On the law of the iterated logarithm for weighted sums of independent random variables in a Banach space
Assume that (X n) are independent random variables in a Banach space, (b n) is a sequence of real numbers, Sn= ∑ 1 n biXi, and Bn=∑ 1 n b i 2 . Under certain moment restrictions imposed on the variablesX n, the conditions for the growth of the sequence (bn) are established, which are sufficient fo...
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| Date: | 1993 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
1993
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/5924 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512146633261056 |
|---|---|
| author | Matsak, I. K. Plichko, A. M. Мацак, І. К. Плічко, А. М. |
| author_facet | Matsak, I. K. Plichko, A. M. Мацак, І. К. Плічко, А. М. |
| author_sort | Matsak, I. K. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-19T09:21:13Z |
| description | Assume that (X n) are independent random variables in a Banach space, (b n) is a sequence of real numbers, Sn= ∑ 1 n biXi, and Bn=∑ 1 n b i 2 . Under certain moment restrictions imposed on the variablesX n, the conditions for the growth of the sequence (bn) are established, which are sufficient for the almost sure boundedness and precompactness of the sequence (Sn/B n ln ln Bn)1/2). |
| first_indexed | 2026-03-24T03:24:09Z |
| format | Article |
| fulltext |
Y,r:!.!<:.519.21
I. K. MauaK, Kann. ¢i3.-MaT. nayK (ln-T nerK. npoM-cTi, KHiB),
A. M. IlJii'IKO, Kann. ¢i3.-~1ar. nayK (ln-T npHKJI. npo6JJ. MexaniKH i MaTeMaTHKH AH YKpainH,
JlbBiB)
IIPO 3AKOH DOBTOPHOro JIOrAPHcDMA )lJijl 3BA)KEHHX
CYM HE3AJIE)KHHX BHilA,llKOBHX BEJIHqun
Y 6AHAXOBOMY IlPOCTOPI
Assume that (Xn) are independent random variables in a Banach space, (bn) is a sequence of real
numbers, s. = L~ b, X;, and B. = L~ b,'. Under some moment restrictions imposed on the variables
X., the conditions for the growth of the sequence (b.) are found, which are sufficient for boundedness
and precompactness of the sequence (S. /(B.ln In Bn)v2) almost surely.
Hexalt (X.) - neJaJIC)Kni BHnanKoai aeJJH'IHIIH B 6a11axoa0My npocTOpi, (b.) -nocJJi.nOBHiCTb
," ," 2 niltcnHX •rnceJJ, Sn = "-'1 b; x,, B n = "-'1 b, . TipH MOMCHTnHX o6Me)Kenn.11x Ha BeJIH'IHHH Xn
3Haltneni YMOBH H3 picT noc JJ iAOBHOCTi (bn), nocTaTni AJJ.ll o6Me)KeHocTi lt nepeAKOMnaKTnocTi
nocJJinoanOCTi (Sn I (Bn In In Bn)112 ) M3lt)Ke 11ane a110.
1. Bcryn i oc110011i pe3yJ11>TaTn. HexaA E - cenapa6enbttHA 6attaxis npocTip 3
HOpMOIO 11-11 i Cllp.ll.)l(eHHM E". t.fepe3 Xn, 11 = 1, 00 , ll03Ha'IHMO £-3HalfHi He33.lle
.)l(Hi BHlla,ll.KOBi BeJIH'IHHH (H.B. B.), 3a,naui ua AMOBiptticHOMY npoCTopi (Q, 'B,P) [1,
c. 201; 2], bll, 11 = 1, oo, - llOCJiit-\OBHiCTb n.iACHHX lfHCeJI, SIi = L~ biXi, Bn =
= L~ bl, L(t) = In t npH t > e i L(t) = 1 apu t :s; e, L 2(t) = L(L(t)), X(t) =
= (2tLi(t)) 1l 2.
6yn.eMO rOBOpHTH, mo llOCJiiAOBHiCTb b11Xn 3a,ll.OBOJibH.ll€ 3aKOH nOBTOpuoro JIO
rapmpMa (3IIJI), .llKmo Ma0.)l(e ttaaeBH0 (M. H.)
A(b, X) : lim II Sn II I x<Bn) < OO (1)
"
i M. H.
{S,.f X(B,J, II~ 1} rrepen.K0MIIaKTHa B £. (2)
IlpH HaKJia,ll.eHHX B ttac-rynHOMY a63aQi YMOBaX j3 3aK0tty O a6o 1 BHUJIHBU€, mo
A(b, X)- ueBuaa,n.Kosa BeJIH'IHna.
Or JI.llA pe3y Jib TaTiB i D.OCHTb rroBny 6i6niorpac.pi10 apo 3IIJI y 6attaxoBHX npo
CTopax MO.)l(Ha 3HaATH B po6oTax [3-6]. y n.aniA CTaTTi Ae.llKi BiAOMi pe3yJibTaTH apo
3IIJI AJI.ll on.ttaKOBO po3non.ineu0x senu,m:u y 6auaxoBOMY ripocTopi nepeueceui ua
383.)Keui cyMH . .Uani BBa)KUTHMeMO, mo M x,. = 0, D x,. = (MIIXn f) 112 < oo, Bn j 00
i B,.lb;, ➔ OO rrpH /I ➔ 00 • HexaA r(b, X): lim,. M IIS11lllx(Bn), a en:._
CHMeTpH'IHi H.B. B. 6epnynni, P(e,. = ± 1) = 1 / 2.
TeopeMa 1. Hexaa npu oe11Kux o > 0 i 1 < p < oo
b,; = O(B 11 I (L(Bn)f1+i>)!(p-l)), (3)
sup M II X11 112P < oo. (4)
"
Tooi Oil.fl d = sup11 D X 11
r(b, X) :s; A(b, X) :s; d + r(b, X). (5)
Hac✓iiiJoK 1. Hexaa IIN - noc11iiJOlmicmb AiHiuHux o6MeJKeHux cKi11'te1111oou
MipHux onepamopio y npocmopi E i QN x = x - IIN x .. JlKu~o r(b, X) = 0 i
© I. K. MAUAK. A. M. TIJWIKO, 1993
ISSN 0041 -6053 . YK.p. Mam. JKYpH., 1993, m. 45, N• 9 1225
1226 I. K. MAUAK, A. M. IIJIP-IKO
sup D(Q,, X,,) ➔ 0 npu N ➔ oo, (6)
fl
moo yM00ax meopeJ.at 1 cnpaoeiJauouii 3ll.ll (1), (2).
BiA3Ha•mMO OAJUI Ba)KJIHBHlt 'laCTKOBlilt BHllaJ].OK, KOJIH BHKOIIYITTbC.Sl YMOBa (6).
Hexalt R: E"' ➔ E - KOBapiau.iltttHlt o neparnp B. B. Y, MY = 0, DY< 00 ; HR -
riJib6ep-ris niAnpoCTip npocrnpy E, acou.iltoBaHHlt 3 R [1, c. 126], T06TO nonos
He1111.S1 R(E*) B HOpMi CKam1pHoro A06yTKy (Rf, Rg) = Mf(Y) g(Y), f, g e E"'. He
xalt (fk) C E* - TOTaJibHa Ha £ opTOI'OHaJibHa BiJJ,HOCHO BKa3anoro CKaJI.Slpnorn
A06yTKY nocJiiAOBHiCTb. 51.Kmo OOKJiaCTH n,,x = L,~=11;,(x) Rfk, TO D(Q,,Y) ➔ 0
npH N ➔ oo [4, 5]. TaKHM 'IHHOM, KOJIH BeJIH'IHHH Xn OA!IaKOBO p03llOAiJieHi (OT
)Ke MaJOTb cniJibHHlt KOBapiar~ilt1mlt oneparop R) , TO Jl,JI.Sl nOCJiiL~OBHOCTi (fk) YMO
Ba (6) BHKOI-IyeTbC.H.
llJI.H B. B. Y 3 KOBapiau.ilt1-mM oneparnpoM R nOKJIMeMo
cr(R) = sup{j(Rf..f)l 112: 11/11= I} =sup{(Ml/(Y)2 1) 112: 11/11= I} .
TeopeMa 2 . .JIKil{O (3 yMooax meopeMU I oe/ltt'IUHU x,, MG/Olllb cnillbf/UU K06api
ai1iii11ui1 onepamop R i 11p11 oeJ1Kii1 momallbHiii 1ta E opmow1-ia11b11ii1 11oc11iooo110-
cmi (fk) OAJI oe11u'IUH QNX;, = X,, - L,:=I fk(X,,)Rfk ouK011ana yMooa (6), mo
max ( cr(R), r(b , X)) ~ A(b, X) ~ cr(R) + f'(b, X).
HaraJJ,a€MO, mo E na3HBa€TbC.H npOCTOpOM nmy 2, KOJIH L(JI.H KO)KHOl llOCJii AOB
HOCTi (xn) e E 3 yMOBH I,~ II x,, IP < 00 mmmrnae 36i)l(HiCTb M. H. PRAY I,~ e,,x,,,
i npOCTOpOM KOTHny 2, KOJIH j3 36i)KH0CTi M. H. PHAY Li~ e,,x,, BHilJllfBa€
L,~ ll x,,W< 00 [l,c.251] .
Hac.11iiJoK 2 . .JIKU/O E - npocmip muny 2, mo o yMo6a.x meopeMu 1 ouK0Hy
€mbCJ1 cniooiiJHmue1mJ1 (l), mo'INiwe A(b, X) ~ d, a o )'M06a.x meopeMu 2 811Ko11y
€nlbCJ1 3fl.ll (1), (2) i A(b, X) = cr(R).
HaraJJ,a€MO, w:o KOBapia1.1,ilt1-mfi onepaTop R Ha3HBa6TbC.Sl raycciBCbKHM, KOJIH
icuy€ raycciBCbKa B. B. G(R), .SIKa Ma€ R CBOIM KOBapiau.iltIHiM onepaTOpOM.
Hac.11iiJoK 3. flKU/ O E - npocmip Komuny 2, a 8 )'M08ax meope,ltu 2 011epa
mop R wycci8cbKUi1, mo cnpa8eo11118ui1 3ll.ll (1), (2) i A(b. X) = cr(R).
llJIR owrnKOBO po3nOL1iJie1mx H. B. B. X,, yMOBY (3) MO)KHa TpoxH nocJia61-1TH.
TioK11aJJ.eMo 'l'(t) = t I L 2(t ).
TeopeMa 3. H e.xaii R - K08apial(ii1Huii onepamop X 1 , N. o. 8 . X,, oiJ11aKooo
p03/100illeHi, 6UKO/ta/W )'MOO{l (4) i
(7)
Tooi eipHa meop eMa 2 i npu f'(b, X)= 0 3ll.ll ( 1), (2).
Teoper.1a 4. H exaii It . o. o. X,, 001-taK080 p o311ooi,1eHi i c11Memp11't11i, a E -
npocmip muny 2. a11.J1 cnpa8eO.II.U80Cmi 3ll.ll (1). (2) i pio1-t0cmi A (b' X) = cr(R)
oocmam1t11 6yob-11Ka i3 11acmymuix zpyn yMoe:
i) b~ .J.. npu n i oo i
(8)
ISSN ·0041 -6053. YKp . Mam. JKYP"- · /993 . m . ./5, N" 9
TIPO 3AKOH TIOBTOPHoro nor APH<I>MA JlJI51 3BA)KEHHX CYM ...
ii) 0/lJl OeJ!K020 1 < p < 00 OUKOHQ/ta .YMOOa (4),
b,; i, B11 I h; i npu fl ➔ oo,
b,; = O(B11 I (L(B,J) 1/(p-ll);
iii) OUKO/tafti )'MOOII (8), (9) ma icny€ maKe h > 0, u~
M exp (h II Xiii) < 00 •
1227
(9)
(10)
2. 3IlJI o R1. Hexall (S11) - noc11inoBtticTb H. B. B. B R 1, M s11 = 0, D Sn= I,
Z" = I.;' b,.f:,,·. Tiin 3TI.f1 B R1 3BHY:a!hm po3yMiHnh BHK01-1amu1 M. n. piBHOCTelt
lirn Z,, / X(B 11 ) = I,
"
Jim Z,JX(B 11 ) = - 1.
11
(11)
(12)
Tocp11.:,Kemu1 1. f111J1 cnpaoeo11uoocmi 3flJ/ (11), (12) oocmanmi y.Moou (3), (4)
(npu E = R1, xii= s,.).
ToepJ.-:lKCHHJI 2. llexail H. 0. 0. (s,,) OOltaKOOO po3nooi11e11i ii 8UKOH)'IOmbCJl
)'MOOU (4), (7) (npu E = R1, X,, = S,.). Tooi cnpaoeO/lU8Lti'i 3flJ/ (11), (12).
Toepiv1<e1111J1 3. Hexaii 11. e. o. (1;,.) 0011aKoeo po3noiJi11eni i c1mempu 1mi. f111J1
c npaoeo11u80cmi 3fl.n (11), (12) iJocmamHJt 6yob-RKa 3 zpyn )'MOB i)-iii) meopeMu
4 (npu E =R1, X,. = 1;").
3ay8ll)KeltHJl 1. YMOBH BHKOiramu1. 3TIJI B R I po3r JUIJla.JIHCb y 6aran,ox po-
6oTax (AHB., uanpHKJiaA, [6 - 8)). y [9] BHBY:a.JlHCb 3Ba)KeHi CYMH OJlHaKOBO po3-
noni11e1-m.x H.B. B. B R1 i np1-1 1 s p < 3 I 2 naKnana.nacb yMoBa, Tpox1-1 c11a6Kiwa
Hi.)!( (7).
JleMa 1. Hexaii /{A} - xapaKmepucmu•ma c/)yHKl.{iR MHO)l,.Uftu A . .JIKU{O O/lR
6)'0b-J!K020 't > 0
(13)
11
(14)
mo /IOC/liOOBHiCmb Z 11 300080/lbHJ!€ 3flJ/ (11), (12).
JleMa 2. B )'M080X moepO)Keft/iJ! 3 npu 8UKOH0ftfti 6yiJb-JlKOi' 3 zpyn )'MOB i)-iii)
r s-1," b2 r- 2 1 MG€MO M. II. "~m- 11 Li=l i '>i = .
JleMa 1 € HaCJii)lKOM KJiaCHY:HOro 3TI.TT KOJIMOropoBa [7, 10) . .fleMa 2 BHOJUIBa€ 3
pe3yJibTaTiB [11).
aoeeiJellltJl meepiJ)KellltJl 1. /lOCHTb BCl_-aHOBHTH ClliBBi)lHOWeHH.ll (13), (14).
Ma€MO
P(I b,, Sn I> 't 'lf(B,.)1 12) s MI s11 b,,12P h 2P 'lf(B")P s Cb,;P h 2P 'lf(B")P.
3 yMOBH (3) BttnJittBae, w.o rrptt JlOCHTb Be11ttKHX fl b,;r I 'lf(B")P s C I b; I B 11 x
x L(B 11 ) 1+6 12 . OcKiJibKH p51.Jl ' h; I B,, L(B11 ) 1+612 36ira€TbC.ll [7 , c . 339), TO yMoBa L,,
( 13) BHKOHY€TbCif. 3aCTOCYBaHH.ll uepiBHOCTi reJib)lepa )la€
M ( 1;; I {I b; s. I> 't 'lf(Bnl 12}) s (MI s;l2P)1 /p(P(I b;s-1>'t'lf(BJ1 12 ))11 q,
ISSN 004J-{J()53. YKp. ,.,am. xyp1<., 1993 , m. 45, N• 9
1228 I. K. MAUAK, A. M. ITJIP-IKO
.ue 1 / p +1 / q = 1. 3BiJJ.CH i 3 cnissi.unowenb (4), (13) smvmaa€ (14).
aoeeiJeHllJl me,epi))KellllJl 2 aIIa..TIOri'IHe:
I,,,P(lb,.~l>'t'lf(B,,)1 12) ~ I,,,P(ls,.l>C,11 11P) < 00 •
36i)KuicTh JJ.pyroro PMY rrptt yMoai (4) i OJJ.HaKoaitt po3rroidne1-1oc-ri se.TIH'IHH JJ.o6pe
BiJJ.OMa.
aooeiJeltltJl moepi))KeltltJl 3. CKOpHCTa€MOCb HaCTYITHOIO BiJJ.OMOIO HepiBHiCTIO
[1 2]:
Jim,, L~ a; e;fx(A,,) ~ 1 M. H.
/].JU{ A,.= I,~' a,2 ➔ 00 rrptt II ➔ oo . )],.TIH CHMeTpH'IHHX H. B. B. 3 Hei: OTpHMY€MO
Jim I; b; s,-lx(A,,@) ~ 1 M. H.,
n
(15)
51:Kll(O M. H.
)'- ' 11 2 ,-2 A,.(..~) = L,i b; ...,; ➔ oo nptt 11 ➔ oo.
Po36i)KI-IiCTh pHJJ.y ' b,7 S~ B yMOBaX TBepL\)KeHlrn: 3 BHIT.TllJBa€ 3 pimIOCTefi L,,,
[13, C. 53]
'Mmin(l, b;.s~) =' (P(b;.s;,<l)+b;,M(s;,I{b,7s;, ~1})) = 00 • ~n ~ n
HepimiiCTb (15) i JieMa 2 rrpHBO/].HTb L\O Ol\il tKH
lim z,. / x(B,,) ~ 1 M. H.
11
(16)
y TBepL\)KeHHi 3 np1-1nycKa€TbCH BHKOHaHOIO YMOBa (8), 3 51:KOI 3 ypaxyammHM
OJ~HaKOBOI p03ITOJJ.i.TieHOCTi Be.TIH'IHH s,, nerafiHO o,nep)Ky€MO (14). Bill.OMO [ 14] , lllO 3
(14) BHII.TIHBa€ uepiBHiCTb Jim,. Z,,/ X(B ,,)~ 1 M. H. 3Bil.{CH 3 ypaxynaHHJIM (16) BH
ITJlliBa€ pimiiCTb ( 11). Pim1iCTb ( 12) OJJ.ep)Ky€MO 3 ( 11). nepeXO/.{H'm JJ.O ae.TIH'IHH - S,·
HacTynue .llOIIOMi)KHe rnep.u)KeHHH MiCTlffbC}I 8 D.OBe.uemii JieMH 3 po6oTH [ 15].
JleMa 3. liexati (~,,) - 11oc,1ioo811icmb 8. 8. 8 R 1 i A,. - oil1cua 11oc.t1ioo8-
uicmb, A,, i oo npu n ➔ oo. 5/Ku{O 0/111 oe.11Ku.x C > 0. 13 > 1, 11 0 i 8Cix t e ( 1, 13)
P(maxl$ k Sn~ ~ fX(A,,)) ~ CL(A,,r11 npu ll > "o• mo M. H. Jim"(..,/ X(A,,) ~ 1.
3. J1.ouene11HJ1 TCOpCM 1 - 4.
aoaeiJe1t1tJl meopeMU ] . 3aq:>iKCY€MO L\OBiJlbHe 'IHCJIO O < 't < 1 / 2
IIOKJia/.{eMO
x'. /{llb X II < d cs )112 }X x" -x x'. · s' - , "bx' s" - " s' 11 = n n - 't 'I' 11 11• ,, - " - " • " - .L., 1 i i , ,, - ,J" - ,.-
)],JTH 0 1\iHKH seJIH'-Ui.HH II s;,11 HaM no-rpi6e11 11ecKi11<te1n10BHMipnm"i aua.nor BiHOMoi
eKCITOHeHL\ia.TibHOI nepi1mOCTi EepnwTefiHa [ 16]:
P(III,;'r;ji -MIII.;'r;jl ~u) ~ exp(-u2 /(2B+211VJ), (17)
( y) E 0, B > , "MIIYII~ V O MIIY,-11"' <_ Ile ; - H. B. B. B •, U. > _ L,1 , - , a > TaKe, w.o
~m!MIIY;ll2 V' .... 2 /2 npH m=2,3 .. .. .
KpiM T()f"O, KOpHCTYBaTHMeMOCb nacTymmMI-I 11epiBHOCTMMH /.{Jl}I II. B. B. ( Y, ) 13 £
[16, 17]:
ISSN 0041 -6053. Y.:p. Mm11. ;,;y/m .. 1993 , m . <15. N" 9
nPO 3AKOH TIOBTOPHOro nor APM<I>MA )lJ15l 38A)KEHl1X CYM . . . 1229
npH MY,= 0,
n
$ I,MIIY;ll2 . (19)
i=I
3 03Ha'-leHIUI x;; Ma€M0
L P(x;; ;t: O) ~ L (b,.hd)2P M II Xn 112P \jl(Bnrp < 00 •
n n
Ocramu1 ttepim1icTh npu yMoni (4) BcrattonJieHa npH il0ne.uenui rnep,rpKenHJJ 1. Toai,
3a JieMOIO EopeJIJJ - K a11reJu1i
sup II s;,' II = S < 00 M . H.
Tl
OT)Ke, AJIJI ,~one)lemrn. npaBOI uepiBI-IOCTi n (5) A0CHTh BCTaH0BITTH ou,iHKY
Jim II s:, II/ X(B,.) ~ d + r(b, X) .
11
3acTOCOBYIO'-lli ou,iHKH (19), (20), oaep)Ky€M0
~ P(M II s:,' II s~v2 ~ ss,~vz + ..fi.).
3ni,ncH i 3 (20) mmmma€
MIIS:,'11 ~ CB1,/2.
P(II s;, II - M II s;, II > u) ~ exp (-u2 / [28,,cF + 2utd\jl(B,,)112]).
(20)
(21)
(22)
ITillCTaBJIJII0'-111 CI0AR II = vd(l + 2-r) 1 l 2X(B,,) npu V e (l + l3), l3 = (2 / (l +
+ 2t)) 112, Ma€M0
(23)
TTpH <PiKCOBalfOMY -r AJl}f /~0CHTh BeJIHKHX fl
MIIS,. 11 :s; (l(b,X)+-r)x(B). (24)
OcKiJJl,KH IMIIS,,II-M II S:,11 I~ Mll s::11. T030qi110K (22)-(24)npHUOCl1Th
ne1mK11x II i v e ( L 13) MaE:M0
P (11s:, II ~ [L(b, X) + 2-r + vd(l + 2-r) 112] X(B,,)) ~ L(B,.r 1?. . (25)
OcKim,KH Ms:, + Ms:,' = Ms,, = 0, TO
MJ1s:, 11 ~ Mlls:.'11 ~ cs,Y2. (26)
3 ou,it-lOK ( 18), (25) i (26) BIIn.llHBa€, 1.1.(0 npu LIOCHTb neml.KHX /l I \I E (l, 13)
~ I ' [!(b . X) + 4-r + d( 1 + 2-r) 1121 X(B,, )) ~ L(B,,rY2.
ISSN 0041 -6053 . YKp ·''""' - )IC_)'/Jlt . , /993 , Ill . ./5, N' 9
1230 I. K. MAUAK. A. M. TTJIP-IKO
3aCTOCOBYIO'fH JieM)' 3, onep)Ky€MO ttepinniCTb
lim,.11 s;. - MS:. II I X(B,,) $ r(b, X) + 4't + d(l + 2't)112
M. H. A11e 't - LlOBiJll,He 'IHCJIO 3 i11Tepsany (0, 1 / 2), TOMY
lim,,IIS:.-Ms:.11 1x(B,,) $ d+ f'(b ,X) M .. II.
3sincH i 3 (26) BHfiJIHBa€ (21), TOOTO npaBa HepiBHlCTb B (5) BCTaHOBJieHa. Jlisa
ttepiBHiCTb B (5) BHfiJIHBa€ 3 JieMH <Dary (LlHB. (5)).
ao6eiJe1t1tJl llQCAiiJKy 1. 3 piBHOCTi r(b, X) = 0 Ta 3 (5) BHfiJJHBa€ (1), a TaKO)K
(npaxosyIO'fH, ll.\O B CKin,1emma1-1Mip110My npocropi TINE
(27)
a TOM)' A: r(b, QNX) = 0)
A(b, X) $ A(b, TINX) + A(b, QNX) $ sup,, D(TINX,,) + sup,, D(QNX,,) ..
Bnac11iAOK (6) npyrnA: nommoK np}IM)'€ AO O npw N ➔ 00 , a nepumA: (apaxoayIO'fH
(4)) /lJl.ll KO)KHOro N MenumA: Bill 00 • ToMy /lJl}I 6yAb-.HKOrO e > 0 MHO)KH!Ja
{S,,I X(B ,,), 11 ~ 1} noKpHBa€TbC.H M. H. cKin'feHnOio KiJJbKicno Ky111> paniyca e,
OT)Ke, nepeAKOMnaKnia.
aoaeiJellltJl meopeMU 2. Jlisa nepiBHiCTb TeopeMH € HaCJJi/lKOM 3fIJ1 B R 1
(Taepn)KeHH.a: 1) i 11eMH <DaTy. llnj{ nosene1rn.H npasoi nonepe/lHbO BCTaHOBHMO
piBHiCTb A(b, X) = cr(R) y CKiH'feHHOBHMipHOM)' BHfiaAKY (£=Rm).
BnacninoK niaoi nepiBHOCTi TeopeMH nocttTb noKaJaTH, ll.\O T = A(b, X) $ cr(R) ..
3 TBep/l)Kettn.a: 1 Ma€MO T < 00. y Haurnx YMOBax j3 3aKOHY O a6o 1 BHnmma€
(/lHB. (3-5]) icuyBaHH.H TaKOI BHMipHOI niLJ.MHO)KIHIH n c C n, ll.\O P(nJ = 1 i
lim,, II S,,(w)II/ X(B,,) = T np0 0) e nc OcKiJJbKH B R"' 06Me)KeHa MHO)KHHa nepeA
KOMnaKTHa, TO /lJUf KO)KHOro (J) e n c icHylOTb (1mnanKOBi) noc11inonHiCTb Ilk i TO
'fKa X, /l.Tl.ll .a:KHX
Ii,r1 IIS,,/w)ll / x(B,,k) = T i Ii,r1 IIS,,/w)/x(B,,k)-xll = 0,
OT)Ke II x II= T. 3rinno 3 mepn)KemurM 1 AJI.a: 6yn1>-.a:Koro f e £*, llfll = 1 Ma€MO
(M IJ(X1)12) 112 = lim,J'5'11)/ X(B,,) ~ limkf(S,,)I X(B,,k) ~f(x).
Toni
cr(R)= sup {(Mlf(X1)l2) 112: llfll= 1} ~ sup { IJ(t)I: llfll= l} ~ llxll= T.
TaKHM 'IHHOM, nn.a: 6yAh-.H:Koro N A (b, TINX)= cr(R N), ne RN - (cniJJbHHli)
KOBapiaL(iliHHlt onepaTop BeJIH'fHH TINX,,. 3 ocraIIHbOI piBHOCTi Ta (5) Ma€MO
A(b, X) $ A(b, ITNX) + A(b, QN>..') $ cr(RN) + supD(QNX,,) + r(b, QNX) ..
"
OcKiJJbK~ cr(RN) $ cr(R) (5) i BHaCJTillOK (27) 1(b, QNX) = r(b, X), TO, nepeXOl(j{'f[-{
no rpaHJIL(i no N Ta BHKOpHCTOBYIO'fH (6), 0/.(ep)Ky€MO npaay 1-1epim1iCTb TeopeMH.
Y npocropi rnny 2 M II s,, Ir $ C(E) cf-B,., as npocropi KOTHny 2
M II S,. 112 $ C(E) M II G(R) 112B,, [2].
OT)Ke, B o6ox mman.Kax f'(b, N) = 0 . Uj{ pi011iCTb 3a6e3ne'fy€ cnpaaenmmicTb
ISSN 0041-6053 .. YKp . ..Alf.Im. JKJfJ1t. , 1993. m . 45, N• 9
TTPO 3AKOH TTOBTOPI-IOro JI Or APl1Cl>MA J].Jl .sI 38A)KEHHX CYM . . . 1231
nacnin.KiB 2, 3.
,aoeeiJe1t1tJl meope.Mu 3 ai1a.nori'lue a.ooea.emuo Teope~m 2 (nepexia. Bill. yMOB
(3) ll.0 (7) MiCTHThCR B ll.OBea.em-ti rnepa.)KeHHR 2).
,aoeeiJel/ltJl meope.MU 4. 3ria.no 3 3TIJ1 B RI (TBepa.)KeHU,I 3), l).Jlj{ KO)KUOrO
fe E* M . H.
lim/(Sll)/x(Bll) = (Mlf(X1)1 2) 112. (28)
11
Toni 11.nj!_ 6yn.1>-,1Koro N M. 11.
(29)
(ll.HB. AOBe1.1.emrn TeopeMH 2).
Y npocTopi T«ny 2 1.1.Jlil CHMeTpH'IHHX I-1. B. B. (X,,)
lim 11s,,111x( I~ b,211x,1r) :,; C< 00 M. 1-1.
11
(30)
npH
r; b,2 IIX,lr i 00 , /1 ➔ 00 M. H. (31)
(JJ.HB. [ 15]; cnpaBeD,JIHBiCTb YMOBH (31) BCTaHOBJlelia y rnepJ\)KeHHi 3).
13 JieMH 2 Ta Ol..\iHKH (30) BHnn«Ba€ A(b, X) :,; Cd. Jla.ni noBTOp!OIOThCR apry
MeHTH AOBeD,emui. TeopeMH 2 3 BI-IKOpHCTaHlutM l\i€1 HepiBHOCTi Ta (29).
3ayeaJKe1mR 2. Hexalt K - OAHHH'!Ha KYJIR ri11h6epTOBOro npocTopy HR·
BiD.OMO [3], IQO KOJllf BHKOHat-Ii cniBBiAHOllleHH,I (2), (28) (30KpeMa, BOIUf BHKOHaHi B
yMooax TeopeM 2-4 npw r(b, X) = 0), TO M. u. inf {II S,JX(B ,.)- x II : x e K} ➔ 0
npH II ➔ oo , a KOJII-1 AOWlTKOBO I-/ R - necKiH'leH1-IOBI-IMip1rnfi npocTip , TO M. H.
C( {S,,/ X(B11)}) = K, He C( {x,.}) - MIIO)KHI-ta rpaHH'IHHX TO'IOK noc11iD.OBHOCTi {x,,}.
I. BaxaHZu H. II., Tap11e11aiJ3e B. H., '-106a11.1m C. A. B epoin11ocTHble pacnpeneneHH.ll a 6a11axoa1,1x
npocTpa11cTsax. -M.: llayKa, 1985. - 368 c.
2. Araujo A .. Cine E. The central limit theorem for real and Banach valued random variables. - New
York : Wiley, 1980. - 233 p.
3. Kuelbs J. A strong convergence theorem for Banach space valued random variables // Ann. Probab.
- 1976. -4, N° 5. - P. 744-771.
4. Goodman \I.. Kuelbs J., Zi1111 J. Some results on the LIL in Banach space with applications to
weighted empirical processes// Ibid. - 1981. -9, N° 5. - P. 713-752.
5. Acosta A., Kuelbs J., Ledoux M. An inequality for the law of the iterated logarithm // Leet. Notes
Math. - 1983. - 990. - P. 1-29.
6. Bingam N. H. Variants on the law of the iterated logarithm// Bull. London Math. Soc. - 1986. -
18, N° 5. - P. 433-467.
7. llempoo B. B. CyMMLI 11e3aBHCHMLIX CJ1y'laA11L1x BCJIH'!Hlt. - M.: HayKa, 1972. - 414 c.
8. MapmuKLlltH.elt A. 11 .. llempoo B. B. 0 11eo6xonm11,1x H /lOCTaTO'lllblX ycJJOBIUIX J.IJUI 3aKOIH1
IIOBTOp11oro 11orapHcp~rn // TcopHll nepOlllH0CTCA 11 cc npHMCHCHHll. - 1977. - 22. Bhlll. 1. - C.
IS-26.
9 . K11eco1J 0. 11. 3aKOH nonT op11o r ·o Jl01"'3J)Hl"j:>Ma /111>1 B38CJUClllll,IX cyMM IIC3aBHCHMblX O/.IHll3K0B0
pacnpellC JICHHblX cJJy'!af.1111.x BCJIH'IHH // TaM )KC . - 1986. - 31, Bblrl. 2. - C. 389- 391.
10. Tomkins R. J. Lindebcrg functions and the law of the iterated logarithm// Z. Wahrscheinlichkeits
theor. und verw. Geb. - 1983. - 65 , N" I. - P. 135-143.
11. Mai,aK 11. K. () cyMMHJJOBallHH HC:XIBHCHMI.X CJ1y'laA111,1x BCJIH'IHII MCTO/l0M Pucca // YKp. MaT.
)Kyp11. - 1992. - 44, N° 5. - C. 641---647 .
12. Weis.f M. On the law of the iterated logarithm// J. Math. Mech. - 1959. - 8, N° 2. - P. 121-132.
13. Kaxa1t )f(_ .. [l _ C J1y'!af.1111,1e <py11Ku11011aJ1bHble pl!Ltbl. - M.: M1-1p, 1973. - 304 c.
14. M apmUKGllH.elt A . lf. 06 0/ll!OCT0J)OIIIICM 3aK0IIC J"I0BT0pHoro J1or·ap11,pMa // TcopH>t BCJ)O>ITH0C-
1 Cf.I H e.e npHMCIIClllill. - 1985. - 30, Bhlfl. 4. - C. 694-705.
15. Maw1K lf . K. , 17,111.,Ko A. H. I-Ie.paue 11cTna Xmt'!Hlla 11 ac11MnTon1•1ecKoc noneneHHe cyMM
2, f." xn ll 6a11axon1,1x pe-U!CTKax // Y Kp. MaT. )Kypt1. - 1990. - 42, N° 5. - C. 639---644.
16. llw,e,111c lf. <!>., Caxa11e1<Ko A. lf. 3aMC'laHHll o 11epa11e.11cT11ax lJJJll nepo,i-r11ocTeA 6om,w1-1x
YKJI0IICl-!Htl // TcopHll nepOllT II0Cl Cl! Ii cc npHMCIIClllill. - 1985. - 30, llbln. l. - C. [27-131.
17. Caxmwm:o A. 11. () 11epane11cTnax JleBH - Ko11~10roposa lJJJll c J1yYaA111,1x llCJJHYHtt co 3HaYCHlill
MH n 6a11axo1JOM npoc-rpaHCTllC / / TaM JKC. - 1984. - 29, Bblrl. 4. - C. 793-799.
Onep)Kal!o 02. 04. 91
ISSN 0041-6053. YKp .. ,,am )l(_l'fl"·· 1993 . m . .J5, N" 9
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| id | umjimathkievua-article-5924 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:24:09Z |
| publishDate | 1993 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/96/cd96ae6e79be3dbb902097720f295f96.pdf |
| spelling | umjimathkievua-article-59242020-03-19T09:21:13Z On the law of the iterated logarithm for weighted sums of independent random variables in a Banach space Про закон повторного логарифма для зважених сум незалежних випадкових величин у банаховому просторі Matsak, I. K. Plichko, A. M. Мацак, І. К. Плічко, А. М. Assume that (X n) are independent random variables in a Banach space, (b n) is a sequence of real numbers, Sn= ∑ 1 n biXi, and Bn=∑ 1 n b i 2 . Under certain moment restrictions imposed on the variablesX n, the conditions for the growth of the sequence (bn) are established, which are sufficient for the almost sure boundedness and precompactness of the sequence (Sn/B n ln ln Bn)1/2). Нехай $(X_n)$ — незалежні випадкові величини в банаховому просторі, $(b_n)$ — послідовність дійсних чисел, $S_n = ∑_1^n b_i X_i,$ i $B_n = ∑_1^n b_i^2$. При моментних обмеженнях на величини $X_n$ знайдені умови на ріст послідовності $(b_n)$, достатні для обмеженості й передкомпактності послідовності $(S_n/B_n \ln \ln B_n)^{1/2})$ майже напевно. Institute of Mathematics, NAS of Ukraine 1993-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5924 Ukrains’kyi Matematychnyi Zhurnal; Vol. 45 No. 9 (1993); 1225–1231 Український математичний журнал; Том 45 № 9 (1993); 1225–1231 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/5924/8531 https://umj.imath.kiev.ua/index.php/umj/article/view/5924/8532 Copyright (c) 1993 Matsak I. K.; Plichko A. M. |
| spellingShingle | Matsak, I. K. Plichko, A. M. Мацак, І. К. Плічко, А. М. On the law of the iterated logarithm for weighted sums of independent random variables in a Banach space |
| title | On the law of the iterated logarithm for weighted sums of independent random variables in a Banach space |
| title_alt | Про закон повторного логарифма для зважених сум незалежних випадкових величин у банаховому просторі |
| title_full | On the law of the iterated logarithm for weighted sums of independent random variables in a Banach space |
| title_fullStr | On the law of the iterated logarithm for weighted sums of independent random variables in a Banach space |
| title_full_unstemmed | On the law of the iterated logarithm for weighted sums of independent random variables in a Banach space |
| title_short | On the law of the iterated logarithm for weighted sums of independent random variables in a Banach space |
| title_sort | on the law of the iterated logarithm for weighted sums of independent random variables in a banach space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5924 |
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