Bounded law of the iterated logarithm for multidimensional martingales normalized by matrices
We investigate a bounded law of the iterated logarithm for matrix-normalized weighted sums of martingale differences in $R^d$. We consider the normalization of matrices inverse to the covariance matrices of these sums by square roots. This result is used for the proof of the bounded law of the itera...
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| Date: | 2006 |
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| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509611113578496 |
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| author | Koval, V. A. Коваль, В. О. |
| author_facet | Koval, V. A. Коваль, В. О. |
| author_sort | Koval, V. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:56:18Z |
| description | We investigate a bounded law of the iterated logarithm for matrix-normalized weighted sums of martingale differences in $R^d$. We consider the normalization of matrices inverse to the covariance matrices of these sums by square roots. This result is used for the proof of the bounded law of the iterated logarithm for martingales with arbitrary matrix normalization. |
| first_indexed | 2026-03-24T02:43:51Z |
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K�O�R�O�T�K�I���P�O�V�I�D�O�M�L�E�N�N�Q
UDK 519.21
V.�O.�Koval\ (Ûytomyr. texnol. un-t)
OBMEÛENYJ ZAKON POVTORNOHO LOHARYFMA
DLQ BAHATOVYMIRNYX MARTYNHALIV,
NORMOVANYX MATRYCQMY
We investigate a bounded law of the iterated logarithm for matrix-normalized weighted sums of
martingale differences in R
d. We consider the normalization by the square roots of matrices inverse to
covariance matrices of these sums. We use this result to prove the bounded law of the iterated logarithm
for martingales with arbitrary matrix normalization.
DoslidΩu[t\sq obmeΩenyj zakon povtornoho loharyfma dlq matryçno normovanyx zvaΩenyx
sum martynhal-riznyc\ v R
d
. Rozhlqnuto normuvannq kvadratnymy korenqmy z matryc\, ober-
nenyx do kovariacijnyx matryc\ cyx sum. Danyj rezul\tat vykorystovu[t\sq dlq dovedennq
obmeΩenoho zakonu povtornoho loharyfma dlq martynhaliv z dovil\nymy matryçnymy normu-
vannqmy.
Zakon povtornoho loharyfma (ZPL) dlq matryçno normovanyx sum nezaleΩnyx
vypadkovyx vektoriv vstanovleno v roboti [1]. Dlq zvaΩenyx sum bahatovy-
mirnyx martynhal-riznyc\ ZPL u dewo inßij formi, niΩ v [1], dovedeno v [2].
Tut nakladalys\ Ωorstki obmeΩennq na vahovi koefici[nty. V3danij roboti
doslidΩu[t\sq obmeΩenyj ZPL dlq zvaΩenyx sum bahatovymirnyx martynhal-
riznyc\, normovanyx matrycqmy. Pry c\omu vykorystovu[t\sq pidxid, zapropo-
novanyj v [1].
Nexaj Rd — evklidiv prostir vektor-stovpciv i ( Zn , n ≥ 1 ) — martynhal-riz-
nycq v Rd vidnosno fil\traci] ( Fn , n ≥ 1 ) . Poklademo
Sn = D Zi i
i
n
=
∑
1
, Bn = D Di i
T
i
n
=
∑
1
, n ≥ 1,
de ( Di , i ≥ 1 ) — poslidovnist\ nevypadkovyx matryc\ rozmiru d × d ; T — znak
transponuvannq. ZauvaΩymo, wo koly E Z Zn n
T( ) = I , n ≥ 1, de I — odynyçna
matrycq, to Bn — kovariacijna matrycq vektora Sn . Budemo prypuskaty, wo
pry deqkomu n0 ≥ 1 matrycq Bn0
[ nevyrodΩenog. Todi pry vsix n ≥ n0 vyzna-
çeno oberneni matryci Bn
−1. Poznaçymo çerez Bn
−1 2/ kvadratnyj korin\ z Bn
−1.
Dlq3 evklidovo] normy vektora abo matryci vykorystovu[mo poznaçennq || ⋅ || , a
dlq vyznaçnyka matryci A — | A | . Poklademo
tn = ln ln /
+ +( )Bn
1 2, n ≥ 1, de ln+ x = ln max { x , e } , x ≥ 0.
Teorema 1. Prypustymo, wo E Zn
p < ∞ , n ≥ 1, pry deqkomu p > 2 i vyko-
nugt\sq nastupni umovy:
sup
n
n nZ
≥
−( )
1
2
1E F < ∞ m.n.; (1)
Bn → ∞ , n → ∞ ; (2)
pry deqkomu τ > p – 2
E B D Zn i i
p
i
n
−
=
∑ 1 2
1
/ = O t Bn n
τ ln( )( )−1 , n → ∞ . (3)
© V.3O.3KOVAL|, 2006
1006 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
OBMEÛENYJ ZAKON POVTORNOHO LOHARYFMA DLQ BAHATOVYMIRNYX … 1007
Todi
limsup /
n
n n nt B S
→∞
− −1 1 2 < ∞ m.n. (4)
Dovedennq. Z umovy (2) ta teoremy 3.6.3 [3] vyplyva[, wo poslidovnist\
(| Bn |, n ≥ 1) monotonno zrosta[ do neskinçennosti. Tomu zhidno z lemog33.3 [4]
znajdet\sq stroho zrostagça poslidovnist\ natural\nyx çysel ( nj , j ≥ 1 )
taka,3wo
2 8
1 1B B Bn n nj j j
≤ ≤
+ + , j ≥ 1. (5)
Budemo prypuskaty (ne obmeΩugçy zahal\nosti), wo n1 ≥ n0 . Zhidno z lemog
Borelq – Kantelli dlq dovedennq (4) potribno pokazaty, wo
P max /
n n n n n n
j j j
t B S M
< ≤
− −
=
∞
+
>
∑
1
1 1 2
1
< ∞ , (6)
de M — deqka skinçenna majΩe napevno vypadkova velyçyna.
Vraxovugçy pravu nerivnist\ v (5), dista[mo
tn j +1 ∼ tn j+1
, j → ∞ .
Tomu znajdet\sq j1 ≥ 1 take, wo pry vsix j ≥ j1
tn j +1 > 1 2
1
/( )
+
tn j
. (7)
Pry vsix m ≤ n ma[ misce nerivnist\ [1] (lema 1)
B Bm n
− −1 2 1 2/ / ≤ d B Bn m| | | |( )/ /1 2 .
Vraxovugçy danu nerivnist\, pravu nerivnist\ v (5) ta (7), otrymu[mo
P max /
n n n n n n
j j
t B S M
< ≤
− −
+
>
1
1 1 2 ≤
≤ P max max/ / /
n n n
n n
n n n
n n n
j j
j
j j
j j
B B B S M t
< ≤
−
< ≤
−
+
+
+
+
+
>
1
1
1
1
1 2 1 2 1 2
1 ≤
≤ P d B S M t
n n n n n n
j j
j j
2 2 1 2
1
1 1
1 2max //
< ≤
−
+ + +
> ( )
≤
≤ P max / /
1
1 2
1
1 1
4
≤ ≤
−
+ + +
> ( )
n n n n n
j
j j
B S M d t , j ≥ j1 . (8)
Nexaj An — bud\-qkyj rqdok matryci Bn
−1 2/ , n ≥ n0 . Dlq dovil\noho fiksova-
noho N ≥ n0 ta i = 1, 2, … , N poklademo
Ui = AN Di Zi ,
Yi = U I U t U I U ti i N i i N i≤ ( )( ) − ≤ ( )( )[ ]−1 2 1 2 1/ /E F ,
de I ( ⋅ ) — indykator vypadkovo] podi].
Rozhlqnemo odnovymirnyj martynhal Y n Ni
i
n
=
∑ ≤ ≤
1
1, vidnosno fil\traci]
( Fn , 1 ≤ n ≤ N ) . Oskil\ky ce martynhal z nul\ovym serednim ta | Yi | ≤ 1 / tN , i =
= 1, 2, … , N , to, vykorystovugçy vidpovidnu nerivnist\ Kolmohorova (dyv., na-
pryklad, [5, s. 209]), dlq bud\-qkoho α > 0 ta λ > 0 takoho, wo λ ≤ tN , ma[mo
P Emax
1 1
2
1
12
1
≤ ≤ =
−
=
∑ ∑> +
( ) +
n N i
i
n
N
i i
i
N
Y
t
Yλ λ αF ≤ 2e–αλ. (9)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
1008 V.3O.3KOVAL|
Zhidno z umovog (1) vypadkovi velyçyny
E Zi i
2
1F −( ), i ≥ 1, obmeΩeni skin-
çennog majΩe napevno vypadkovog velyçynog L . Todi
E Yi i
i
N
2
1
1
F −
=
( )∑ ≤ E A D ZN i i i
i
N
( )[ ]−
=
∑ 2
1
1
F ≤
A D ZN i i i
i
N
2 2
1
1
E F −
=
( )∑ ≤
≤ L B DN i
i
N
−
=
∑ 1 2 2
1
/ = L B D D BN i i
T
N
i
N
tr − −
=
∑
1 2 1 2
1
/ / = L d ,
de tr ( ⋅ ) — slid matryci.
Vraxovugçy danu nerivnist\, a takoΩ pokladagçy v (9) λ = tN , α = 2tN i
poznaçagçy M ′ = L d + 2, otrymu[mo
P max
1 1≤ ≤ =
∑ > ′
n N i
i
n
NY M t ≤ 2 2 2exp −( )tN . (10)
Poklademo
Yi = Ui – Yi , i = 1, 2, … , N .
Vraxovugçy, wo
E Ui iF −( )1 = 0 majΩe napevno, ma[mo
Yi = U I U t U I U ti i N i i N i> ( )( ) − > ( )( )[ ]−1 2 1 2 1/ /E F .
Oskil\ky poslidovnist\ Y n Ni
i
n
=
∑ ≤ ≤
1
1, utvorg[ martynhal z nul\ovym se-
rednim, to, vykorystovugçy nerivnist\ Kolmohorova ta vykonugçy neskladni
ocinky, dista[mo
P P Emax max
1 1 1 1
2 2
1≤ ≤ = ≤ ≤ =
−
=
∑ ∑ ∑> ′
≤ >
≤
n N i
i
n
N n N i
i
n
N N i
i
N
Y M t Y t t Y ≤
≤ t U I U t t t UN i i N
i
N
N N
p
i
p
i
N
−
=
− −
=
> ( )( )[ ] ≤ ( )∑ ∑2 2
1
2 2
1
1 2 2E E/ =
= 2 22 4
1
2 4 1 2
1
p
N
p
N i i
p
i
N
p
N
p
N i i
p
i
N
t A D Z t B D Z− −
=
− − −
=
∑ ∑≤E E / = : g ( N ) . (11)
Z nerivnostej (10) ta (11) vyplyva[
P Pmax max
1 1 1
2 2
≤ ≤ ≤ ≤ =
> ′
= > ′
∑n N N n N n N i
i
n
NA S M t U M t ≤
≤ P Pmax max
1 1 1 1≤ ≤ = ≤ ≤ =
∑ ∑> ′
+ > ′
n N i
i
n
N n N i
i
n
NY M t Y M t ≤
≤ 2 2 2exp −( )tN + g ( N ) . (12)
Poklademo M = 8d M ′ i poznaçymo çerez A A An n n
d( ) ( ) ( ), , ,1 2 … vidpovidni rqdky
matryci Bn
−1 2/ . Todi na pidstavi (12) otryma[mo
P max / /
1
1 2 4
≤ ≤
− ≥ ( )
n N N n NB S M d t ≤
≤ P
=
max ( )
11
2
≤ ≤
> ′
∑ n N N
k
n N
k
d
A S M t ≤ 2 2 2d tNexp −( ) + d g ( N ) .
Na osnovi dano] nerivnosti ta nerivnosti (8) robymo vysnovok, wo
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
P max /
n n n
n n n
j j
t B S M
− < ≤
− − >
1
1 1 2 ≤ 2 2 2d tnj
exp −( ) + d g ( nj ) . (13)
Vykorystovugçy livu nerivnist\ v (6), pry vsix j ≥ 1 dista[mo
B Bn n
d j d
j
≥ ( ) ⋅| |
1
2 2
1
/
/ / . (14)
Zvidsy vyplyva[, wo
exp −( )
=
∞
∑ 2 2
1
tn
j
j
< ∞ . (15)
Vykorystovugçy (3), ma[mo
g n t B D Zj
j
p
n
p
n i i
p
i
n
j
j j
j
( ) =
=
∞
− − −
==
∞
∑ ∑∑
1
2 4 1 2
11
2 E / ≤
≤ C t Bp
n
p
n
j
j j
2 2 4
1
− − −
=
∞ ( )∑ τ ln < ∞ (16)
na pidstavi nerivnosti (14) ta umovy p – τ < 2 ( C — skinçenna stala z umovy
(3)). Zi spivvidnoßen\ (13), (15) ta (16) vyplyva[ (6).
Teoremu 1 dovedeno.
Nexaj ( An , n ≥ 1 ) — dovil\na poslidovnist\ matryc\ rozmiru k × d ( An ≠ 0,
n ≥ 1 ) . Poznaçymo çerez || ⋅ ||2 spektral\nu normu matryci.
Teorema 2. Qkwo vykonugt\sq umovy (1) – (3), to
limsup /
n
n n
n n n
T
n
A S
A B A t→∞
2
1 2 < ∞ m.n.
Dovedennq vyplyva[ z teoremy 1 z uraxuvannqm nerivnosti
A S A B B Sn n n n n n≤ −1 2
2
1 2/ /
ta totoΩnosti A B A B An n n n n
T1 2
2 2
1 2/ /
= .
ZauvaΩymo, wo koly E Z Zn n
T( ) = I , n ≥ 1, to A B An n n
T — kovariacijna matry-
cq vektora A Sn n.
1. Koval V. A new law of the iterated logarithm in R
d
with application to matrix-normalized sums of
random vectors // J. Theor. Probab. – 2002. – 15, #31. – P. 249 – 257.
2. Lai T. L. Some almost sure convergence properties of weighted sums of martingale difference
sequences // Almost Everywhere Convergence, II. – Boston: Acad. Press, 1991. – P. 179 – 190.
3. Lankaster%P. Teoryq matryc: Per. s anhl. – M.: Nauka, 1978. – 2803s.
4. Wittmann R. A general law of iterated logarithm // Z. Wahrscheinlichkeitstheor. und verw. Geb. –
1985. – 68, #34. – S. 521 – 543.
5. Duflo M. Random iterative models. – Berlin: Springer, 1997. – 385 p.
OderΩano 13.01.2005
|
| id | umjimathkievua-article-3510 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:43:51Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a4/3fc065cbb66845493c283b6b7fab09a4.pdf |
| spelling | umjimathkievua-article-35102020-03-18T19:56:18Z Bounded law of the iterated logarithm for multidimensional martingales normalized by matrices Обмежений закон повторного логарифма для багатовимірних мартингалів, нормованих матрицями Koval, V. A. Коваль, В. О. We investigate a bounded law of the iterated logarithm for matrix-normalized weighted sums of martingale differences in $R^d$. We consider the normalization of matrices inverse to the covariance matrices of these sums by square roots. This result is used for the proof of the bounded law of the iterated logarithm for martingales with arbitrary matrix normalization. Досліджується обмежений закон повторного логарифма для матрично нормованих зважених сум мартингал-різниць в $R^d$. Розглянуто нормування квадратними коренями з матриць, обернених до коваріаційних матриць цих сум. Даний результат використовується для доведення обмеженого закону повторного логарифма для мартингалів з довільними матричними нормуваннями. Institute of Mathematics, NAS of Ukraine 2006-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3510 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 7 (2006); 1006–1008 Український математичний журнал; Том 58 № 7 (2006); 1006–1008 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3510/3757 https://umj.imath.kiev.ua/index.php/umj/article/view/3510/3758 Copyright (c) 2006 Koval V. A. |
| spellingShingle | Koval, V. A. Коваль, В. О. Bounded law of the iterated logarithm for multidimensional martingales normalized by matrices |
| title | Bounded law of the iterated logarithm for multidimensional martingales normalized by matrices |
| title_alt | Обмежений закон повторного логарифма для багатовимірних мартингалів, нормованих матрицями |
| title_full | Bounded law of the iterated logarithm for multidimensional martingales normalized by matrices |
| title_fullStr | Bounded law of the iterated logarithm for multidimensional martingales normalized by matrices |
| title_full_unstemmed | Bounded law of the iterated logarithm for multidimensional martingales normalized by matrices |
| title_short | Bounded law of the iterated logarithm for multidimensional martingales normalized by matrices |
| title_sort | bounded law of the iterated logarithm for multidimensional martingales normalized by matrices |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3510 |
| work_keys_str_mv | AT kovalva boundedlawoftheiteratedlogarithmformultidimensionalmartingalesnormalizedbymatrices AT kovalʹvo boundedlawoftheiteratedlogarithmformultidimensionalmartingalesnormalizedbymatrices AT kovalva obmeženijzakonpovtornogologarifmadlâbagatovimírnihmartingalívnormovanihmatricâmi AT kovalʹvo obmeženijzakonpovtornogologarifmadlâbagatovimírnihmartingalívnormovanihmatricâmi |