On types of distributions of sums of one class of random power series with independent identically distributed coefficients
By using the method of characteristic functions, we obtain sufficient conditions for the singularity of a random variable. $$ξ = \sum_{k=1}^{∞} 2^{−k}ξ_k,$$ where $ξ_k$ are independent identically distributed random variables taking values $x_0, x_1$, and $x_2$ $(x_0 < x_1 < x_2)$ with...
Збережено в:
| Дата: | 1999 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4591 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510738406178816 |
|---|---|
| author | Litvinyuk, A. A. Литвинюк, А. А. |
| author_facet | Litvinyuk, A. A. Литвинюк, А. А. |
| author_sort | Litvinyuk, A. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:09:14Z |
| description | By using the method of characteristic functions, we obtain sufficient conditions for the singularity of a random variable.
$$ξ = \sum_{k=1}^{∞} 2^{−k}ξ_k,$$
where $ξ_k$ are independent identically distributed random variables taking values $x_0, x_1$, and $x_2$ $(x_0 < x_1 < x_2)$ with probabilities $p_0, p_1$ and $p_2$, respectively, such that $p_i ≥ 0,\; p_0 + p_1 + p_2 = 1$ and $2(x_1 − x_0)/(x_2−x_0)$ is a rational number. |
| first_indexed | 2026-03-24T03:01:46Z |
| format | Article |
| fulltext |
Y]~K 519. 21
A. A. JIHTmmmK (I-Iat~ hen. yH-T, KHin)
IIPO THHH PO3HO~IJHB CYM O]],HOFO KJIACY
BHIIA~KOBHX CTEIIEHEBHX PH]IIB 3 H E 3 A J I E ~
O]],HAKOBO PO3IIO~[HJIEHHMH KOE|
By using the method of characteristic functions, we obtain sufficient conditions of the singularity of a
random variable
-- " 2" t ,
k=l
where ~t are independent equally distributed random variables taking the values x o, x l, and x 2 (x 0 <
<x I < x2) with the probabilities Po, Pt, and P2, respectively, such that pi>0, Po + Pl + P2 = 1, and
2(x I - x o ) l ( x 2 - :Co) is a rational number.
MeTogOM xapazTepucTHqHUX qbyHzuiR o~ep)KaH0 ]~OCTaTHi yMOBH CHHryJ]apHOCTi BHrla/IKOBOi Be-
JIHqHHH
= ~2-~[~,
k=l
Re ~ k - He3ZJIe~KHi ORHaKOBO poano/tD~eHi BHHa~KOBi BeJIH'-IHHK gKi Ha6yBalOTb 3HaqCHb X0, Xl, X 2
(X.o< xt < x~) a i~oaipnocTaMU Po ,P~ ,P~ ni~noBiaHo, p i~ 0, P0 + P~ + P2 = 1, npu ttbo~y
2(x I - Xo) l ( x ~ - x0) r pauioHa~bHmd qHCJmM.
P o 3 r a ~ a c T ~ C S BI4naIIKOBa BCJIHqIa_Ha (B.B.):
= ~ 2 - k ~ k , (1)
kffil
l lc ~k ~ Hc3a~e~v~Hi o ~ a K o s o po3no/Ii~IeHi B. B., ~lKi Ha~yBalO'l'b 3Ha,4eHb x 0, x 1, x 2
( x o < x I < x 2 ) a i~osipHoc'r~Mx~ PO, P l , P 2 si~noBi~HO, p i > O, PO + P l + P2 = 1.
OCHOBHOIO 3a~aqcIo e ~ocai~Y~CHHS cTpyzTypH poano~iJ1y [I] B.B. ~. 3 TcOpCMI4
~,3KeCr - BiHT~Cpa [I] BHn:mBae qHCTOTa poano~iay ~ 3a TeOpeMOIO I'[. J'Icsi [1]
MaTHMe ]~HCKpCTHPI~ p o a n o ~ i a TOni i TiJ'IbKH TO~i, KOJIH m a x {Po, P l , P 2 } = 1. B H -
IIa~OK X i ..~ i p o 3 r J L ~ a B c s cno~aTKy S po6oTi [2], ~C 6yJm ovpx~aHi B TOpMiHax BH-
IIa]IKOBHX MaTpHI.U~ ]~C~lKi IIOCTaTHi yMOBH CHHry.vt$1pHOCTi ~, azi rrpaKTH~IHO ncpc-
BipHTH sa~zo, a n0TiM S pO6OTi [3 ] 3a~aqa 6yJm nonHic'no po3B'~3aHa ~4CTO]ZOM
xapaKTCpHCTHqHHX cl3yHKRi/~l. BHnaRoK, ZOJm X i ~ ~OBiJIbHi ~i~cHi ~HcJm, IIpHBO-
]~wr~ ~O ~aWaHX yCKaa~HCU~. Y ~auiR pO6OTi npono~y~oT~ca pc~3yJlbTaTH /~OCJli-
IDK~I~, KOSIH 2 (X I -- X 0 ) �9 paI~OHaOIbm4M ~I4C~IOI~, rlo~aHHI~ y BI4r~I~i He~IcOpOT-
x2 - xo
uoro ~ao6y re~a, np~i ubo~y BHK0pHCTOByeTbC.q M~'rO/IHKa, 3anponoHoBaHa B[3],
Oqe~lllHO, III0 B. a.
k=l k f l k=l k=l
A. A. ~IHTBHHIOK, 1999
128 I$SN 0041-6053. YKp. ~aam. ~gypn., 1999, m. 51, N ~ I
rIPO THHH PO3HO/2IJIIB CYM O/~IOl'O ILFIACY ... 129
0 - X 0 I :l otx',, I I
Mae TOI~ )Ke THrl poano~ai~y, mo I1 B.B. ~. IIpH Ubo~y ~" e B.n. Bnr~a/Iy (1), npa x 0 =
= 0. O/~a~OBa~ T~n poano~ai~y MZaOT~ i ~ n. ~ ' Ta ~", ae
X2 -- XO k=l ~. 2 -- XO) k=l
~ ' 0 m 2
a
Pi Po Pi P2
OTXe, { ' i ~ " Mazo~ O~HaKoBi ~ n H poano~iay (,, ~sa poarlo~ian F 1 i F 2 B R 1
Ha3tlBaIOT~r poano~iaaMn o ~ o r o tt TOrO 3K T~ny, sKmo F2(x) = F 1 ( a x + ~),
a > 0. YIapaMeTp a HaaHBaerbca MacmTa6HHM MHOaCaHKOM, a napaMeTp [3
t~eHTpy1oqo~ CTa~Xo~o (a6o napaMeTp poaMimeHHX)") [6, C. 163].
J-IeMa 1. Xaparmepucmu~na qbynrtci~ (x.~.) posnobiny e .e . ~ 3anucyembc.~
y euenabi
f~ = f ( t ) = I-Iq~tc(t), (2)
k=l
be
itm i2t
OPt(t) = Po + Pl ea2k + P2e 2k , (3)
i bah ro~rnozo k r N 3aboooabt~e qbyHgt(ionaAbl~e pienann.a
f ( t ) = f r , (4)
be
r = P0 + Pt cos + +
+ i ( P l S i n ( ~ 2 n ) +P2sin(21-nt)) . (5)
fl[ooebenn~ 3a oaHaqeHH.qM X. C~. i BJ'IaCTHB0CT~MH ldaTeMaTI4qHOrO CIIO~iBaH-
H~ Mae'MO
f ( t ) = Me it~ = M e it~t2-t{t = (6)
= = M e ' 2 ' M e i 2 ~ = r ~ f ~ , (7)
~e ~t i ~ - - H ~ r a ~ i ~ - o ~ a x o n o p o a n o ~ i a e ~ i . 3 ( 6 ) B ~ n m m a r
Ocxia~ml pinaic 'n, (7) nHXOm/eT~a ~ ;~onia~Horo t, TO, aaMimmmn t Ha
t12, MaTm4eMo
ISSN 004 J -6053. Yr, p. 7~wn. ~'yps.. 1999, m. 51,1~ 1
130 A.A..rIHTBHHIOK
I'liaca'a~HBmH ocramle cniBni~lomcmla B (7), o~cpw,_nMO
t t t
OCTalIH~ piBHica~ cnpaBc~al4na ~aa KOXHOrO t r R. ToMy aaMiea t Ha" t/2 i
IIi~CTaliOBKa BHpaay f(t/2) y (7) ripttBC,llC go
t t t t
a Hacry~ i aHa~orbmi KpOKH ~ ~O qbymc~ioua~HOi piBHOCTi (4).
$1eMy I ~o~e~eHo.
B p ~ I m a ~CTOTy poanoniay ~, BHB~O noBeniHKy MOny~ x.cla B. S. ~ Ha
He, CKiI-F-ICHHOCTi, TO6TO oI~iHHMO Be3IH~H,IHy
L~ = , lim suplf (,) I
I r i s "
Bi~oMo [4, 5], mo KO.rm f ( t ) - - X. qb. ,-mc'ro: 1) ~.xc~pe'moro poano~i.ay, TO L~ =
1; 2) a6co.moT~o Hcnepepe~oro, L~ = 0; 3) '-reCTO cm-wy.napHoro po~no~i.ny, TO L~
~oacc Ha6y~aT~ ecix aHaqCHS [0; 1]. To~y yMona 0 < L~ < 1 pieHoc~.a~Ha qHCTiI~
CHHI~JLqpHOCTi poano~iay B. B. ~.
J'IeMa 2..,r vucao m - - napne, mo L~ > If(a~)l, a r u ~ o m - - nenapne,
mo & > I f ( a ~ ) l l ~ - 2pll.
,~ooei~enns~. Poar.rtane~o H0c.ni~oBHiC'I~ t k= 2~ax. OHCBH~HO, mo tk--> ~,
k ---> ~ . BpaxoB3nO~I (5) ~sta k > n, ~acMo
I q~(~n/2 = ' P o +PlCOS(2k-nm~)+P2COS(2~+l-na~) +
+ i(P l sin(2k-nm~) + p~sin(2~+~-"a~))[ ~ =
= [Po + Pl c ~ + P212 =
f l npH k > n i napHoMy m,
( 1 - 2 p [ ) 2 rrpH k=n iHenapHo~y m.
Bpaxosy~'~H Te, ~O pinHiCa~ (4) Mae ~ictIr ~a~ KOaC.HOrO k r N i 6y~tr~-aKoro t,
aOKpeMa i ~ t = 2 ~a~, nic~a ni~erarxonr.H ~aTH~CMO
Lf (2kax) l = L:(, , ,Ol = I : ( a ~ ) l l ~ o ( " , O ! =
-- ~] f ( a~ ) [ ~:,,IRO m rlapi-ie.,
IJf(a~)[l l-2PI~ aKmO m HenapH~
OcKi.m,lClt ~X
L~ ~ limlf(2ka~)l,
k-..)oo . . . .
a f(2kax) Heaaae~wrbBia k, To cnpa~aam~r~pa~emla ae~m.
JIcMy 2 ~O~r
ISSN 0041-6053. Yxp. ,llam. ,vcypn., 1999 m. 51. !r e I
IIPO THIII4 PO31"IOB//IIB CYM O]IHOFO K.rlACY ... 131
Hae.aiOog. .,'truto max {Po, P l ,P2 }< 1 i f (arQ ~ O, m o ~ ~ar gucmo cuney-
nnpnutl poanoOia npu y:uoei, u4o m ~ napne a6o pl # 1 / 2.
TeopeMa. Atrugo m--,~uc.ao napne a6o Pl # 1/2 i O.ax 6yOb-,~rozo k ~ {1, 2,
3 . . . . . k o - 1 }, Oe k o - - t-tati,~tenute namypa.abue guc.ao mare , u4o 2 ~~ > M =
=max {m,a},
Po + Pl c~ + P2 c~ ~ 0 (8)
a6o
Pl sin(2qgm) + P2 sin(2-kna) # 0,
mo Lg >_ [f(ga)[ > 0, a om.,we, ~ ;*tae cuney.axpnu~posnoai.a.
] loeeOenn~ Ilona~o [f(na)[ y Barnxai
/re
q~ = P0 + Pl c~ + P2 c~ +
+ i (p 1 sin(2-kmn) + P2sin(21-kan)).
]IoBe~eMo nacryrmy nepiBnicr~:
(an'~
q'txrJ o (9)
2-krcm<g i 0 < 21 ocKiY~BIGt 0 <
a ov~xe, Pl sin(2-kmg) + P2 sin(21-ka~) > 0.
9IK Bi~oMo, a6conmrna a6i~xUOCTb necKiH~eHHoro ~ao6yvKy (9)pi~Hoc~r, Ha
a6comoa~ai~ a6iwa~oeri pa/xy
~(~0(2-kag) -11,
k=ko
ro6ro piBaocHsmna 36iwaaocri pzRy
~1~(2 - ' a~ ) -1 [ . (10)
tc=~
3o6paam~o [9(2-kax)[ 2 y sara .~i
l~~ = ( P o - 1 + PlCOS2-kmn + P2COS21-ka•) 2 +
+ (p! sin 2-kmn + P2 sin 21-kax) 2 =
= (P0 - 1)2 + pI2 + p~ + 2(P0 - 1)Pl c~ 2-km~) +
ISSN 0041-6053. Yr, p. ~mm, ,~ypn.. 1999, m. 51, IV ~ 1
132 A.A. JIHTBHHIOK
+ 2(Po - 1 )P2 cos (2 l - k a ~ ) + 2pip2 cos 2-k(m~ - 2a~) =
= ( P o - 1 + p l + p 2 ) 2 - 2 ( p 0 - 1 ) P l -
- 2 ( P o - 1 )P2 - 2PIP2 + 2('Po - 1 )Pl cos (2-kmTr) +
+ 2 ( P o - 1)P2COS(21-kax) + 2PlP2COS(2-tTC(m - 2 a ) ) =
= 4((Po - 1)pl sin2(2 - k - l r a g ) + (Po - 1)P2 s in~(2-kag) +
+ piP2 sin 2 ( 2 -~ -1~ (m - 2a))) .
B p a x o s ~ o ~ Te, mO sin: A sm ~ k - , a/~OBiJl~Ha CKiHqeHHa KiJrbKiCTb ~neHis
pJt]~y He BII~IBar Ha floro 36i~caiCTb, i TC, mo
�9 rL4 r~4
0 < s m ~ - < ~k- npa k > k ( A ) ,
3a oaaaKOm IIopiBHZIIH~I po6a~O BHCHOBOK npo 36b~icT~ pa~ay (I0)i ~aO6yTKy (9)�9
3(8)nrmnnBar npn k e {I ..... k o- I}. ToNy f(a~)~O, mo
BHaCJIi]~OK JIeMI4 2 piBnOCHnbnO L~ > 0 i crmrynspnocTi pozno/~iny B. B. ~.
T e o p e ~ ]IOBe]IeHO.
1. Typ6un A, 0., llpat~eaumtati H. B. ~pazTanbntae Mno~ecT~, qbynzt~H, pacnpejleJ1eHHa.- K,eB:
Hayz. ~ty~xa, 1992. - 208 e.
2. Bunnuutun 51. 0., MopoKa B.A. Hpo Tun dpynKRii pozno~iJiy nnnajixosoro eTeneneeoro pn~y//
ACHMHTOTHqHHIt anani3 BrlnallKOBHX esoJnottitl. -Kais III-T MaTeMaTuKtl HAH YKpairm, 1994. -
C. 65 - 73,
3. 17pa~bosumu~ M. B. Po3no/finH cyM BHIla/IKOBHX creneHemtx ps~tin//]1orion. HAH YKpaiHH. --
1996.-Ng5.-C. 32-37.
4. Pratsevytyi N. V. Fractal, superfractal and anomalously fractal distribution of random variables
with independent n-adic digits, an infinite set of which is fixed / A. V. Skorokhod, Yu. V. Borovs-
kikh (Eds). Exploring stochastic laws. Festschrift in Honor of 70-th Birthday of Acad.
V. S. Korolyulc- VSP, 1995.-P. 409-416.
5. ,]'/ygaq E. XapaKTepUcTHqeCKtle dpyllKtl.Hrl. -M.: Hayxa, 1979. -424 c.
6. @ennep B. B~/leHue n Teopmo eepoaTHOCTetl rlee npt~.aO~eHH,q. - M.: Mrlp, 1984.- T. 2. -
738 c.
O~tep.,,KaHo 04.08.97
I$5N 0041.6053. Y~p. uam. ~'ypn.. 1999 m. 51o N e I
|
| id | umjimathkievua-article-4591 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:01:46Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b4/10124c5b41bbd00fa3a36b61a0edf8b4.pdf |
| spelling | umjimathkievua-article-45912020-03-18T21:09:14Z On types of distributions of sums of one class of random power series with independent identically distributed coefficients Про типи розподіяив сум одного класу випадкових степеневих рядів з незалежними однаково розподіленими коефіцієнтами Litvinyuk, A. A. Литвинюк, А. А. By using the method of characteristic functions, we obtain sufficient conditions for the singularity of a random variable. $$ξ = \sum_{k=1}^{∞} 2^{−k}ξ_k,$$ where $ξ_k$ are independent identically distributed random variables taking values $x_0, x_1$, and $x_2$ $(x_0 < x_1 < x_2)$ with probabilities $p_0, p_1$ and $p_2$, respectively, such that $p_i ≥ 0,\; p_0 + p_1 + p_2 = 1$ and $2(x_1 − x_0)/(x_2−x_0)$ is a rational number. Методом характеристичних функцій одержано достатні умови сингулярності випадкової величини $$ξ = \sum_{k=1}^{∞} 2^{−k}ξ_k,$$ де $ξ_k$, - незалежні однаково розподілені випадкові величини, які набувають значень $x_0, x_1$ та $x_2$ $(x_0 < x_1 < x_2)$ з імовірностями $p_0, p_1$, та $p_2$, відповідно, $p_i ≥ 0,\; p_0 + p_1 + p_2 = 1$, при цьому $2(x_1 − x_0)/(x_2−x_0)$ є раціональним числом. Institute of Mathematics, NAS of Ukraine 1999-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4591 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 1 (1999); 128–132 Український математичний журнал; Том 51 № 1 (1999); 128–132 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4591/5886 https://umj.imath.kiev.ua/index.php/umj/article/view/4591/5887 Copyright (c) 1999 Litvinyuk A. A. |
| spellingShingle | Litvinyuk, A. A. Литвинюк, А. А. On types of distributions of sums of one class of random power series with independent identically distributed coefficients |
| title | On types of distributions of sums of one class of random power series with independent identically distributed coefficients |
| title_alt | Про типи розподіяив сум одного класу випадкових степеневих рядів з незалежними однаково розподіленими коефіцієнтами |
| title_full | On types of distributions of sums of one class of random power series with independent identically distributed coefficients |
| title_fullStr | On types of distributions of sums of one class of random power series with independent identically distributed coefficients |
| title_full_unstemmed | On types of distributions of sums of one class of random power series with independent identically distributed coefficients |
| title_short | On types of distributions of sums of one class of random power series with independent identically distributed coefficients |
| title_sort | on types of distributions of sums of one class of random power series with independent identically distributed coefficients |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4591 |
| work_keys_str_mv | AT litvinyukaa ontypesofdistributionsofsumsofoneclassofrandompowerserieswithindependentidenticallydistributedcoefficients AT litvinûkaa ontypesofdistributionsofsumsofoneclassofrandompowerserieswithindependentidenticallydistributedcoefficients AT litvinyukaa protipirozpodíâivsumodnogoklasuvipadkovihstepenevihrâdívznezaležnimiodnakovorozpodílenimikoefícíêntami AT litvinûkaa protipirozpodíâivsumodnogoklasuvipadkovihstepenevihrâdívznezaležnimiodnakovorozpodílenimikoefícíêntami |