On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. II
We establish conditions under which wavelet expansions of random processes from Orlicz spaces of random variables converge uniformly with probability one on a bounded interval.
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| Date: | 2008 |
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| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2008
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509244643606528 |
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| author | Kozachenko, Yu. V. Perestyuk, M. M. Козаченко, Ю. В. Перестюк, М. М. |
| author_facet | Kozachenko, Yu. V. Perestyuk, M. M. Козаченко, Ю. В. Перестюк, М. М. |
| author_sort | Kozachenko, Yu. V. |
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| datestamp_date | 2020-03-18T19:48:06Z |
| description | We establish conditions under which wavelet expansions of random processes from Orlicz spaces of random variables converge uniformly with probability one on a bounded interval. |
| first_indexed | 2026-03-24T02:38:02Z |
| format | Article |
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UDK 519.21
G. V. Kozaçenko, M. M. Perestgk (Ky]v. nac. un-t im. T. Íevçenka)
PRO RIVNOMIRNU ZBIÛNIST|
VEJVLET-ROZKLADIV VYPADKOVYX PROCESIV
IZ PROSTORIV ORLIÇA VYPADKOVYX VELYÇYN. II
Conditions are established under which wavelet expansions of random processes from the Orlicz spaces
of random variables converge uniformly with probability one on a bounded interval.
Najden¥ uslovyq, pry kotor¥x vejvlet-razloΩenyq sluçajn¥x processov yz prostranstv Or-
lyça sluçajn¥x velyçyn sxodqtsq ravnomerno s veroqtnost\g edynyca na ohranyçennom ynter-
vale.
Vstup..Cq stattq [ prodovΩennqm roboty [1], tomu numeracig punktiv i formul
v nij prodovΩeno.
U p’qtomu punkti rezul\taty perßo] çastyny pro povedinku vypadkovyx pro-
cesiv iz prostoriv Orliça vypadkovyx velyçyn zastosovano do procesiv iz prosto-
riv Orliça eksponencial\noho typu.
V ßostomu punkti dovedeno teoremu pro rivnomirnu zbiΩnist\ vejvlet-roz-
kladiv.
U s\omomu punkti navedeno umovy, za qkyx vejvlet-rozklady vypadkovyx
procesiv iz prostoriv Orliça vypadkovyx velyçyn zbihagt\sq rivnomirno na ob-
meΩenomu intervali z imovirnistg odynycq. Okremo rozhlqnuto vejvlet-roz-
klady vypadkovyx procesiv z L p( )Ω , p ≥ 1, ta prostoriv Orliça eksponencial\-
noho typu.
5. Vypadkovi procesy iz prostoriv Orliça eksponencial\noho typu.
Oznaçennq 5.1 [2]. Prostir Orliça LU ( )Ω nazyva[t\sq prostorom Orli-
ça eksponencial\noho typu, qkwo vin porodΩu[t\sq funkci[g vyhlqdu U x( ) =
= exp ( )ψ x{ } – 1, de ψ( )x — deqka C-funkciq. Zhidno z [2], prostory Orliça
eksponencial\noho typu poznaçatymemo Expψ ( )Ω , a normu v c\omu prostori
— ⋅ Eψ
.
Teorema 5.1 [2]. Nexaj ψ = ψ( )x{ , x R∈ } — dovil\na C-funkciq ta ξ ∈
∈ Expψ ( )Ω . Todi dlq vsix x > 0 ma[ misce nerivnist\
P ξ ≥{ }x ≤ 2exp –ψ
ξ
ψ
x
E
.
Qkwo dlq deqkyx C > 0, D > 0 ta vsix x > 0 vykonu[t\sq
P ξ >{ }x ≤ C x
D
exp –ψ
{ },
to ξ ∈ Expψ ( )Ω ta ξ
ψE ≤ (1 + C)D.
Teorema 5.2. Nexaj X = X t( ){ , t T∈ } , T = a b,[ ], – ∞ < a < b < + ∞, — sepa-
rabel\nyj vypadkovyj proces iz prostoru Expψ ( )Ω . Nexaj isnu[ taka funkciq
σ = σ( )h{ , 0 ≤ h ≤ b a– }, wo σ( )h [ neperervnog, monotonno zrosta[, σ( )0 =
= 0, ta
sup ( ) – ( )
–
, ,
t s h
t s a b
EX t X s
≤
∈[ ]
ψ
≤ σ( )h .
© G. V. KOZAÇENKO, M. M. PERESTGK, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6 759
760 G. V. KOZAÇENKO, M. M. PERESTGK
Nexaj dlq deqkoho 0 < ε < b – a zbiha[t\sq intehral
ψ
σ
ε
(– )
(– )ln
–
( )
1
0
12
1∫ +
b a
u
du < ∞.
Todi vypadkova velyçyna sup ( )
, ,t s a b
X t
∈[ ]
naleΩyt\ prostoru Expψ ( )Ω i dlq
bud\-qko] toçky t a b0 ∈[ ], ta bud\-qkoho
0 < θ < min ,
–
exp ( ) –
1
2 2 1
1
0
σ
ψ
b a
w{ }( )
ma[ misce rivnist\
sup ( )
,t a b E
X t
∈[ ]
ψ
≤ X t E( )0 ψ
+
+ exp ( )
( – )
ln
–
( )
(– )
(– )ψ
θ θ
ψ
σ
θ
2 1
1 2
11
0
1
0
{ } +
∫
w
b a
u
du = D tψ ( )0 , (5.1)
de w0 = σ sup –
,t a b
t t
∈[ ]
0 ≤ σ( – )b a .
Vypadkovyj proces X t( ) [ vybirkovo neperervnym z imovirnistg odynycq.
Dlq vsix ε > 0 vykonu[t\sq nerivnist\
P sup ( )
,t a b
X t
∈[ ]
>
ε ≤ 2
0
exp –
( )
ψ
ψ
x
D t
. (5.2)
Dovedennq. Nerivnist\ (5.1) ta tverdΩennq pro neperervnist\ procesu vy-
plyvagt\ vidpovidno z naslidkiv 2.1 ta 2.2, oskil\ky funkciq U x( ) =
= exp ( )ψ x{ } – 1 zadovol\nq[ umovu g z Z0 = 2, K = 1, A = 1 (dyv. [2]), tobto
CU = exp ( )ψ 2{ }, U z(– )( )1 = ψ(– ) ln( )1 1z +( ), z > 0.
ObmeΩennq na θ vyplyvagt\ z obmeΩen\ na θ z naslidku 2.1. Nerivnist\ (5.2)
vyplyva[ z (5.1) ta teoremy.5.1.
Teorema 5.3. Nexaj X = X t( ){ , t T∈ } , T = a b,[ ], – ∞ < a < b < + ∞, — sepa-
rabel\nyj vypadkovyj proces iz prostoru Expψ ( )Ω . Nexaj dlq deqkoho δ <
< 1, C > 0 vykonu[t\sq nerivnist\
sup ( ) – ( )
–
, ,
t s h
t s a b
EX t X s
≤
∈[ ]
ψ
≤ C
h
ψ δ(– )
–
ln1
1
1 1+
. (5.3)
Todi vypadkova velyçyna sup ( )
, ,t s a b
X t
∈[ ]
naleΩyt\ prostoru Expψ ( )Ω i dlq
bud\-qko] toçky t a b0 ∈[ ], ta
0 < θ < min ,
ln sup –
ln
exp ( ) –
–
(– )
,
–
(– )
1
1
1
2 2 1
1
0
1
1
1
ψ
ψ ψ
δ
+
+
{ }( )
∈[ ]t a b
t t
b a
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
PRO RIVNOMIRNU ZBIÛNIST| VEJVLET-ROZKLADIV VYPADKOVYX … 761
ma[ misce nerivnist\
sup ( )
,t a b E
X t
∈[ ]
ψ
≤ D tψ δ, ( )0 ,
de
D tψ δ, ( )0 = X t E( )
ψ
+ exp ( )ψ 2{ } ×
×
1
1
1
2
1
1
1
1
1
0 0
1
–
ln max ,
–
– –
(– ) – –
θ
ψ
θ
θ
δ
δ δ δb a
w C w
+
,
w0 = C t t
t a b
ψ
δ
(– )
,
– –
ln sup –1
0
1
1
1 +
∈[ ]
.
Vypadkovyj proces X t( ) [ vybirkovo neperervnym z imovirnistg odynycq. Dlq
vsix ε > 0 vykonu[t\sq nerivnist\
P sup ( )
a t b
X t
≤ ≤
>
ε ≤ 2exp –
( ),
ψ ε
ψ δD t0
.
Dovedennq. Lehko baçyty, wo σ(– ) ( )1 u = exp –
–
ψ
δC
u
1
1
. OtΩe,
ψ
σ
(– )
(– )ln
–
( )
1
12
1
b a
u
+
=
= ψ ψ
δ
(– ) ln
–
exp –1
2
1 1
b a C
u
+
≤
≤ ψ ψ
δ
(– ) ln max ,
–
exp1 1
2
b a C
u
=
= ψ ψ
δ
(– ) ln max ,
–1 1
2
b a C
u
+
≤
≤ ψ(– ) ln max ,
–1 1
2
b a
+ C
u
δ
. (5.4)
Ostannq nerivnist\ v (5.4) vyplyva[ z toho, wo [2] pry x > 0, y > 0 ψ(– ) ( )1 x y+ ≤
≤ ψ(– ) ( )1 x + ψ(– ) ( )1 y . Takym çynom,
0
1
1
0
2
1
w
b a
u
du
θ
ψ
σ∫ +
(– )
(– )ln
–
( )
≤ ψ θ(– ) ln max ,
–1
01
2
b a
w
+
+
0
0w
u du C
θ
δ δ∫ – = ψ θ(– ) ln max ,
–1
01
2
b a
w
+ C wδ δ δ
δ
θ1
1 0
1 1
–
– – .
OtΩe, z (5.1) vyplyva[, wo sup ( )
,t a b E
X t
∈[ ]
ψ
≤ D tψ δ, ( )0 .
Vsi inßi tverdΩennq [ naslidkamy teoremy.5.2.
Pryklad 5.1. Nexaj ψ( )x = L x α
, 1 ≤ α, de L > 0 — deqka konstanta. Ne-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
762 G. V. KOZAÇENKO, M. M. PERESTGK
xaj b – a ≥ 2. Todi umova (5.3) nabyra[ vyhlqdu : nexaj isnugt\ konstanty δ < 1
ta D > 0 taki, wo
sup ( ) – ( )
–
, ,
t s h
t s a b
EX t X s
≤
∈[ ]
ψ
≤ D
h
ln 1 1 1
+
αδ .
Todi
Dψ δ, = X t E( )0 ψ
+ exp
–
ln
–
–
– –
L
b a
L
w
D
L
w2
1
1
2 1
1
1
0 1 0
1α
δ
α
δ
θ δ
θ
α
δ{ }
+
,
de
0 < θ < min ,
ln sup –
ln
exp –
–
,
–
1
1
1
2 2 1
0
1
1
1
+
+
⋅{ }( )
∈[ ]t a b
t t
L
b a
δα
α δα
,
w0 = D
t t
t a b
ln
sup –
,
–
1 1
0
1
+
∈[ ]
δα
.
Teorema 5.4. Nexaj X = X t( ){ , t R∈ } — separabel\nyj vypadkovyj proces
iz prostoru Expψ ( )Ω . Prypustymo, wo vykonugt\sq nastupni umovy:
Bk — intervaly a ak k, +[ ]1 taki, wo – ∞ < ak < ak +1 < + ∞, ak +1 – ak > 2,
k Z∈ ,
Bk
k Z∈
∪ = R ;
na koΩnomu z intervaliv Bk isnugt\ taki funkci] σk = σk h( ){ , 0 ≤ h ≤
≤ ak +1 – ak}, wo σk h( ) — neperervni monotonno zrostagçi funkci], σk ( )0 =
= 0, ta sup ( ) – ( )
–
,
t s h
t s B
E
k
X t X s
≤
∈
ψ
≤ σk h( ) ;
dlq deqkoho ε > 0 vykonu[t\sq umova
ψ
σ
ε
(– )
(– )ln
–
( )
1
0
1
12
1∫ + +
a a
u
duk k
k
< ∞;
c = c t( ){ , t R∈ } — deqka neperervna funkciq, taka, wo c t( ) > 0, rk =
= inf ( )
–
t Bk
c t
∈
1
, t k0 — deqka toçka z intervalu Bk ;
θk — dovil\ni çysla taki, wo
0 < θk < min ,
–
exp ( ) –
1
2 2 1
11
0
σ
ψ
a a
w
k k
k
+
{ }( )
, de w k0 = σk
t a a
k
k k
t tsup –
,∈[ ]+
1
0 ;
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
PRO RIVNOMIRNU ZBIÛNIST| VEJVLET-ROZKLADIV VYPADKOVYX … 763
D t kψ ( )0 = X t k E
( )0
ψ
+ exp ( )
( – )
ln
–
( )
(– )
(– )ψ
θ θ
ψ
σ
θ
2 1
1 2
11
0
1
1
0
{ } +
∫ +
k k
w
k k
k
k k a a
u
du ,
w k0 = σk
t a ak k
t tsup –
,∈[ ]+
1
0 ;
dlq deqkoho s > 2 0max ( )
k Z
k kr D t
∈
ψ zbiha[t\sq rqd
exp –
( )
ψ
ψ
s
r D tk kk Z 0
∈
∑ . (5.5)
Todi dlq vsix ε > 2s ma[ misce nerivnist\
P sup ( )
( )t R
X t
c t∈
>
ε ≤ 2
0
exp – exp –
( )
ψ ε ψ
ψs
s
r D tk kk Z
{ }
∈
∑ (5.6)
ta isnu[ vypadkova velyçyna ξ > 0, P ξ < ∞{ } = 1, taka, wo z imovirnistg ody-
nycq X t( ) < ξc t( ) dlq vsix t R∈ .
Dovedennq. Qk i pry dovedenni teoremy.3.1, ma[mo
P sup
( )
( )t R
X t
c t∈
>
ε ≤
k Z t Bk
X t
c t∈ ∈
∑ >
P sup
( )
( )
ε ≤
k Z t B kk
X t
r∈ ∈
∑ >
P sup ( )
ε
. (5.7)
Z teoremy.5.2 vyplyva[, wo
P sup ( )
t B kk
X t
r∈
>
ε ≤ 2
0
exp –
( )
ψ
ψ
s
r D tk k
, (5.8)
de θk zadovol\nqgt\ nerivnosti z formulgvannq teoremy. Oskil\ky pry vsix
x > 2, y > 2 x y ≥ x + y , z toho, wo [2] ψ x y+( ) ≥ ψ( )x + ψ( )y , dlq
s
r D tk kψ ( )0
. > 2 ta
ε
s
> 2 ma[mo
ψ ε
ψr D tk k( )0
= ψ ε
ψ
s
r D t sk k( )0
≥ ψ ε
ψ
s
r D t sk k( )0
+
≥
≤ ψ
ψ
s
r D tk k( )0
+ ψ ε
s
.
OtΩe, dlq tyx Ωe ε ta s
exp –
( )
ψ ε
ψr D tk k0
≤ exp – exp –
( )
ψ ε ψ
ψs
s
r D tk k
0
, (5.9)
tobto z (5.7) — (5.9) vyplyva[ (5.6) ta tverdΩennq teoremy.
ZauvaΩennq 5.1. Oskil\ky nerivnist\ xy ≥ x + y, x > 0, y > 0, ma[ misce pry
x > 1 ta y > x
x – 1
, to z dovedennq teoremy.5.3 vyplyva[, wo ]] tverdΩennq ma[
misce pry
s
r D t
k Z
k ksup ( )
∈
ψ 0
> x > 1 ta
ε
s
> x
x – 1
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
764 G. V. KOZAÇENKO, M. M. PERESTGK
ZauvaΩennq 5.2. Zhidno z zauvaΩennqm 2.3 w k0 u vyrazi dlq D t kψ ( )0
moΩna zaminyty na 2 sup ( )
,t a b
E
k k
X t
∈[ ] ψ
.
Teorema 5.5. Nexaj X = X t( ){ , t R∈ } — separabel\nyj vypadkovyj proces
iz prostoru Expψ ( )Ω . Prypustymo, wo vykonugt\sq nastupni umovy:
Bk — intervaly a ak k, +[ ]1 taki, wo – ∞ < ak < ak +1 < ∞, ak +1 – ak > 2,
k Z∈ ,
Bk
k Z∈
∪ = R ;
na koΩnomu z intervaliv Bk vykonu[t\sq nerivnist\
sup ( ) – ( )
–
,
t s h
t s B
E
k
X t X s
≤
∈
ψ
≤ C
hk ψ δ(– )
–
ln1
1
1 1+
, Ck > 0, δ < 1; (5.10)
c = c t( ){ , t R∈ } — deqka neperervna funkciq taka, wo c t( ) > 0 , rk =
= inf ( )
–
t Bk
c t
∈
1
, t k0 — deqka toçka z intervalu Bk ;
θk — dovil\ni çysla taki, wo
0 < θk < min ,
ln sup –
ln
exp ( ) –
–
(– )
–
–
(– )
1
1
1
2 2 1
1
0
1
1
1
1
1
ψ
ψ ψ
δ
+
+
{ }( )
∈[ ]
+
+t a a
k
k k
k k
t t
a a
;
D t kψ δ, ( )0 = X t k E( )0 ψ
+ exp ( )
–
ln
–(– )ψ
θ
ψ2
1
1 2
1 1
0{ }
+
k
k k
k
a a
w +
+ C wk k k
δ δ δ
δ
θ1
1 0
1
–
– –
;
dlq deqkoho s > 2 0max ( )
k Z
k kr D t
∈ ψδ zbiha[t\sq rqd
exp –
( )
ψ
ψδ
s
r D tk kk Z 0
∈
∑ . (5.11)
Todi dlq vsix ε > 2s ma[ misce nerivnist\
P sup
( )
( )t R
X t
c t∈
>
ε ≤ 2
0
exp – exp –
( )
ψ ε
δ
ψ
ψδ
{ }
∈
∑ s
r D tk kk Z
(5.12)
ta isnu[ vypadkova velyçyna ξ > 0, P ξ < ∞{ } = 1, taka, wo z imovirnistg ody-
nycq X t( ) < ξc t( ) dlq vsix t R∈ .
Teorema vyplyva[ z teorem.5.4 ta 5.3. ZauvaΩennq 5.1 stosu[t\sq takoΩ i ci[]
teoremy.
ZauvaΩymo, wo oskil\ky pry c ≥ 1, x > 0 ψ (– ) ( )1 cx ≤ c xψ(– ) ( )1
ta pry c ≥
≥ 1 ln
–
1
1
+
+
c
a ak k
≤ c
a ak k
ln
–
1
1
1
+
+
, to nerivnist\ (5.12) vykonu[t\sq dlq
vsix takyx θk , wo
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
PRO RIVNOMIRNU ZBIÛNIST| VEJVLET-ROZKLADIV VYPADKOVYX … 765
0 < θk < min , exp ( ) – –1 2 2 1
1
ψ δ{ }( )( )
,
w k0 = C
t t
k
t a a
k
k k
ψ
δ
(– )
,
–
ln
sup –
1
0
1
1
1
1
+
∈[ ]+
.
Pryklad 5.2. Nexaj v teoremi.5.5 ψ( )x = L x α
, 1 ≤ α, de L > 0 — deqka
konstanta, ak +1 – ak > 2. Nexaj isnugt\ taki konstanty Dk > 0 ta δ < 1, wo
vykonugt\sq nerivnosti
sup ( ) – ( )
–
,
t s h
t s B
E
k
X t X s
≤
∈
ψ
≤
D
h
k
ln 1 1 1
+
δα . (5.13)
Todi, zhidno z prykladom 5.1 (qkwo t k0 = ak ),
D akψδ( ) = X ak E( )
ψ
+
+ exp
–
ln
–
( – )
– –L
a a
L
w
D
L
w
k
k k
k
k
k k⋅{ }
+
+
2
1
1
2 1
1
1
1
0 1 0
1α
α
δ
α
δ δ
θ δ
θ , (5.14)
de w k0 = D
a ak
k k
ln
–
–
1 1
1
1
+
+
αδ
.
Oskil\ky pry 0 < x < 1
2
ln( )1 + x ≥ x 2 3
2
ln , to qkwo poklasty v (5.14) θk =
= θ, de θ — bud\-qke çyslo, take, wo
0 < θ < min , exp –
–
1 2 2 1
1
L α δ{ }( )( )
, (5.15)
to z (5.14) otryma[mo nerivnist\
D akψδ( ) ≤ X ak E( )
ψ
+ exp
–
ln
––
L L
a ak k⋅{ }
+
2
1
1 2
1
1
1
α α α
θ
×
× D
a ak
k k
ln
–
–
1
1
1
1
+
+
δα
+
D
L
D
a a
k
k
k k
δ
α
δ δ
δ
δα
δ
θ1
1
1
1
1
1
1
1
( – )
ln
–
– –
–
–
+
+
=
= X ak E( )
ψ
+ exp
–
ln
–
ln
–
/ –
L
D
L
a a
a a
k k k
k k
⋅{ }
+
+
+
2
1
1 2
1
1
1
1
1
1
1
α
α
α δα
θ
+
+
1
1
1
1
1
1
θ δδ
δ
δα
( – )
ln
–
–
–
+
+a ak k
≤ X ak E( )
ψ
+ exp
–
L
D
L
k⋅{ } /2 1
1 1
α
αθ
×
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766 G. V. KOZAÇENKO, M. M. PERESTGK
× 1
2 3
2
21
1
1
1
1
ln
ln
–
( – )
+
+
δ
α αδa a
a ak k
k k +
+
1
1 2 3
2
1 1
1
( – ) ln
( – )–
–
δ θδ
δ
αδ
δ
αδ
+a ak k . (5.16)
Nexaj teper Dk = D > 0, c t( ) — monotonno zrostagça pry t > 0 parna funk-
ciq, t k0 = ak , rk = 1
c ak( )
. Dlq c\oho prykladu umova (5.11) nabyra[ vyhlqdu :
dlq deqkoho s > 2 0max ( )
k Z
k kr D t
∈
ψδ zbiha[t\sq rqd
exp –
( )
L s
r D tk kk Z ψδ
α
0
∈
∑ < ∞. (5.17)
Lehko pereviryty, wo umova (5.17) bude vykonuvatys\, qkwo pry dosyt\ velykyx
k vykonu[t\sq umova
r D ak kψδ( )( ) < C
kln( )
1
α
, (5.18)
de C > 0 — deqka konstanta. Rqd (5.17) bude zbihatysq pry s takyx, wo
Ls
c
α
α >
> 1. Dlq toho wob vykonuvalas\ umova (5.18), dosyt\, wob pry velykyx k vyko-
nuvalys\ umovy
X a rk E k( )
ψ( ) ≤ C
kln( )
1
α
ta r
a a
a ak
k k
k kln
–
( – )
+
+
1
1
1
1
2
α δα < C
kln( )
1
α
.
(5.19)
Qkwo pry dosyt\ velykomu k poklasty ak = e k
, to lehko peresvidçytys\,
wo druha umova v (5.19) vykonu[t\sq, qkwo pry dosyt\ velykyx t funkciq
c t( ) ma[ vyhlqd
c t( ) = ln ln lnt t t( ) ( )
1 1 1
α α αδ . (5.20)
OtΩe, ma[ misce, napryklad, taka teorema.
Teorema 5.6. Nexaj X = X t( ){ , t R∈ } — separabel\nyj vypadkovyj proces
iz prostoru Expψ ( )Ω , ψ( )x = L x α
, α ≥ 1. Nexaj dlq X t( ) vykonu[t\sq
umova (5.13) pry Dk = D , c t( ) zadano v (5.20), X t E( )
ψ
≤ t t
1 1
αδ αln ×
× ln ln t( )
1
α , t > e2
. Todi isnu[ taka vypadkova velyçyna ξ > 0, P ξ < ∞{ } = 1,
wo z imovirnistg odynycq X t( ) < ξc t( ) dlq vsix t R∈ .
Pryklad 5.3. Nexaj Zβ = Z tβ( ){ , t R∈ , 0 < β < 1} — WSSSI-proces iz pros-
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PRO RIVNOMIRNU ZBIÛNIST| VEJVLET-ROZKLADIV VYPADKOVYX … 767
toru S Expψ ( )Ω (dyv. oznaçennq 2.9 ta 2.7). Ce oznaça[ zokrema, wo isnu[ kon-
stanta D > 0 taka, wo dlq vsix t, s R∈
Z t
Eβ
ψ
( ) ≤ D E Z tβ( )( )( )2
1
2 ≤ D t β, (5.21)
Z t Z s
Eβ β
ψ
( ) – ( ) ≤ D E Z t Z sβ β( ) – ( )( )( )2
1
2 t ≤ D t s– β . (5.22)
Zastosu[mo teoremu.5.4 do c\oho procesu.
Nexaj t k0 = ak , θk = θ , de θ — çyslo, dlq qkoho vykonu[t\sq neriv-
nist\.(5.15). Todi σk h( ) = Dhβ
, σk u(– ) ( )1 = u
D
1
β
, w k0 ≤ D a ak k( – )+1
β
,
Z ak Eβ
ψ
( ) ≤ D ak( )β. Lehko baçyty, wo
ψ
σ
θ
(– )
(– )ln
–
( )
1
0
1
1
0
2
1
w
k k
k
k a a
u
du∫ + +
≤ ψ
β
(– ) ln
–1
0
1
1
0
2
1
w
k k
k a a
u
D
du∫ +
+
=
= ψ
ν
ν
β
β
β(– )
( – )
–
ln ( – )1
0
1 1
0 1
1
1
2
1
w D a a
k k
k k k
d D a a
+( )
+∫ +
≤ D a a Ik k( – )+1
β
β ,
de
Iβ = ψ
ν
νβ
(– ) ln1
0
1
1
1
2
1∫ / +
d .
Ostannij intehral zbiha[t\sq, oskil\ky pry dosyt\ velykyx x ψ(– )( )1 x ≤ Cx ,
C > 0 [2], a pry x ≥ 0 ln( )1 + x ≤ 1
α
αx , 0 < α ≤ 1.
OtΩe, pry bud\-qkomu 0 < α ≤ 1 ta dosyt\ malyx ν
ψ
ν β
(– ) ln1
1
1
2
1/ +
≤ C ln 1
2
11ν β/ +
≤ C
α
να
α
β1
2
–
.
Zrozumilo, wo pry α < β Iβ < ∞, tomu v c\omu vypadku z (5.21), (5.22) vyplyva[,
wo D akψ ( ) ≤ D ak
β + d I D akβ ( +1 – ak )β
, de d = exp ( )
( – )
ψ
θ θ
2 1
1
{ } . Lehko ba-
çyty, wo umova (5.15) vykonu[t\sq, qkwo dlq dosyt\ velykyx k > 0
D a
c a
k
k
ψ ( )
( )
<
<
s
b kψ (– ) (ln )1 ( )
, b > 1, s > 0. Qkwo poklasty ak = e k
, k > 0, to zrozumilo, wo
cq umova vykonu[t\sq dlq funkci] c t( ) = t t bβ ψ(– ) ln ln ,1 ( )( ), t > e2
, b > 1.
Takym çynom, ma[ misce taka teorema.
Teorema 5.7. Nexaj Zβ = Z tβ( ){ , t R∈ , 0 < β < 1} — separabel\nyj
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768 G. V. KOZAÇENKO, M. M. PERESTGK
WSSSI-proces iz prostoru Expψ ( )Ω . Todi isnu[ taka vypadkova velyçyna ξ >
> 0, P ξ < ∞{ } = 1, wo z imovirnistg odynycq Z tβ( ) < ξ βc tb ( ) , de c tbβ( ) =
= t b tβ ψ(– ) ln ln1 ( )( ), t > e2
, b > 1.
Pryklad 5.4. Nexaj X = X t( ){ , t R∈ } — separabel\nyj kvazistacionarnyj
proces iz prostoru Expψ ( )Ω (dyv. oznaçennq 2.10), X t E( )
ψ
≤ EX . Nexaj isnu[
taka neperervna monotonno zrostagça funkciq σ = σ( )h{ , h > }0 , wo
sup ( ) – ( )
–t s h
X t X s
≤
≤ σ( )h . (5.23)
Prypustymo, wo zbiha[t\sq intehral
ψ
σ
(– )
(– )ln
( )
1
0
1
1 1
EX
u
du∫ +
< ∞.
Zastosu[mo do c\oho procesu teoremu.5.4. Nexaj Bk = a ak k, +[ ]1 — taki interva-
ly, wo – ∞ < ak < ak +1 < ∞, ak +1 – ak > 2,
k Z
kB
∈
∪ = R ; c = c t( ){ , t R∈ } — deqka
neperervna funkciq taka, wo c t( ) > 0, rk = inf ( )
(– )
t Bk
c t
∈
1
. Vraxuvavßy zauva-
Ωennq 5.2, poklademo w k0 = 2 EX . Todi pry θ takyx, wo
0 < θ < min ,
exp ( ) –
1 1
2 1
σ
ψ{ }
(my vraxuvaly, wo ak +1 – ak ≥ 2), ma[mo
D akψ ( ) ≤ EX + exp ( )
( – )
ln
–
( )
(– )
(– )ψ
θ θ
ψ
σ
θ
2
1
1 2
11
0
2
1
1{ } +
∫ +
E
k k
X a a
u
du . (5.24)
Oskil\ky ψ(– )( )1 x y+ ≤ ψ(– )( )1 x + ψ(– )( )1 y , to
ψ
σ
(– )
(– )ln
–
( )
1 1
12
1
a a
u
k k+ +
≤ ψ
σ
(– )
(– )ln
–
ln
( )
1 1
11
2
1
1+
+ +
+a a
u
k k
≤
≤ ψ(– ) ln
–1 11
2
+
+a ak k
+ ψ
σ
(– )
(– )ln
( )
1
11 1+
u
.
OtΩe, z (5.24) otrymu[mo nerivnist\
D akψ ( ) ≤ EX + exp ( )
( – )
ln
( )
(– )
(– )ψ
θ θ
ψ
σ
θ
2
1
1
1
11
0
2
1{ } +
∫
EX
u
du +
+ exp ( )
–
ln
–(– )ψ
θ
ψ2
1
1
2 1
2
1 1{ } +
+
E
a a
X
k k
= ˆ ( )D akψ .
Teper dlq toho, wob vykonuvalas\ umova (5.5), dosyt\, wob pry s >
> 2 max ˆ ( )
k Z
k kr D a
∈
ψ zbihavsq rqd
exp – ˆ ( )
ψ
ψ
s
r D ak kk Z
∈
∑ < ∞. (5.25)
Zrozumilo, wo umova (5.25) bude vykonuvatys\, qkwo dlq deqkoho B > 0 zbihaty-
met\sq rqd
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PRO RIVNOMIRNU ZBIÛNIST| VEJVLET-ROZKLADIV VYPADKOVYX … 769
exp –
ln
–(– )
ψ
ψ
s
Br
a a
k
k kk Z 1 11
2
+
+∈
∑ < ∞. (5.26)
Lehko pereviryty, wo umova (5.26) bude vykonuvatys\, qkwo dlq dosyt\ velykyx
k r
a a
k
k kψ(– ) ln
–1 11
2
+
+ ≤ C
b k
1
1ψ(– ) ln( )( )
, de b > 1, a C > 0 — deqka
konstanta ( napryklad, C = s
B
. Qkwo dlq dosyt\ velykyx k > 0 poklasty
ak = ek
(dlq vid’[mnyx k vzqty symetryçne rozbyttq, a c t( ) — parna funkciq),
to lehko peresvidçytys\, wo ostanni nerivnosti vykonugt\sq dlq funkci], wo
pry dosyt\ velykyx t ma[ vyhlqd
c t( ) = C t b tψ ψ(– ) (– )ln ln ln1 1( ) ( ), b > 1. (5.27)
OtΩe, ma[ misce, napryklad, taka teorema.
Teorema 5.8. Nexaj X = X t( ){ , t R∈ } — separabel\nyj kvazistacionarnyj
proces iz prostoru Expψ ( )Ω , dlq qkoho vykonu[t\sq umova (5.23). Todi isnu[
taka vypadkova velyçyna ξ > 0, P ξ < ∞{ } = 1, wo z imovirnistg odynycq
X t( ) < ξc t( ), de c t( ) vyznaçeno v (5.27).
6. Rivnomirna zbiΩnist\ vejvlet-zobraΩen\. Nexaj ϕ = ϕ( )t{ , x R∈ } —
funkci] z prostoru L R2( ) , ˆ ( )ϕ y = e x dxiyx
R
– ( )ϕ∫ — peretvorennq Fur’[ funk-
ci] ϕ, ϕ0k x( ) = ϕ( – )x k . Nexaj vykonugt\sq umovy:
a1)... ˆ ( )ϕ πy k
k Z
+
∈
∑ 2
2
= 1 majΩe skriz\;
a2)...isnu[ taka periodyçna z periodom 2π funkciq m x0( ), m x0( ) ∈
∈ L 2 0 2; π[ ]( ) , wo majΩe skriz\ ˆ ( )ϕ y = m
y y
0 2 2
ϕ̂ ;
a3)... ˆ ( )ϕ 0 ≠ 0 ta ˆ ( )ϕ y [ neperervnog v nuli.
Todi funkciq ϕ nazyva[t\sq f-vejvletom.
Nexaj δ( )x — taka funkciq, wo ]] peretvorennq Fur’[ ma[ vyhlqd
ˆ( )δ y = m
y
i
y y
0 2 2 2
+
{ }
π ϕexp – ˆ .
Funkciq δ( )x nazyva[t\sq m-vejvletom, wo vidpovida[ f-vejvletu ϕ. Nexaj
ϕ jk x( ) = 2 22i i x k/ –ϕ( ) , δ jk x( ) = 2 22i i x k/ –δ( ), j R∈ , k R∈ .
Vidomo (dyv., napryklad, [3 – 6]), wo systema funkcij ϕ0k{ , δ jk , j = 0, 1,
2, … , k Z∈ } [ ortonormovanym bazysom v L R2( ) . Bud\-qku funkcig f ∈ L R2( )
moΩna zobrazyty u vyhlqdi rqdu, wo zbiha[t\sq u seredn\omu kvadratyçnomu:
f x( ) = α ϕ0 0k k
k Z
x( )
∈
∑ +
j
jk jk
k Z
x
=
∞
∈
∑ ∑
0
β δ ( ), (6.1)
de α0k = f x x dx
R k( ) ( )∫ ϕ0 , β jk = f x x dx
R jk( ) ( )∫ δ ta α0
2
kk Z∈∑ +
+
j jkk Z=
∞
∈∑ ∑0
2β < ∞.
ZobraΩennq (6.1) nazyva[t\sq vejvlet-zobraΩennqm.
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770 G. V. KOZAÇENKO, M. M. PERESTGK
Oskil\ky intehraly, wo vyznaçagt\ α0k i β jk , isnugt\ ne lyße dlq funk-
cij z L R2( ) , moΩna otrymaty vejvlet-zobraΩennq dlq bil\ß ßyrokoho klasu
funkcij, niΩ prostir L R2( ) , qki budut\ zbihatysq v pevnyx normax.
Znajdemo umovy, za qkyx zobraΩennq (6.1) budut\ zbihatysq rivnomirno na
pevnomu intervali α β,[ ].
Oznaçennq 6.1 [3, 6]. Nexaj ϕ — f-vejvlet. Dlq ϕ vykonu[t\sq umova S,
qkwo isnu[ parna funkciq Φ = Φ( )x{ , x R∈ } taka, wo Φ( )0 < ∞, Φ( )x mo-
notonno spada[ pry x ≥ 0, Φ x dx
R
( )∫ < ∞ ta ϕ( )x ≤ Φ x( ) dlq x R∈ .
Dali nam bude potribna taka lema.
Lema 6.1. Nexaj dlq f-vejvletu ϕ vykonu[t\sq umova S z funkci[g Φ
ta δ( )x — m-vejvlet, wo vidpovida[ ϕ. Todi pry vsix x R∈ ma[ misce ne-
rivnist\
δ( )x ≤ B
xΦ 2 1
4
–
,
de 0 < B < ∞ — deqka konstanta.
Dovedennq. Zhidno z lemog 4.1 [6] (dyv. takoΩ [4]), funkci] ϕ( )x ta δ( )x
dopuskagt\ taki zobraΩennq:
ϕ( )x = 2 2h x kk
k Z
ϕ( – )
∈
∑ ,
de hk = 2 2ϕ ϕ( ) ( – )u u k du
R∫ ta hkk Z∈∑ 2
< ∞,
δ( )x = 2 2λ ϕk
k Z
x k( – )
∈
∑ , (6.2)
de λk = (– ) –
–1 1
1
k
kh .
Nexaj k ≥ 0. Todi
hk = 2 2
R
u u k du∫ ( ) ( )Φ Φ – = 2 2
3
–
–
∞
∫ ( ) ( )
k
u k u duΦ Φ +
+ 2 2
3k
u k u du
∞
∫ ( ) ( )Φ Φ – ≤ Φ Φk u du
k
3
2
3
( )
∞
∫
–
+
+ Φ Φk u k du
k3
2 2
3
( )
∞
∫ – ≤ 2
3
2Φ Φ Φk u du u k du
RR
( ) + ( )
∫∫ – =
= Φ Φ
k C
3
,
de CΦ = 3
2 R
u du∫ ( )Φ < ∞.
Pry k ≤ 0 analohiçno otrymu[mo hk ≤ Φ Φ
k
C
3
.
OtΩe, pry vsix k Z∈ ma[mo
hk ≤ C
k
Φ Φ
3
. (6.3)
Teper z (6.2) ta (6.3) vyplyva[, wo dlq koΩnoho x R∈
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PRO RIVNOMIRNU ZBIÛNIST| VEJVLET-ROZKLADIV VYPADKOVYX … 771
δ( )x ≤ 2 2 1ϕ( – ) –x k h
k Z
k
∈
∑ ≤ C x k
k
k Z
Φ Φ Φ2 2
1
3
–
–( )
∈
∑ =
= C x k
k
k Z
Φ Φ Φ2 2 1
3
– –( )
∈
∑ = I x( – )2 1 . (6.4)
Poznaçymo 2x – 1 = 0. Todi dlq u > 0
I u( ) ≤ C u k
k
u k
k
k u k u
Φ Φ Φ Φ Φ2
3 33
4
3
4
– –( )
+ ( )
≤ ≥
∫ ∫ =
= C A u A uΦ 2 1 2( ) ( )+( ).
Lehko baçyty, wo
A u2( ) ≤ Φ Φu
u k
k u
4 3
4
( )
≥
∫ – ≤ Φ Φu u k
k Z
4
( )
∈
∫ – ,
A u1( ) ≤ Φ Φu k
k u
4 33
4
≤
∫ ≤ Φ Φu k
k Z
4 3
∈
∫ .
OtΩe, dlq u > 0
I u( ) ≤ Φ ΦΦ
u C u k
k Z4
2
( )
∈
∑ – +
k Z
k
∈
∑
Φ
3
. (6.5)
PokaΩemo, wo isnu[ konstanta D, 0 < D < ∞, taka, wo dlq vsix u > 0
k Z
u k
∈
∑ ( )Φ – +
k Z
k
∈
∑
Φ
3
< D.
Lehko baçyty, wo
k u
k u
≥ +
∑
1
Φ( – ) =
k u k
k
k u d
≥ +
∑ ∫
1 1–
( – )Φ ν ≤
k u k
k
u d
≥ +
∑ ∫
1 1–
( – )Φ ν ν ≤
≤
u
u d
∞
∫ Φ( – )ν ν =
0
∞
∫ Φ( )ν νd < ∞.
Analohiçno otrymu[mo
k u
u k≤∑ –
( – )
1
Φ ≤
0
∞
∫ Φ( )ν νd ,
k Z
u k
∈
∑ ( )Φ – ≤ 2 0
0
Φ Φ( ) ( )+
∞
∫ ν νd = D1 < ∞, (6.6)
k Z
k
∈
∑
Φ
3
< D2 < ∞. (6.7)
OtΩe, z (6.5) – (6.7) vyplyva[, wo dlq u > 0
I u( ) ≤ C D D u
Φ Φ2
41 2( )+
. (6.8)
Dlq u < 0 Φ u k+( ) = Φ – –u k( ) = Φ u k–( ) , tobto dlq u < 0
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772 G. V. KOZAÇENKO, M. M. PERESTGK
I u( ) ≤ C u k
k
k Z
Φ Φ Φ2
3∈
∑ ( )
– ≤ C
u
D DΦ Φ2
4 1 2
+( ) . (6.9)
Teper lema vyplyva[ z (6.4), (6.5), (6.8), (6.9).
Teorema 6.1. Nexaj dlq f-vejvletu ϕ vykonu[t\sq umova S z funkci[g
Φ , c = c x( ){ , x R∈ } — taka parna funkciq, wo c x( ) > 1, x R∈ , c x( ) mono-
tonno zrosta[ pry x > 0 ta
R
c x x dx∫ ( )( ) Φ < ∞. Krim toho, isnu[ taka
funkciq 0 < A u( ) < ∞, u > 0, wo dlq dosyt\ velykyx x
c ax( ) ≤ c x A a( ) ( ) , a > 0. (6.10)
Nexaj f = f t( ){ , x R∈ } — taka vymirna na R funkciq, wo f x( ) ≤ c x( ),
x R∈ ; f x( ) neperervna na intervali ( , )a b , – ∞ < a < b < + ∞.
Todi
f xm( ) = α ϕ0 0k k
k Z
x( )
∈
∑ +
j
m
jk jk
k Z
x
= ∈
∑ ∑
0
1–
( )β δ → f x( )
pry m → ∞ rivnomirno na koΩnomu intervali α β,[ ] � ( , )a b ,
α0k = f x x dx
R
k( ) ( )∫ ϕ0 , β jk = f x x dx
R
jk( ) ( )∫ δ .
Dovedennq. Teoremu.6.1 dovedeno v roboti [7] z dodatkovog umovog, wo m-
vejvlet δ( )x takoΩ zadovol\nq[ umovu S z funkci[g Φ. Ce dodatkove prypu-
wennq potribno dlq ob©runtuvannq isnuvannq β jk ta rivnosti
f xm( ) = f y Z x y dy
R
m( ) ( , )∫ , (6.11)
de Z x ym( , ) = ϕ ϕ0 0kk Z kx y( ) ( )∈∑ +
j
m
jk jkk Z
x x= ∈∑ ∑0
1–
( ) ( )ψ δ .
β jk isnu[, oskil\ky z lemy 6.1 vyplyva[, wo
β jk ≤ 2 22j
R
jc x x k dx( ) –∫ ⋅( )δ = 2
2
2– j
j
R
c
u k
u du
+
( )∫ δ ≤ (6.12)
≤ 2
2
2 1
4
2– –j
j
R
c
u k u
du
+
∫ Φ = 2 2 2 1
2
2 1– –j j
R
c t k t dt( )+ + +
( )∫ Φ ≤
≤ 2 2 1
2
2 1– j
R
c t k t dt( )+ + +
( )∫ Φ . (6.12)
Oskil\ky dlq dosyt\ velykyx t
c t k2 1
2
+ +
≤ c t k2 1
2
+ +
≤ c t4( ) ≤ c t A( ) ( )4 ,
to ostannij intehral v (6.12) zbiha[t\sq, tobto β jk isnu[. Rivnist\ (6.11) [ nas-
lidkom teoremy Fubini, qkwo dovesty, wo dlq koΩnoho x R∈
ϕ ϕ0 0k
Rk Z
ku f u du x( ) ( ) ( )∫∑
∈
< ∞ (6.13)
ta
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
PRO RIVNOMIRNU ZBIÛNIST| VEJVLET-ROZKLADIV VYPADKOVYX … 773
δ δjk
Rk Z
jku f u du x( ) ( ) ( )∫∑
∈
< ∞. (6.13′ )
Nerivnist\ (6.13) dovedeno v roboti [7]. Dovedemo nerivnist\ (6.13′ ).
Z lemy 6.1 vyplyva[
δ jk y( ) = 2 22j j y kδ –( ) ≤ 2
1
2
2
1
2
2j jB y kΦ – –
.
OtΩe,
k Z
jk jku x
∈
∑ δ δ( ) ( ) ≤ 2
1
2
2
1
2
1
2
2
1
2
2j j
k Z
jB u k x kΦ Φ– – – –
∈
∑ . (6.14)
V lemi 8.2 z knyhy [6] (dyv. takoΩ [4] ) dovedeno take tverdΩennq. Nexaj F =
= F x( ){ , x > }0 — taka funkciq, wo F( )0 < ∞, F x( ) monotonno spada[ pry x >
> 0 ta F x dx
R
( )∫ < ∞. Todi isnugt\ konstanty c1 > 0, c2 > 0 taki, wo
k Z
F x k F y k
∈
∑ ( ) ( )– – ≤ c F c x y1 2 –( ) .
Qkwo vybraty F x( ) = Φ x
2
, to z (6.14) vyplyva[
k Z
jk jku x
∈
∑ δ δ( ) ( ) ≤ 2 21
2
2
1j jc B c u xΦ ⋅( )– – . (6.15)
Z (6.15) vyplyva[
δ δjk
Rk Z
jku f u du x( ) ( ) ( )∫∑
∈
≤ 2 21
1
2
j
R
jc f u c u x du∫ ( )( ) ––Φ ≤
≤ 2 21
1
2
j
R
jc c u c u x du∫ ( )( ) ––Φ < ∞.
Ostannq nerivnist\ dovodyt\sq tak, qk i obmeΩenist\ intehrala v (6.12).
7. Rivnomirna zbiΩnist\ vejvlet-rozkladiv vypadkovyx procesiv. Na-
stupna teorema da[ zahal\ni umovy rivnomirno] zbiΩnosti vejvlet-rozkladiv dlq
vypadkovyx procesiv iz prostoriv Orliça vypadkovyx velyçyn.
Teorema 7.1. Nexaj X = X t( ){ , t R∈ } — separabel\nyj vypadkovyj proces
iz prostoru Orliça LU ( )Ω , de U x( ) zadovol\nq[ umovu g. Nexaj dlq proce-
su X t( ) vykonu[t\sq umova teoremy.3.1 vidnosno funkci] c t( ) > 0, a dlq pro-
cesu X n a deqkomu intervali a b,[ ] — umova naslidku 2.1. Krim toho, nexaj
ϕ — deqkyj f-vejvlet, δ — vidpovidnyj jomu m -vejvlet i dlq ϕ vykonu[t\-
sq umova S z funkci[g Φ. Qkwo dlq funkci] c x( ) isnu[ funkciq 0 < A u( ) <
< ∞, u > 0, taka, wo dlq dosyt\ velykyx x, a > 0 c ax( ) ≤ c x A a( ) ( ) ta
R
c x x dx∫ ( )( )Φ < ∞, to dlq bud\-qkoho intervalu α β,[ ] � ( , )a b X tn( ) → X t( )
pry n → ∞ rivnomirno po t ∈. α β,[ ] z imovirnistg odynycq,
X tn( ) = ξ ϕ0 0k
k Z
k x
∈
∑ ( ) +
j
n
ik ik
k Z
x x
= ∈
∑ ∑
0
1–
( ) ( )η δ , (7.1)
ξ0k = X t t dtk
R
( ) ( )ϕ0∫ , ηik x( ) = X t t dtik
R
( ) ( )δ∫ . (7.2)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
774 G. V. KOZAÇENKO, M. M. PERESTGK
Dovedennq. Teorema vyplyva[ z teoremy.6.1. Dijsno, qkwo dlq X na inter-
vali a b,[ ] vykonugt\sq umovy naslidku 2.1, to zhidno z naslidkom 2.2 vypadko-
vyj proces [ vybirkovo neperervnym z imovirnistg odynycq na a b,[ ].
Zhidno z teoremog.3.1, z imovirnistg odynycq dlq t R∈ X t( ) ≤ ξ0 c t( ), de
P ξ0 < ∞{ } = 1, tobto dlq X na intervali a b,[ ] z imovirnistg odynycq vykonu-
gt\sq vsi umovy teoremy.6.1.
Nastupna teorema da[ zahal\ni umovy rivnomirno] zbiΩnosti vejvlet-rozkla-
div vypadkovyx procesiv iz prostoriv LU ( )Ω , p ≥ 1.
Teorema 7.2. Nexaj X = X t( ){ , t R∈ } — separabel\nyj vypadkovyj proces
iz prostoru LU ( )Ω , p ≥ 1. Nexaj dlq procesu X t( ) vykonu[t\sq umova teo-
remy.4.2 vidnosno funkci] c t( ) > 0 ta na deqkomu intervali a b,[ ] vykonu-
gt\sq umovy teoremy.4.1. Nexaj ϕ — deqkyj f-vejvlet, δ — vidpovidnyj
jomu m -vejvlet i dlq ϕ vykonu[t\sq umova S z funkci[g Φ . Qkwo dlq
funkci] c t( ) vykonu[t\sq umova (6.10) ta
R
c t x dx∫ ( )( )Φ < ∞ , to dlq bud\-
qkoho intervalu α β,[ ] � ( , )a b X tn( ) → X t( ) pry n → ∞ rivnomirno po t.∈
∈. α β,[ ] z imovirnistg odynycq ( X tn( ) zadano v (7.1), (7.2)).
Analohiçno moΩna otrymaty umovy rivnomirno] zbiΩnosti vejvlet-rozkladiv,
qkwo dlq procesu X zamist\ umov teoremy.4.2 vymahaty vykonannq naslidku 4.1
abo teoremy.4.4. Z teoremy.4.3 ta 6.1 vyplyva[ nastupna teorema.
Teorema 7.3. Nexaj X = X t( ){ , t R∈ } — separabel\nyj WSSSI-proces iz
prostoru SL p( )Ω , p > 1
α
, ϕ( )x — f-vejvlet, dlq qkoho vykonu[t\sq umova S
z funkci[g Φ t( ) = 1
t tδ α γ+ ( )ln
, γ > 1
p
, δ > 1
, dlq dosyt\ velykyx t . To-
di dlq bud\-qkoho intervalu a b,[ ] X tn( ) → X t( ) pry n → ∞ rivnomirno po
t.∈. α β,[ ] z imovirnistg odynycq ( X tn( ) zadano v (7.1), (7.2)).
Dovedennq. Dlq WSSSI-procesiv pry p > 1
α
vykonugt\sq umovy teore-
my.4.1, tobto vin [ vybirkovo neperervnym z imovirnistg odynycq na bud\-qkomu
intervali a b,[ ]. Dali, teorema vyplyva[ z teoremy.4.3, oskil\ky z imovirnistg
odynycq X t( ) < ξc t( ), de ξ — vypadkova velyçyna, P ξ < ∞{ } = 1, c t( ) — taka
funkciq, wo pry dosyt\ velykyx t c t( ) = t tα γln( ) , γ > 1
p
.
Analohiçno, z naslidku 4.2 vyplyva[ nastupna teorema.
Teorema 7.4. Nexaj X = X t( ){ , t R∈ } — kvazistacionarnyj separabel\nyj
vypadkovyj proces iz prostoru L p( )Ω , dlq qkoho vykonu[t\sq umova
sup ( ) – ( )
–t s h
p p
E X t X s
≤
( )1 ≤ Chδ , δ > 1
p
, C > 0.
Todi qkwo dlq f- vejvletu ϕ vykonu[t\sq umova S z funkci[g Φ t( ) =
= 1
t tν γln( )
, ν > 1 + 1
p
, γ > 1 + 1
p
, to dlq bud\-qkoho intervalu a b,[ ]
X tn( ) → X t( ) pry n → ∞ z imovirnistg odynycq rivnomirno na a b,[ ] ( X tn( )
zadano v (7.1), (7.2)).
Analohiçno z punktu 5 moΩna otrymaty umovy rivnomirno] zbiΩnosti vejv-
let-rozkladiv dlq vypadkovyx procesiv iz prostoriv Orliça eksponencial\noho
typu. Napryklad, ma[ misce taka teorema.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
PRO RIVNOMIRNU ZBIÛNIST| VEJVLET-ROZKLADIV VYPADKOVYX … 775
Teorema 7.5. Nexaj X = X t( ){ , t R∈ } — separabel\nyj vypadkovyj proces
iz prostoru Orliça Expψ ( )Ω . Nexaj dlq deqkoho intervalu a b,[ ] vykonu-
gt\sq umovy teoremy.5.2 ta dlq X — umovy teoremy.5.4 vidnosno funkci]
c t( ) > 0. Nexaj dlq f-vejvletu ϕ vykonu[t\sq umova S z funkci[g Φ . Todi
qkwo dlq Φ vykonu[t\sq umova (6.10) ta
R
c x x dx∫ ( )( )Φ < ∞ , to dlq bud\-
qkoho intervalu α β,[ ] � ( , )a b X tn( ) → X t( ) pry n → ∞ rivnomirno po t ∈
∈ α β,[ ] z imovirnistg odynycq ( X tn( ) zadano v (7.1), (7.2)).
Z teoremy.5.8 vyplyva[, wo dlq separabel\noho procesu z prostoru Expψ ( )Ω
na bud\-qkomu intervali X tn( ) → X t( ) pry n → ∞ rivnomirno pry t ∈ a b,[ ] z
imovirnistg odynycq, qkwo dlq f-vejvletu ϕ vykonu[t\sq umova S z funkci[g
Φ t( ) = 1
1 1t t b ts ψ ψ(– ) (– )ln ln ln( ) ( )
, s > 1.
Vysnovky. V roboti doslidΩeno vlastyvosti vypadkovyx procesiv X t( ) iz
prostoru Orliça vypadkovyx velyçyn. Zokrema, pobudovano taki funkci] c t( ) >
> 0, wo z imovirnistg odynycq sup
( )
( )t R
X t
c t∈
< ∞, ta znajdeno ocinky dlq jmovir-
nostej P sup
( )
( )t R
X t
c t∈
>
ε . Otrymani rezul\taty zastosovugt\sq do znaxodΩennq
umov rivnomirno] zbiΩnosti z imovirnistg odynycq vejvlet-rozkladiv cyx proce-
siv na obmeΩenyx intervalax. Peredbaça[t\sq vyvçennq rozpodiliv inßyx funk-
cionaliv vid procesiv X t( ) ta znaxodΩennq umov zbiΩnosti z imovirnistg ody-
nycq vejvlet-rozkladiv cyx procesiv v inßyx normax (zokrema, v L Tp( )).
1. Kozaçenko G. V., Perestgk M. M. Pro rivnomirnu zbiΩnist\ vejvlet-rozkladiv vypadkovyx
procesiv iz prostoriv Orliça vypadkovyx velyçyn // Ukr. mat. Ωurn. – 2007. – 59 , # 12. –
S..1647 – 1660.
2. Buldygin V. V., Kozachenko Yu. V. Metric characteristics of the random variables and random
processes. – Providence, Rhode Island: Amer. Math. Soc., 2000. – 257 p.
3. Daubechies I. Ten lecture on wavelets. – Philadelphia: Soc. Industrial and Appl. Math., 1992. –
324 p. (Dobeßy Y. Desqt\ lekcyj po vejvletam: per. s anhl. – M.; YΩevsk: RXD, 2001. –
463 s.)
4. Härdle W., Kerkyacharian G., Picard D., Tsybakov A. Wavelets, approximation and statistical
applications. – New York: Springer, 1998. – 265 p.
5. Walter G., Shen. X. Wavelets and other orthogonal systems. – London: Chapman and Hall / CRC
2000. – 370 p.
6. Kozaçenko G. V. Lekci] z vejvlet analizu. – Ky]v: TVIMS, 2004. – 147 s.
7. Kozachenko Yu. V., Perestyuk M. M., Vasylyk O. I. On uniform convergence of wavelet expansion
of ϕ-sub-Gaussian random processes // Random Oper. and Stochast. Equat. – 2006. – 14, # 3. –
P. 209 – 232.
8. Braverman M. Í. Ocenky dlq summ nezavysym¥x sluçajn¥x velyçyn // Ukr. mat. Ωurn. –
1991. – 43, # 2. – S. 173 – 178.
OderΩano 20.02.07,
pislq doopracgvannq — 21.05.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
|
| id | umjimathkievua-article-3194 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:38:02Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/9d/9013b9939fb181df9979c099fd7db39d.pdf |
| spelling | umjimathkievua-article-31942020-03-18T19:48:06Z On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. II Про рівномірну збіжність вейвлет-розкладів випадкових процесів із просторів Орліча випадкових величин. II Kozachenko, Yu. V. Perestyuk, M. M. Козаченко, Ю. В. Перестюк, М. М. We establish conditions under which wavelet expansions of random processes from Orlicz spaces of random variables converge uniformly with probability one on a bounded interval. Найдены условия, при которых вейвлет-разложения случайных процессов из пространств Орлича случайных величин сходятся равномерно с вероятностью единица на ограниченном интервале. Institute of Mathematics, NAS of Ukraine 2008-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3194 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 6 (2008); 759–775 Український математичний журнал; Том 60 № 6 (2008); 759–775 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3194/3134 https://umj.imath.kiev.ua/index.php/umj/article/view/3194/3135 Copyright (c) 2008 Kozachenko Yu. V.; Perestyuk M. M. |
| spellingShingle | Kozachenko, Yu. V. Perestyuk, M. M. Козаченко, Ю. В. Перестюк, М. М. On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. II |
| title | On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. II |
| title_alt | Про рівномірну збіжність вейвлет-розкладів випадкових процесів із просторів Орліча випадкових величин. II |
| title_full | On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. II |
| title_fullStr | On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. II |
| title_full_unstemmed | On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. II |
| title_short | On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. II |
| title_sort | on the uniform convergence of wavelet expansions of random processes from orlicz spaces of random variables. ii |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3194 |
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