, 1), r,ne HCKOTOpbiC 1<03cpcpH
ll.HCHTbl Ai 0TpHII.3TC.TibHbl, T3iC 'ITO OHH HC 6y,nyT X . cp. KOMTI03 HII.HH ycToii
'IHBbiX 33KOHOB. Ho 9TO - C}l.HHCTBeHHbri1 BH.LJ: a JibTe pHaTHI3HbiX runoTe3,
COBMCCTHbiX C COOTHOIIICHHHMH (59,3).
0TMCTHM T3IOKC, 'ITO Bbi60p CT3THCTHKH ( 59,2) HC BCer,na Bb!rO.LJ:HO
.LJ:C.TJ3Tb DO Dp3BHJI3M § 56, T3K K3K npH 60JlbWOM 'IHCJIC HCH3BCCTHbiX napa
l\I CTpOB Ai noJiy'IHTCH 4pe3MepHo 6oJibwoe 'IHCJIO rrepeMeHHbiX r.
287
0TMeTHM erue CT3THCTHqecKOe TOJIKOB3HHe COOTHOllie~IHH (59,3) .
Mo)lmo Bhi6opKy (58, 1) pa36HTb Ha qeTbipe qacTH H cocTaBHTb q e Tbipe
HOBbiX He3aBHCHMbiX BhJ6opKH H3 3HaqeHHH cpopM L 1 (x), L 2 (x) , L 3 (x), L 4 (x).
Tor)J.a npH)J.eTcH npHMeHHTb KpHTepHii O)J.HOpO)J.HOCTH q eTbipex He3aBHCHMbiX
BbJ6opoK.
Mo.>KHO T 3K)l<e cocTaBHTb H3 (58, 1) )J.Be He3aBHCHMbiX Bbi6opKH H3 cpopM
L1 (x) H L 2 (x) H 3aTeM H3 Toil .>Ke B'bJ6opKH )J.Be He3aBHCHMbiX Bbi6opKH H3
cpopM L 3 (x) H L4 (x). 9TH IIOCJie)J.HHe Bbi6opKH oy)J.yT, O)J.H3KO, 3aBHCeTb OT
nepBbiX.
Mbi Mo.>KeM TpaKTOBaTb Bonpoc o6 O)J.Hopo)J.HOCTH Bbi6opoK H 3 L1 (x)
H L 2 (x) npH IIOMOIUH KpHTepHH 5. B. fHe)J.eHKO H B. C. KopoJIJOKa [18].
DycTb Mbl Ha3HaqHM IIPH .9TOM KpHTHqecKyJo 3 0HY zi o6oeMa aJ.
J].JIH Bbi6opoK H 3 cpopM L 3 (x) H L 4 (x) Mhi MO)l<eM TaK)l<e Ha3HaqHTb
KpHTHqecKyJO 30HY Z 2 06oeMa a2. floiia)J.3HHH COOTBeTCTBYJOIUHX Bbi6opoqHbiX
TOqeK B 30Hbl zi H z2 6y)J.yT co6biTHHMH 3aBHCHMbiMH. DycTb 3TO 6y)J.yT
co6b!THH ~ H 58. Tor)J.a Mbi 6y)J.eM OTBepraTh mnoTe3y Ho, eCJI H npo
H30ii)J.eT co6bJTHe 'll u ~- DpH 3TOM HIIIeeM
P('ll. u ~) <: P('ll) + P(m) = a1 + az,
qTo )J.aeT B03MO.>KHOCTb oiJ,eHHTb o6oeM KpHTHqecKoi'I 30Hhi )J.JIH Ho H npH
MeHHTb KpHTepHif O)J.HOpO)J.HOCTH )J.JIH )J.BYX Bbi60pOK.
§ 60
3aMeqaHHe B KOHIJ,e § 2 II03BOJIHeT CTpOHTb TaiGKe KpHTepHH, OCHOBaH
Hbie Ha O)J.HHaKOBOH pacnpe)J.eJieH HOCTH CTaTHCTHKH
sup (a1x1, a2x2 , .. . , a,x,) H sup (b1xl, b2x2, .. . , b,x,).
T aKHM o6pa30M, MO)KHO CTpOIHb KpHrepHH corJiaCHH Bbi6opKH (58, 1)
c rurroTe30H Ho o TOM, qTo saKOH pacrrp e,rr.eJieHHH xi orrpe)J.e.TiemrhiM o6pa-
30M CBH33 H C rrpe)J.eJibHbi MH 3aKOH3MH )J..TIH qJi eHOB BapHaU,HOHHOrO pH)J.3
11 MO.>KeT CO)J.ep)l<aTb .TII06oe qHCJIO H eH3BeCTHbiX rrapaM e TpOB.
3aMeTHM erue, qTo MOIKHO ITOCTaBHTb BOITpOC 0 ITOCTpOeHHH KpHTepHeB
COrJiaCHH Bbi60pKH
pacrrp e)J.eJINIHOCTH
H g2(x1, ... , x,)
O.!UWpO)J.HOCTH.
CO CJIO)KIWH rHITOTe30i1, OCHOB3HHb! X Ha O)J.HH3KOBOH
)J.BYX COOTBeTCTBYIOIUHX CTaTHCTHK U1 (Xi, ... , X,)
H Ha CBe)J.eHifH, TaKHM o6pa30M, BOIIpoca K KpHTepHIO
§ 61
B HaCTOHIUeM rr aparpacpe Mbi )J.a)J.HM rrpHJIO)l<e~me TeopeM I, II Jl III
K BbiBO.!!Y H 3BeCTHoro saKoH a MaKcBeJIJi a .!IJIH pacnpC.!!eJiei-IHH cKopocre i1
MOJieKyJI B KJiaCCHqeCKOi1 cpH3HKe ( CM. IIO 3TOMY ITOBO.!IY 3aMeTKY aBTOpa [1]) .
PaccMoTpHM MHmKeCTBO xaoTHq ecKH .!!BmKyiUHXCH qacTHU..
DycTb HMeiOT MecTo .!!Be rHrroTe3bi :
I. CymecTBYIOT TPH opToroHaJibHbie OCH OX!, ox2, OXs, T3I<He, qTo
IIpOeKIJ,HH CKOpOCTH ~Iay.!!aqy B35ITOH qaCTHU.bl Ha 3TH OCH, Xi, X 2 H X 3, CT3TH
CTHqeCKH He3aBHCH~!bi H O.!IHHaKOBO pacrrpe)1.eJieHbl.
II. CyruecTBYI.OT erue )J.Be ocH OL1 H OL2 (He o6H3 aTeJihHO opToroHaJih
Hbre) - T3KHe, qTo rrp oeKIJ,HH CKOpOCTH TOil )Ke qaCTHIJ,bl Ha OCH, l1 H f2
O.!IHH3KOBO pacnpe)J.eJieHbJ.
288
EcJIH o6o3Hal!HM az, a2, a3 H COOTBeTCTBeHHO bz, b2, b3 KOCHHYCbi yrJIOB
OLz H OL2 c OCHMH OXz, ox2 H OX3, TO IIOJIYllHM
(61,1)
rHrroTe3a II 03HallaeT, liTO [z ~ [ 2.
ECJIH JIHHeHHbie q)QpMbi lz H l2 y)J.oB.TJeTBOpH!OT yCJioBHHM TeopeMbi I,
TO 3TO CBOHCTBO 3KBHBaJieHTHO HOpMaJibHOCTH pacrrpe)J.eJieHHH Xt, X2, X3, liTO
H COCT3BJI5IeT CO)J.ep)K3HHe 33KOH3 ~aKCBeJIJia.
0)J.HaKo yCJioBHH TeopeMbi I HeCKOJibKO rpoM03)J.KH. Mbi MO)KeM rrpHMe
HHTb TeopeMy II, )J.JIH llero rroTpe6yeM BbiDOJIHeHHH eme O)J.HOH mrroTe3bJ:
III. Cpe)J.HHH lleTBepTaH cTerreHb cKopocTH llaCTHU. KOHel!Ha.
B TeopeMe II IIOJIO./KHM m = 2. TiycTb ocH OLz H OL2 T3KOBbi, liTO
sup(la1 !, las l, laai)=Fsup(lbll, lb2 l, lbsl)- (61,2)
PaccMoTpHM
a (z) = I all%+ I a2lz + I aslz- I bllz- I b2lz - I bslz. (61,3)
Ol!eBH)J.HO, a (2) = 0. EcJIH ocH OLz H OL2 TaKOBbi, liTO MaKCHMaJibHbiH
peaJibHbiH KOpeHb 0 (z), r < 4, TO HMeeM m = r ~ + 1] H, ITO rHIIOTe3e III,
CJIY43HHble BeJIHl!HHbl Xz, X2, X3 HMeiOT lleTBepTbiH MOMeHT H 2m = 4. BBH)J.y
3TOrO BeJIHl!HHbl Xz, X2, Xa HOpMaJibHbl, T. e . HMeeT MeCTO 33KOH ~aKCBeJIJia.
EcJIH yroJI <p Me)K)J.Y OCHMH OLz H OL2 MaJI, TO az, a2, aa 6JIH3KH K bz,
b2, ba H H3H60JiblllHH peaJibHbiH KopeHb a(z), y 6y)J.eT 60Jibllle 4.
f1oCMOTpHM, K3KYIO . rHIIOTe3y 0 CKOpOCTH MOJieK)'JI HpKHO C)J.eJI3Tb,
liT06bi H B 3TOM cJiyl!ae rHrroTe3bi I H II rrpHBeJIH K 3aKoHy MaKcBeJIJia.
TiycTb ocb OLz c HarrpaBJimomHM e)J.HHHl!HbiM BeKTopoM azT + a2T + a3k
pacrroJIO)Kei-Ia B IIOJlO.IKHTeJibHOM OKTaHTe H az =sup (a 1, a2, a3 ) • .UorrycTHM,
liTO az > a2 H az > a3. TiycTb )J.pyraH OCb OL2 c HarrpaBJI5I!OllJ.HM e)J.HHHll
HbiM aeKTopoM bzT + b2J + b3h cocTaBJIHeT c OL 1 M3JibiH yroJI 'P · Tor)J.a,
eCJIH <p )J.OCT3T04HO M3JIO, 6y)J.eM HMeTb: b; =SUp (bt, b2, b3); b; > 0.
TiycTb I az - bz I= {;; )J.JIH orrp e)J.eJieHHOCTH rrycTb az = bz + s, {; > 0.
Tor)J.a HMeeM
EcJIH z rroJIO)I<HTeJibHOe l!HCJio, paBHoe x, rroJiyl!HM
a(x) >a~ -3b~ = b~ [ ( 1 + iJ -3] > b~ [ ;~~ -2 j
4b
npH )J.OCTaTOl!HO MaJioM yrJie qJ. I1pH x > T, a(x) =F 0. 1-'haK, HaH60Jib-
. 4b
rnmi IIOJIO./KHTeJibHbiii KopeHb r cj:>yHKU,HH a (z) 6y)J.eT < T H
2m = 2[2; 1 + 2 J .
ECJIH cpe)J.H5I5I 2m-5I CTerreHb CKOpOCTH 43CTHU,bl KOHel!Ha, TO HMeeT MeCTO
33 KOH ~aKCBeJIJia.
289
JIHTEPATYPA
I. IO. B. JI H n n n K, 3aMe4aHHH no nono:1y KJJaccwiecimro nbiBO:la 3aimna M aKc
ncJIJia, JlAH, 1952, T. 85, N~ 6, 1251 - 1254.
2. J. M arc i n k i e w i c z, Sur une propriete de !a loi de Gauss, Math. Zeit
schrift, Bd. 44, H. 4-5, 1938, 612--618.
3. A. )l. M bl w K H c, JluHe~iHbi C tlH(p(jJepcHIJ.HaJibHbiC ypanHeHIHI c 3ana3;I.biBaiOli.I.HM
apryMeHTO!'<I, fTTH, 1951.
4. A . 0 . r e JIb cjJ 0 H A, JinneiiHbiC .LIHcjl<pepeHI..\HaJibHb!C ypanHCHHH c nOCTOHHHblMII
K03cjHpHI..\HCHTaMH 6eCKOHC'III01"0 HO[ll!J.I,Ka II aCHMHTOTII'ICCKHC IICpHO.L\bl 1..\CJi biX QJYHKI..\HH,
Tp. MHAH, T. XXXVIII, 1951, 42-67.
5. A . 0. r e JI b cjJ 0 H .LI, 0 1..\CJIO'IHCJICHHblX aH<WHTii'ICCKJIX cjlyHKI..\HHX, JlAH, T. 81,
1951, 341 - 344.
6. G. Doetsch, Theorie und Anwendung der Laplace-Ttransformation, 1937.
7. E. T H T 4 Map ill , T eopHH (IJ YIIKI..\HH, rTTH, 1951.
· 8. r. K p a Me p, CJiy'laiiHble neJIH4Hllbi n pacnpe:teJieHHH uepoHTHOcTei~i. fHHJI,
1947.
9. n. M 0 H T e JIb, HopMaJibHble CCMCHCTBa aHaJIHTH'ICCKHX cjlyHKI..\Htl , OHTH, 1936.
10. H. M. P bilK 11 K, T a6JIHI..\bi HHTerpaJiou, cyM M, pHnoB H npoH3ue;iei'IHII, fTTH,
1943.
II. 5. B. r II e ;.I.e H K 0 H A . H. K 0 JI M 0 r 0 p 0 n, f1p eJJ_CJibllb!C pacnpe,leJieHIIH :lJI~
cyMM He3aUHCHMbiX BCJIH'IHH, fTTH, 1949.
12. K y F a n, Les fonctions definies positives et les fonctions corn:pletement
monotones, Memorial de Sc. math., 1950.
13. G. Po I y a, Remarks on characteristic functions, Proc. of the Berkeley,
Symposium of Math. Stat. and Probability, 115-123, 1949.
14. Lagu e r r e, Oeuvres, t. 1. Paris, 1898.
15. B. H. Po M a HoB c K 11 ii, f1pnMeHeHHH MaTeMaTH'IecKoi-i cTaTHCTHKH s onbiTHOM
.L\CJie, fTTH, 1947.
16. A. K. K y T a ii n .LIp., CTaTHCTH'IeCKHe MCTO.L\bl anaJIH3a H KOHTpoJIH Ka'leCTBa
MawHHOCTpOHTCJibHOii npo,n:yKI..\HH, Mawm3, 1949.
17. B. C. f1 a C K e B H '1, 0 HCKOTOpb!X BOnpocax ~IaTeMaTH'ICCKOfi CTaTHCTHKH, IIMCIO
li.I.HX OTHOWCHHC K CTaTHCTH'ICCKOMY aHaJIH3Y H KOHTpOJIIO, 1952.
18. 5. B. r II e .11: e H K 0 H B. C. K 0 p 0 JI 10 K, 0 MaKCHMaJibHOM pacXOlKACHHH ,n:nyx
3MnHpH'ICCKHX pacnpe,n:eJieHHii, JlAH, T. 80, N~ 4, 525- 528.
19. IO. B. JI H H H H K, .TluneiiHble CTaTIICTHKH H nopMaJibHbiii 3aKOH pacnpe,n:eJieHHH,
.LI.AH, T. 83, N~ 3, 1952, 353-355.
f1oJiy<Jena 16 nnsapn 1953 r .
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| id | umjimathkievua-article-7733 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:33:22Z |
| publishDate | 1953 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/73/dbbb634b672d75b6a50395ed84fd3273.pdf |
| spelling | umjimathkievua-article-77332023-08-15T10:12:02Z Линейные формы и статистические критерии. ІІ Линейные формы и статистические критерии. ІІ Линник, Ю. В. Линник, Ю. В. - Institute of Mathematics, NAS of Ukraine 1953-08-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7733 Ukrains’kyi Matematychnyi Zhurnal; Vol. 5 No. 3 (1953); 247-290 Український математичний журнал; Том 5 № 3 (1953); 247-290 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/7733/9430 Copyright (c) 1953 Ю. В. Линник |
| spellingShingle | Линник, Ю. В. Линник, Ю. В. Линейные формы и статистические критерии. ІІ |
| title | Линейные формы и статистические критерии. ІІ |
| title_alt | Линейные формы и статистические критерии. ІІ |
| title_full | Линейные формы и статистические критерии. ІІ |
| title_fullStr | Линейные формы и статистические критерии. ІІ |
| title_full_unstemmed | Линейные формы и статистические критерии. ІІ |
| title_short | Линейные формы и статистические критерии. ІІ |
| title_sort | линейные формы и статистические критерии. іі |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7733 |
| work_keys_str_mv | AT linnikûv linejnyeformyistatističeskiekriteriiíí AT linnikûv linejnyeformyistatističeskiekriteriiíí |