Weakly Sub-Gaussian Random Elements in Banach Spaces
We give a survey of properties of weakly sub-Gaussian random elements in infinite-dimensional spaces. Some new results and examples are also given.
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| author | Vakhaniya, N. N. Kvaratskheliya, V. V. Tarieladze, V. I. Вахания, Н. Н. Кварацхелия, В. В. Тариеладзе, В. И. Вахания, Н. Н. Кварацхелия, В. В. Тариеладзе, В. И. |
| author_facet | Vakhaniya, N. N. Kvaratskheliya, V. V. Tarieladze, V. I. Вахания, Н. Н. Кварацхелия, В. В. Тариеладзе, В. И. Вахания, Н. Н. Кварацхелия, В. В. Тариеладзе, В. И. |
| author_sort | Vakhaniya, N. N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:02:00Z |
| description | We give a survey of properties of weakly sub-Gaussian random elements in infinite-dimensional spaces. Some new results and examples are also given. |
| first_indexed | 2026-03-24T02:46:58Z |
| format | Article |
| fulltext |
UDK 519.21
N. N. Vaxanyq, V. V. Kvaracxelyq, V. Y. Taryeladze
*
(Yn-t v¥çyslyt. matematyky AN Hruzyy, Tbylysy)
SLABO SUBHAUSSOVSKYE SLUÇAJNÁE ∏LEMENTÁ
V BANAXOVÁX PROSTRANSTVAX
A survey of properties of weakly sub-Gaussian random elements in infinite-dimensional spaces is given.
Some new results and examples are also presented.
Navedeno ohlqd vlastyvostej slabko subhaussovyx vypadkovyx elementiv u neskinçennovymirnyx
prostorax, a takoΩ dekil\ka novyx rezul\tativ ta prykladiv.
1. Vvedenye. Subhaussovskye sluçajn¥e velyçyn¥ vveden¥ Û. P. Kaxanom v
1960 h. v [1]. V πtoj rabote v kaçestve motyvacyy vvedenyq πtoho ponqtyq
nazvan yzvestn¥j cykl rabot Pπly y Zyhmunda 1932 h. (sm. [2]). Vposledstvyy
πto ponqtye b¥lo yspol\zovano (qvno yly neqvno) mnohymy avtoramy,
yzuçavßymy rehulqrnost\ traektoryj haussovskyx (y ne tol\ko haussovskyx)
sluçajn¥x processov v termynax povedenyq yx pryrawenyj [3 – 6].
Bolee detal\noe yzuçenye sobstvenno subhaussovskyx sluçajn¥x velyçyn
b¥lo vozobnovleno v rabotax V. V. Buld¥hyna, G. V. Kozaçenko [7] y
E.<Y.<Ostrovskoho [8]. Posle πtyx rabot voznyk ynteres k yzuçenyg analohov
ponqtyq subhaussovskoj sluçajnoj velyçyn¥ dlq sluçajn¥x πlementov so zna-
çenyqmy v obwyx banaxov¥x prostranstvax. Otmetym v πtoj svqzy rabot¥ Fu-
kuda [9] y Antonyny [10]. ∏ty rabot¥ vmeste s upomqnut¥my v¥ße posluΩyly
motyvacyej dlq dannoho yssledovanyq.
2. Predvarytel\n¥e svedenyq. Budem yspol\zovat\ ob¥çn¥e dlq veroqt-
nostnoj lyteratur¥ sohlaßenyq y oboznaçenyq. V çastnosty, budem sçytat\,
çto fyksyrovano nekotoroe, dostatoçno bohatoe, osnovnoe veroqtnostnoe
prostranstvo ( Ω, A, P ). S proyzvol\noj sluçajnoj velyçynoj f : Ω → R svq-
Ωem funkcyg Mf : R → [ 0, ∞ ]:
Mf ( t ) = Eetf
, t ∈ R,
hde E — symvol matematyçeskoho oΩydanyq.
Ysxodn¥m dlq posledugweho yzloΩenyq qvlqetsq sledugwee svojstvo
centryrovann¥x haussovskyx sluçajn¥x velyçyn: esly g — centryrovannaq (t.
e. E g = 0 ) haussovskaq sluçajnaq velyçyna, to
Mg ( t ) = et g2 2 2E /
, t ∈ R.
Napomnym, çto haussovskaq centryrovannaq sluçajnaq velyçyna g s E g
2 = 1
naz¥vaetsq standartnoj haussovskoj sluçajnoj velyçynoj.
Sleduq [1], budem hovoryt\, çto sluçajnaq velyçyna f : Ω → R qvlqetsq
subhaussovskoj, esly dlq nekotoroho a, 0 ≤ a < ∞, v¥polnqetsq neravenstvo
Mf ( t ) ≤ et a2 2 2/
, t ∈ R.
Ymeq v vydu otmeçenn¥e svojstva centryrovann¥x haussovskyx sluçajn¥x
velyçyn, moΩno skazat\, çto sluçajnaq velyçyna f : Ω → R qvlqetsq subhaus-
Çastyçno podderΩan hrantom MGYT BFM2003-05878.
© N. N. VAXANYQ, V. V. KVARACXELYQ, V. Y. TARYELADZE, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9 1187
1188 N. N. VAXANYQ, V. V. KVARACXELYQ, V. Y. TARYELADZE
sovskoj v tom y tol\ko v tom sluçae, esly najdetsq takaq centryrovannaq haus-
sovskaq sluçajnaq velyçyna g : Ω → R, çto
Mf ( t ) ≤ Mg ( t ), t ∈ R.
∏tym nablgdenyem moΩno obæqsnyt\ proysxoΩdenye nazvanyq subhaussovskaq
sluçajnaq velyçyna.
V dal\nejßem dlq proyzvol\noj subhaussovskoj sluçajnoj velyçyn¥ f
çerez τ ( f ) budem oboznaçat\ çyslo, opredelqemoe ravenstvom
τ ( f ) =
inf , : ( )a M t e tf
t a∈ ∞[ ) ≤ ∈{ }/0
2 2 2
dlq vsex R .
Çyslo τ ( f ) budem naz¥vat\ subhaussovskym standartom f (sm. [7, 11]; v [1] ys-
pol\zuetsq termyn haussovskoe otklonenye).
PredloΩenye 2.1. Pust\ f : Ω → R — sluçajnaq velyçyna.
A. Esly r > 0 — takoe çyslo, çto Mf ( r ) + Mf ( – r ) < ∞, to:
a1
) E f t fn exp{ }( ) ≤
n
r t
M rn f
!
( )
( )
−
dlq vsex t ∈ ( – r, r ) y n ∈ N;
a2
) 1 +
n
n nt f
n=
∞
∑
1
E
!
= M tf ( ) ≤ M rf ( ) < ∞
y
Mf ( t ) = 1 +
n
n nt f
n=
∞
∑
1
E
!
, t ∈ ( – r, r );
a3
) Mf ( ⋅ ) qvlqetsq analytyçeskoj funkcyej na ( – r, r ) y ee proyzvodn¥e
M f
n( ) opredelqgtsq ravenstvom
M tf
n( )( ) =
1
n
f t fn
!
expE { }( ) , t ∈ ( – r, r ), n ∈ N.
B . Esly f — subhaussovskaq sluçajnaq velyçyna s subhaussovskym
standartom τ ( f ), to:
b1
) Mf ( t ) ≤ et f2 2 2τ ( ) /
, t ∈ R;
b2
) M tf ( ) ≤ 2
2 2 2et fτ ( ) /
, t ∈ R;
b3
) 1 +
n
n nt f
n=
∞
∑
0
E
!
= M tf ( ) ≤ 2 et f2 2 2τ ( ) / < ∞, t ∈ R;
b4
) E f = 0, E f 2 1 2( ) / ≤ τ ( f );
b5
) esly τ ( f ) = E f 2 1 2( ) / , to E f 3 = 0 y E f 4 ≤ 3 2 2
E f( ) ;
b6
) Eetf 2
≤
1
1 2 2− t fτ ( )
, t ∈ 0
1
2 2,
( )τ f
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SLABO SUBHAUSSOVSKYE SLUÇAJNÁE ∏LEMENTÁ V BANAXOVÁX … 1189
Dokazatel\stvo. Spravedlyvost\ utverΩdenyj a1
) – a3
), b1
) – b3
) lehko
moΩno proveryt\. Yspol\zuq a2
) y b1
), zapys¥vaem
n
n nt f
n=
∞
∑
1
E
!
≤
n
n n
t
n
f
=
∞
∑
1
2 2
2!
( )τ
,
y tak kak πto neravenstvo v¥polnqetsq dlq kaΩdoho t ∈ R, yz neho moΩno v¥-
vesty utverΩdenyq b4
) y b5
). Dlq dokazatel\stva utverΩdenyq b6
) poloΩym,
çto t ∈ 0
1
2 2,
( )τ f
y γt — symmetryçeskaq haussovskaq mera v R s dyspersy-
ej 2
t. Tohda dlq vsex ω ∈ Ω
etf 2 ( )ω =
R
∫ e d xxf
t
( ) ( )ω γ .
Otsgda s uçetom utverΩdenyq b1
) y teorem¥ Fubyny poluçaem
Eetf 2
=
R
E∫ e d xx f
tγ ( ) ≤
R
∫ /e d xx f
t
2 2 2τ γ( ) ( ) =
1
1 2 2− t fτ ( )
.
UtverΩdenyq b5
) y b6
) b¥ly dokazan¥ ranee v [11].
Sleduq [7], budem hovoryt\, çto subhaussovskaq sluçajnaq velyçyna f qvlq-
etsq stroho subhaussovskoj, esly τ ( f ) = E f 2 1 2( ) / .
Oboznaçym çerez G ( Ω, A, P ), yly G ( Ω ), mnoΩestvo vsex centryrovann¥x
haussovskyx sluçajn¥x velyçyn g : Ω → R ; çerez S S G ( Ω , A , P ), yly
S S G ( Ω ), mnoΩestvo vsex stroho subhaussovskyx sluçajn¥x velyçyn f : Ω → R;
çerez S G ( Ω, A, P ), yly S G ( Ω ), mnoΩestvo vsex subhaussovskyx sluçajn¥x
velyçyn f : Ω → R.
Lehko vydet\, çto G ( Ω ) ⊂ S S G ( Ω ) ⊂ S G ( Ω ).
V [7, 11] ustanovleno, çto:
a) S G ( Ω ) qvlqetsq banaxov¥m prostranstvom otnosytel\no norm¥ τ ( ⋅ );
dlq f ∈ S G ( Ω ) çyslo τ ( f ) naz¥vagt takΩe subhaussovskoj normoj f ;
b) S G ( Ω ) sovpadaet s podprostranstvom prostranstva Orlyça LΦ ( Ω ), hde
Φ ( x ) = ex2
– 1.
Odnako mnoΩestva G ( Ω ) y S S G ( Ω ) ne qvlqgtsq vektorn¥my podpros-
transtvamy S G ( Ω ); bolee toho, summa dvux centryrovann¥x ortohonal\n¥x ha-
ussovskyx sluçajn¥x velyçyn moΩet ne b¥t\ stroho subhaussovskoj sluçajnoj
velyçynoj (sm. prymer 3.9 nyΩe).
Dlq al\ternatyvnoj xarakteryzacyy subhaussovosty nam ponadobqtsq dve
obwye lemm¥. Predvarytel\no vvedem sledugwye oboznaçenyq. Dlq proyz-
vol\noj sluçajnoj velyçyn¥ f oboznaçym
ϑ1 ( f ) = sup
( )!!n
n
n
n
f
∈ −
/
N
E
1
2 1
2
1 2
, ϑ2 ( f ) = sup
n
n
n
n
n
f
∈
/
N
E
1 2
1 2
,
hde
( 2 n – 1 ) !! = 3 ⋅ 5 … ( 2 n – 1 ) =
( )!
!
2
2
n
nn , n = 1, 2, … .
Dlq yndykatora mnoΩestva A yspol\zuem ob¥çnoe oboznaçenye 1A .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1190 N. N. VAXANYQ, V. V. KVARACXELYQ, V. Y. TARYELADZE
Lemma 2.1. Pust\ f : Ω → R — sluçajnaq velyçyna. Tohda:
A) ϑ2 ( f ) ≤ ϑ1 ( f ) ≤
e
f
2 2ϑ ( );
B) esly t > 0, to E f p ≤
p
e
t e
p
p t f
−
E , p > 0;
C) esly f — takaq sluçajnaq velyçyna, çto pry nekotor¥x a ≥ 0 y b ≥ 1
Eet f ≤ bea t2 2 2/
, t ≥ 0, (2.1)
to spravedlyv¥ sledugwye sootnoßenyq:
c1
) E | f | p ≤ b
p
e
a
p
p
/2
, p > 0;
c2
) ϑ2 ( f ) ≤
2 1 2b
e
a
/
;
c3
) ϑ1 ( f ) ≤ b a1 2/
;
c4
) P f x>[ ] ≤ be x a− /2 22( )
, x ≥ 0;
c5
) E e f
f t
ε 2
1 ≥[ ]( ) ≤
b
a
e
a t
1 2 2
1 2 2 2
−
−( )/
ε
ε ( )
, ε <
1
2 2a
, t ≥ 0.
Dokazatel\stvo. UtverΩdenye A ) sleduet yz sootnoßenyj 2n e n/( ) ≤
≤ ( 2 n – 1 ) !! ≤ n
n
, n = 1, 2, … .
Spravedlyvost\ utverΩdenyq B) sleduet yz neravenstva
x
p ≤
p
e
e
p
x
, x > 0, p > 0. (2.2)
S uçetom utverΩdenyq B) y neravenstva (2.1) dlq lgboho t > 0 ymeem E | f | p ≤
≤ p e t bep p t a/( ) − /2 2 2
. Polahaq v πtom neravenstve t = p a/ , poluçaem
sootnoßenye c1
).
Sootnoßenye c2
) sleduet yz sootnoßenyq c1
) y opredelenyq ϑ 2 , sootnoße-
nye c3
) — yz sootnoßenyj c2
) y A).
V sylu neravenstva Markova s uçetom (2.1) dlq lgb¥x t, x > 0 ymeem
P f x>[ ] ≤ e et x t f−
E ≤ be t x t a− + /2 2 2
. Polahaq v πtom neravenstve t = x / a
2
,
poluçaem sootnoßenye c4
).
Po formule yntehryrovanyq po çastqm y s uçetom sootnoßenyq c4
) pry ε <
< 1 / ( 2 a
2
) y t > 0 ymeem
E e f
f t
ε 2
1 ≥[ ]( ) = e f ttε 2
P ≥[ ] +
+
t
xf x de
∞
∫ >[ ]P
ε 2
≤
1
1 2 2
1 2 2 2
−
−( )/
a
e
a t
ε
ε ( )
,
çto dokaz¥vaet spravedlyvost\ sootnoßenyq c5
).
Sledstvye 2.1. Pust\ f — subhaussovskaq sluçajnaq velyçyna s subhaus-
sovskym standartom τ ( f ). Tohda:
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SLABO SUBHAUSSOVSKYE SLUÇAJNÁE ∏LEMENTÁ V BANAXOVÁX … 1191
a) E f p p( ) /1
≤ βp τ ( f ) dlq kaΩdoho p > 0, hde
βp =
1 2
2 21 1 2
, ,
, ;
esly
esly
p
p e pp
≤
( ) < < ∞
/ //
b) ϑ2 ( f ) ≤
2
e
f
τ( );
c) ϑ1 ( f ) ≤ 2 τ( )f ;
d) P f x>[ ] ≤ 2
2 22
e
x f− ( )/ τ ( )
, x ≥ 0;
e) E e f
f t
ε 2
1 ≥[ ]( ) ≤
2
1 2 2
1 2 2 2
−
− ( )( )/
τ ε
ε τ
( )
( )
f
e
f t
, ε <
1
2 2τ ( )f
, t ≥ 0.
Dokazatel\stvo. UtverΩdenye a) v sluçae p ≤ 2 sleduet yz predloΩenyq
2.1, b4
). Ostal\n¥e utverΩdenyq sledugt yz predloΩenyq 2.1, b2
) y lemm¥ 2.1,
C).
Zameçanye 2.1. Toçn¥e znaçenyq konstant v sluçae sledstvyj 2.1, a) y 2.1,
b) nam neyzvestn¥. Esly g — centryrovannaq haussovskaq sluçajnaq velyçyna
s σ = Eg2 1 2( ) / , to ymegt mesto sledugwye toçn¥e utverΩdenyq:
a) E g p p( ) /1
= π− /
/+
1 2
1
2
1
2
( )p
ppΓ = σ, p > 0;
b) ϑ2 ( g ) = ϑ1 ( g ) = σ;
c) P g x>[ ] ≤ e x− /2 22( )σ , x ≥ 0;
d)
2 1
1
2 22
π σ
σ
+ /
− /
x
e x ( ) ≤ P g x≥[ ] , x ≥ 0 (sm. [12], predloΩenye 2.17);
e) Ee gε 2
=
( ) , ,
, .
1 2
1
2
1
2
2 1 2
2
2
− <
∞ ≥
− /εσ ε
σ
ε
σ
esly
esly
Zameçanye 2.2. Yspol\zuq sledstvye 2.1, d) y zameçanye 2.1, d), moΩno za-
klgçyt\, çto dlq subhaussovskoj sluçajnoj velyçyn¥ f y standartnoj haus-
sovskoj sluçajnoj velyçyn¥ g v¥polnqetsq neravenstvo
P f x>[ ] ≤ P g
x
f
>
2 τ( )
, x ≥ 2 τ ( f )
(sr. [11, c. 15], dokazatel\stvo teorem¥ 1.1.1).
Lemma 2.2. Pust\ f — nekotoraq sluçajnaq velyçyna. Sledugwye utver-
Ωdenyq πkvyvalentn¥:
i) ϑ2 ( f ) < ∞;
ii) ϑ1 ( f ) < ∞;
iii) suwestvuet ε > 0 takoe, çto Ee fε 2
< ∞;
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1192 N. N. VAXANYQ, V. V. KVARACXELYQ, V. Y. TARYELADZE
iv) suwestvuet ε > 0 takoe, çto P f x>[ ] = O e x−( )ε 2
, x → ∞;
v) suwestvugt ε > 0 y c ≥ 1 takye, çto P f x>[ ] ≤ ce x−ε 2
, x > 0;
vi) suwestvugt a ≥ 0 y b ≥ 1 takye, çto Eet f ≤ bet a2 2 2/
, t ≥ 0;
vii) suwestvugt a ≥ 0 y b ≥ 0 takye, çto Eetf ≤ bet a2 2 2/
, t ∈ R.
Krome toho,
a) ϑ2 ( f ) < ∞ ⇒ Ee fε 2
≤
1
1 2
2− ε ϑe f( )
dlq vsex ε <
1
2
2e fϑ ( )
;
b) esly ε > 0 y c ≥ 1 v¥bran¥ tak, çto verno v), to
Ee fε1
2
≤
cε
ε ε
1
1−
dlq vsex ε1 < ε;
s) esly ε > 0 v¥brano tak, çto verno iii), to
Eet f ≤ e et f2 24/( )ε ε
E dlq vsex t ≥ 0;
d) esly ε > 0 v¥brano tak, çto verno iii), to
ϑ2 ( f ) ≤
1 12 1 2
e
e f
E
ε
ε( ) / ;
e) esly ε > 0 y c ≥ 1 v¥bran¥ sohlasno v), to
Eet f ≤
c
etε
ε ε
ε1
1
42
1
−
/( )
dlq vsex ε1 < ε y t ≥ 0.
Dokazatel\stvo. ∏kvyvalentnost\ i) ⇔ ii) v¥tekaet yz lemm¥ 2.1, A).
Ymplykacyq i) ⇒ iii) sleduet yz sformulyrovannoho vo vtoroj çasty πtoj lem-
m¥ utverΩdenyq a). Ona lehko v¥tekaet yz neravenstv E f n2 ≤ n fn nϑ2
2( )( ) ,
n nn / ! ≤ e
n
, n = 1, 2, … , v sylu kotor¥x
Ee fε 2
= 1 +
n
n nf
n=
∞
∑
1
2ε E
!
≤ 1 +
n
n
e f
=
∞
∑ ( )
1
2
2ε ϑ ( ) =
1
1 2
2− ε ϑe f( )
.
Ymplykacyy iii) ⇒ iv) ⇒ v) oçevydn¥. Ymplykacyq v) ⇒ iii) soderΩytsq v ut-
verΩdenyy b), a utverΩdenye b) dokaz¥vaetsq po formule yntehryrovanyq po
çastqm: esly ε1 < ε, to
Ee fε1
2
= 2 1
1
2
ε ε
R
P
+
∫ >[ ]f x x e d xx ≤ 2 1
1
2
ε ε εc e x d xx
R+
∫ −( ) =
cε
ε ε
1
1−
.
Ymplykacyq iii) ⇒ vi) soderΩytsq v utverΩdenyy s), kotoroe sleduet yz oçe-
vydnoho neravenstva
t x ≤
t2
4ε
+ ε x
2
dlq vsex t, x ∈ R y ε > 0 .
Ymplykacyy vi) ⇒ ii) y c) ⇒ d) spravedlyv¥ sohlasno lemme 2.1, pervaq sleduet
yz sootnoßenyq c3
), vtoraq — yz sootnoßenyq c2
), utverΩdenye e) sleduet yz
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SLABO SUBHAUSSOVSKYE SLUÇAJNÁE ∏LEMENTÁ V BANAXOVÁX … 1193
utverΩdenyj c) y b); ymplykacyq vi) ⇒ vii) oçevydna. Ymplykacyq vii) ⇒ vi)
sleduet yz oçevydnoho neravenstva e x ≤ e
x + e
–
x
.
Sledstvye 2.2 (sr. [7], lemma 2, y [11], teorema 1.1.3). Pust\ f — sluçaj-
naq velyçyna, dlq kotoroj ϑ1 ( f ) < ∞. Tohda:
a) esly E f n2 1− = 0, n = 1, 2, … (v çastnosty, esly f — symmetryçnaq
sluçajnaq velyçyna), to f — subhaussovskaq sluçajnaq velyçyna y
τ ( f ) ≤ ϑ1 ( f ) ≤
e
f
2 2ϑ ( );
b) esly E f n2 1− = 0, n = 1, 2, … (v çastnosty, esly f — symmetryçnaq
sluçajnaq velyçyna) y ϑ1 ( f ) = E f 2 1 2( ) / , t o f — stroho subhaussovskaq slu-
çajnaq velyçyna;
s) Eetf ≤ t E f + e t f2 05 22
1
2, ( )ϑ /
, t ∈ R;
d) esly E f = 0, to f — subhaussovskaq sluçajnaq velyçyna y
τ ( f ) ≤ 2 054
1, ( )ϑ f ≤ 2 05
2
4
2, ( )
e
fϑ .
Dokazatel\stvo. Sohlasno lemme 2.2 ymeem 1 +
n
n nt f
n=
∞
∑
1
E
!
= Ee t f <
< ∞, y poπtomu
Eetf = 1 +
n
n nt f
n=
∞
∑
1
E
!
, (2.3)
hde rqd sxodytsq absolgtno.
a) Esly E f n2 1− = 0, n = 1, 2, … , to ravenstvo (2.3) prynymaet vyd
Eetf = 1 +
n
n nt f
n=
∞
∑
1
2 2
2
E
( )!
. (2.4)
Yz (2.4) y sootnoßenyj E f n2 ≤
( )!
!
( )
2
2 1
2n
n
fn
n
⋅
( )ϑ poluçaem neravenstvo τ ( f ) ≤
≤ ϑ1 ( f ).
UtverΩdenye b) sleduet yz utverΩdenyq a).
s) Pust\ x ∈ R y q — proyzvol\noe poloΩytel\noe çyslo. Vvydu toho, çto
x ≤ 1 / ( 4 q ) + q x
2
, dlq kaΩdoho natural\noho n ymeem x
2
n
+
1 ≤ x
2
n
/ ( 4 q ) +
+ q x
2
n
+
2
. Poπtomu dlq vsex q > 0 ymeem
e
x = 1 + x +
n
nx
n=
∞
∑
1
2
2( )!
+
n
nx
n=
∞ +
∑ +1
2 1
2 1( )!
≤ 1 + x + 1
1
12 2
2
+
q
x
+
+
n
n
q n
nq
x
n=
∞
∑ +
+
+
2
2
1
1
4 2 1
2
2( ) ( )!
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1194 N. N. VAXANYQ, V. V. KVARACXELYQ, V. Y. TARYELADZE
Sootnoßenye 1
1
4 2 1
2+
+
+
q n
nq
( )
≤ 1
1
20
4
2
+ +
/
q
q
n
spravedlyvo dlq vsex q >
> 0, esly n ≥ 2. Vmeste s tem neravenstvo 1 +
1
12q
≤ 1
1
20
4
1 2
+ +
/
q
q ymeet
mesto dlq vsex q naçynaq s nekotoroho poloΩytel\noho çysla (naprymer,
1 / 5 ). Sledovatel\no, dlq takyx znaçenyj q poluçaem neravenstvo
e
x ≤ 1 + x +
n
n n
q
q
x
n=
∞
∑ + +
/
1
2 2
1
1
20
4
2( )!
.
Polahaq teper\ q = 1 / 5, poluçaem ocenku
e
x ≤ 1 + x +
n
n
nx
n=
∞
∑ /
1
2
2
2 05
2
( , )
( )!
= + ( )( )x xch 2 054 , , x ∈ R. (2.5)
Polahaq v πtom neravenstve t f vmesto x y yntehryruq, ymeem
Eetf ≤ 1 + t E f +
n
n
nf
n=
∞
∑ /
1
2
2
2 05
2
( , )
( )!
E
, (2.6)
y dokazatel\stvo utverΩdenyq a) zaverßaetsq prymenenyem (2.6) s uçetom nera-
venstv E f n2 ≤
( )!
!
( )
2
2 1
2n
n
fn
n
⋅
( )ϑ , n = 1, 2, … .
UtverΩdenye d) sleduet yz utverΩdenyq s).
Zameçanye 2.3. Sledstvye 2.2, d), po suwestvu, ymeetsq v [1] (predloΩenyq
7 y 8). Ono poluçeno takΩe v [7] s konstantamy e9 16 2/ / (v symmetryçnom
sluçae) y ( , )3 1 21 4 9 16/ / /e (v centryrovannom sluçae). PredloΩennaq zdes\
formulyrovka utverΩdenyq a) v sledstvyy 2.2 predstavlqetsq nam novoj. Pry
dokazatel\stve m¥ sledovaly sxeme [7], odnako v otlyçye ot [7] predvarytel\no
obosnovaly ravenstvo (2.3) y v¥delyly çysto funkcyonal\noe neravenstvo
(2.5).
PredloΩenye 2.2. 1. Sluçajnaq velyçyna f : Ω → R qvlqetsq subhaus-
sovskoj v tom y tol\ko v tom sluçae, esly ϑ2 ( f ) < ∞ y E f = 0.
2. ϑ2 ( ⋅ ) qvlqetsq normoj na S G ( Ω ) y
( , ) ( )2 05
41 4− /
e
fτ ≤ ϑ2 ( f ) ≤
2
e
fτ( ) , f ∈ S G ( Ω ).
3. ϑ1 ( ⋅ ) qvlqetsq normoj na S G ( Ω ) y
( , ) ( )2 05 1 4− / τ f ≤ ϑ1 ( f ) ≤ 2 τ( )f , f ∈ S G ( Ω ).
Dokazatel\stvo. Pervoe utverΩdenye sleduet yz lemm¥ 2.1, A) y sled-
stvyq 2.2, d). Vtoroe y tret\e utverΩdenyq takΩe lehko dokaz¥vagtsq: svoj-
stva norm¥ funkcyonalov ϑ2 y ϑ1 sledugt yz yx opredelenyj, a sootvetstvu-
gwye ocenky v¥tekagt yz lemm¥ 2.1, A) y sledstvyj 2.2, d), 2.1, b) y 2.1, c).
Poluçenn¥e rezul\tat¥ moΩno sformulyrovat\ v vyde sledugwej teorem¥.
Teorema 2.1. Pust\ f : Ω → R — sluçajnaq velyçyna. Sledugwye utverΩ-
denyq πkvyvalentn¥:
i) f — subhaussovskaq sluçajnaq velyçyna;
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SLABO SUBHAUSSOVSKYE SLUÇAJNÁE ∏LEMENTÁ V BANAXOVÁX … 1195
ii) ϑ2 ( f ) < ∞ y E f = 0 ;
iii) suwestvuet ε > 0 takoe, çto Ee fε 2
< ∞ y E f = 0;
iv) suwestvugt ε > 0 y c ≥ 1 takye, çto dlq vsex x ≥ 0 ymeem
P f x>[ ] ≤ c x−ε 2
y E f = 0;
v) suwestvuet ε > 0 takoe, çto P f x>[ ] = O e x−( )ε 2
pry x → ∞ y
E f = 0;
vi) suwestvugt a ≥ 0 y b ≥ 1 takye, çto dlq vsex t ≥ 0 ymeem Eet f ≤
≤ bet a2 2 2/ y E f = 0;
vii) suwestvugt a ≥ 0 y b ≥ 0 takye, çto dlq vsex t ∈ R ymeem Eet f ≤
≤ bet a2 2 2/ y E f = 0.
V samom dele, πkvyvalentnost\ iii) ⇔ iv) ⇔ v) oçevydna, i) ⇔ ii) sleduet yz
pervoho utverΩdenyq predloΩenyq 2.2. ∏kvyvalentnosty ii) ⇔ iii) y ii) ⇔
⇔ vi)<⇔ vii) ymegtsq sohlasno lemme 2.2.
Zameçanye 2.4. ∏kvyvalentnost\ i) ⇔ ii) ⇔ v) formulyruetsq v [13, c. 108]
(hl. VI, upr. 10), πkvyvalentnost\ i) ⇔ v i ) otmeçaetsq zdes\, po-vydymomu,
vperv¥e.
3. Prymer¥ subhaussovskyx sluçajn¥x velyçyn. S pomow\g xaraktery-
zacyonnoj teorem¥ 2.1 lehko pryvesty prymer¥ subhaussovskyx sluçajn¥x ve-
lyçyn. Zdes\ m¥ ukaΩem nekotor¥e yz nyx, poputno otmeçaq neobxodym¥e dlq
dal\nejßeho dopolnytel\n¥e svojstva.
Prymer 3.1. Centryrovannaq haussovskaq sluçajnaq velyçyna g qvlqetsq
stroho subhaussovskoj, τ ( g ) = Eg2 1 2( ) / = ϑ1 ( g ) = ϑ2 ( g ) = 1.
Prymer 3.2. Symmetryçnaq bernullyevskaq (rademaxerovskaq) sluçajnaq
velyçyna ε. V πtom sluçae P ε =[ ]1 = P ε = −[ ]1 = 1 / 2 y
Eet ε =
e et t+ −
2
= ch t ≤ et2 2/
, t ∈ R,
ε — stroho subhaussovskaq sluçajnaq velyçyna y τ ( ε ) = Eε2 1 2( ) / = ϑ1 ( ε ) =
= ϑ2 ( ε ) = 1.
Prymer 3.3. Ravnomerno raspredelennaq na yntervale [ – 1, 1 ] sluçajnaq
velyçyna f. V πtom sluçae
E f n2 =
1
2 1n +
, Eetf =
e e
t
t t− −
2
=
sh t
t
, n ∈ N, t ∈ R.
Prqmoj analyz pokaz¥vaet, çto
sup ( )
n
nn n
∈
− −/ /+
N
1 2 1 22 1 =
1
3
,
sh t
t
≤ et2 6/
, t ∈ R.
Takym obrazom, f — stroho subhaussovskaq sluçajnaq velyçyna s τ ( f ) =
= ϑ2 ( f ) = 1 3/ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1196 N. N. VAXANYQ, V. V. KVARACXELYQ, V. Y. TARYELADZE
Prymer 3.4. Pust\ f — sluçajnaq velyçyna, 0 < a < ∞, P f a≤[ ] = 1,
E f = 0. Tohda f qvlqetsq subhaussovskoj y τ ( f ) ≤ a. Subhaussovost\ oçevydna
v vydu teorem¥ 2.1; ocenka τ ( f ) ≤ a poluçena v [8] (v predpoloΩenyy symmet-
ryçnosty f ona poluçaetsq lehko: Eetf ≤ E e etf tf+( )( )− /2 ≤ Eet f2 2 2/ ≤ et a2 2 2/
,
t ∈ R ).
Sledugwyj prymer pokaz¥vaet, çto ohranyçennaq centryrovannaq (daΩe
dvuxznaçnaq) sluçajnaq velyçyna moΩet ne b¥t\ stroho subhaussovskoj.
Prymer 3.5. Pust\ α, 0 < α < 1, β = 1 – α y fα — sluçajnaq velyçyna s
raspredelenyem P fα β=[ ] = α, P fα α= −[ ] = β. Ymeem:
a) fα — subhaussovskaq sluçajnaq velyçyna y τ ( fα ) ≤ max ( α, 1 – α );
b) fα — stroho subhaussovskaq sluçajnaq velyçyna ⇔ α = 1 / 2.
V samom dele, utverΩdenye a) sleduet yz prymera 3.4. Dalee, oçevydno, çto
pry α = 1 / 2 sluçajnaq velyçyna fα qvlqetsq stroho subhaussovskoj. Obrat-
no, pust\ fα — stroho subhaussovskaq sluçajnaq velyçyna. Tohda E fα
3 = α β
3 –
– β α
3
y sohlasno predloΩenyg 2.1, b5
) α = 1 / 2 y utverΩdenye b) dokazano.
Prymer 3.6. Pust\ fα — sluçajnaq velyçyna s raspredelenyem
P fα =[ ]1 = P fα = −[ ]1 =
α
2
, P fα =[ ]0 = 1 – α,
hde 0 < α ≤ 1. Spravedlyv¥ sledugwye utverΩdenyq:
a) E f p
α = α dlq vsex p > 0;
b) Eetfα = 1 + α ( ch t – 1 ) = 1 + α
n
nt
n=
∞
∑
1
2
2( )!
, t ∈ R;
s) α ≤ τ ( fα ) ≤ 1;
d) α <
1
3
⇒ τ ( fα ) > α , t. e. fα ne qvlqetsq stroho subhaussovskoj slu-
çajnoj velyçynoj;
e) α ≥
1
3
⇒ τ ( fα ) = α , t. e. fα qvlqetsq stroho subhaussovskoj slu-
çajnoj velyçynoj;
f) α ≥
1
4
⇒ ϑ2 ( fα ) = α , t. e. ϑ2 ( fα ) = f Lα 2
;
g) esly
1
4
≤ α <
1
3
, to α = ϑ2 ( fα ) < τ ( fα );
h) esly 0 < α ≤
1
e
, to 2
1 1
e ln
α
−
≤ ϑ α2
2( )f ≤ e ln
1 1
α
−
;
i) lim
( )
α
α
α
τ
→0
2
f
f L
= ∞.
V samom dele, utverΩdenyq a) y b ) oçevydn¥, utverΩdenye s) sleduet yz
prymera 3.5 y yz predloΩenyq 2.1, b4
), utverΩdenye d) — yz predloΩenyq 2.1,
b5
). Dejstvytel\no, esly fα — stroho subhaussovskaq sluçajnaq velyçyna, to
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SLABO SUBHAUSSOVSKYE SLUÇAJNÁE ∏LEMENTÁ V BANAXOVÁX … 1197
α = E fα
4 ≤ 3 2 2
E fα( ) = 3 α
2
, t. e. α ≥ 1 / 3. DokaΩem utverΩdenye e). Lehko vy-
det\ (naprymer, po yndukcyy), çto 2
n
n ! / ( 2 n ) ! ≤ ( 1 / 3 )
n
–
1
. Sledovatel\no, po-
skol\ku α ≥ 1 / 3, to 2
n
n ! / ( 2 n ) ! ≤ α
n
–
1
, t. e. α / ( 2 n ) ! ≤ α
n
/ 2
n
n ! dlq kaΩdo-
ho natural\noho n. Poslednee neravenstvo s uçetom ravenstva b) daet
Eetfα ≤ 1 +
n
n
n
n
n
t
=
∞
∑
1
2
2
α
!
= et2 2α /
, t ∈ R,
t. e. τ ( fα ) ≤ α . Dlq dokazatel\stva utverΩdenyq f) zametym, çto po oprede-
lenyg ϑ2 ( f ) = supn
nn≥ /( ) /
1
1 21 α . Lehko vydet\, çto dlq kaΩdoho natural\no-
ho n yz v¥polnenyq uslovyq α ≥ 1 / 4 v¥tekaet v¥polnenye uslovyq α ≥
≥ α
1/n / n, t. e. ϑ2 ( fα ) = α . UtverΩdenye g) est\ sledstvye utverΩdenyj f)
y<<d). Kak netrudno proveryt\, pry 0 < α ≤ 1 / e dlq kaΩdoho natural\noho
n<<<v¥polnen¥ sledugwye neravenstva: 2 1 1e ln /( )( )−α ≤ supn
n n≥
/ /( )1
1α ≤
≤ e ln 1 1/( )( )−α , çto dokaz¥vaet spravedlyvost\ utverΩdenyq h). Nakonec,
utverΩdenye i) sleduet yz utverΩdenyj a) y h) y yz vtoroho utverΩdenyq pred-
loΩenyq 2.2.
Zameçanye 3.1. Prymer 3.6 po-svoemu prymeçatelen: raspredelenye fα est\
smes\ δ δ( )− +( ) /1 1 2 y δ0 . Oba πty raspredelenyq — stroho subhaussovskye,
odnako, sohlasno utverΩdenyg e) pry α < 1 / 3 yx smes\ ne qvlqetsq takovoj.
Dlq motyvacyy sledugweho prymera pryvedem snaçala odno utverΩdenye.
PredloΩenye 3.1. Pust\ f1 y f2 — nezavysym¥e subhaussovskye sluçaj-
n¥e velyçyn¥. Tohda:
a) τ
2
( f1 + f2 ) ≤ τ
2
( f1 ) + τ
2
( f2 ) [1];
b) τ
2
( f1 ) ≤ τ
2
( f1 + f2 ) [7];
c) esly f1 , f2 — stroho subhaussovskye sluçajn¥e velyçyn¥, to f1 + f2 —
stroho subhaussovskaq sluçajnaq velyçyna [7].
Dokazatel\stvo. UtverΩdenye a) sleduet yz ravenstva Eet f f( )1 2+ =
= E Ee etf tf1 2
, t ∈ R.
Dalee, pust\ t ∈ R. Yz v¥puklosty x � e
t
x
y yz E f2 = 0 sleduet (sm.
lemmu 5.3.4 v [14])
Eetf1 ≤ Eet f f( )1 2+
.
∏to neravenstvo, v sylu proyzvol\nosty t y predloΩenyq, 2.1 b1), vleçet
utverΩdenye b).
UtverΩdenye s) sleduet yz utverΩdenyq a), tak kak τ
2
( f1 ) + τ
2
( f2 ) = E f1
2 +
+ E f2
2 = E( )f f1 2
2+ . ∏to ravenstvo, utverΩdenye a) y predloΩenye 2.1, b4)
dagt τ
2
( f1 + f2 ) = E( )f f1 2
2+ .
Sledugwyj prymer pokaz¥vaet, çto predloΩenyq 3.1, a) y 3.1, s) ne vern¥
bez predpoloΩenyq nezavysymosty, daΩe esly f1 y f2 — standartn¥e haussov-
skye sluçajn¥e velyçyn¥.
Prymer 3.7. Pust\ α, 0 < α < 1, β = 1 – α. Rassmotrym sluçajn¥j vektor
( g1, g2 ) v R
2
s xarakterystyçeskoj funkcyej
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1198 N. N. VAXANYQ, V. V. KVARACXELYQ, V. Y. TARYELADZE
Eei t g t g( )1 1 2 2+ = αe t t− + /( )1 2
2 2 + βe t t− − /( )1 2
2 2
, t1 , t2 ∈ R.
Tohda:
a) g1 , g2 — standartn¥e haussovskye sluçajn¥e velyçyn¥;
b) E g1 g2 = α – β; v çastnosty, esly α = 1 / 2, to E g1 g2 = 0;
c) ζ = g1 + g2 — subhaussovskaq sluçajnaq velyçyna y τ ( ζ ) ≤ 2;
d) ζ = g1 + g2 ne qvlqetsq stroho subhaussovskoj.
V samom dele, utverΩdenyq a) y b) oçevydn¥. DokaΩem utverΩdenyq s) y d).
Dlq kaΩdoho t ∈ R ymeem Eeitζ = αe t−2 2
+ β. Otsgda poluçaem E ζ
2 = 4 α. Yz
neravenstva
αe t2 2
+ β ≤ αe t2 2
+ βe t2 2
= e t2 2
, t ∈ R,
sleduet τ ( ζ ) ≤ 2, a yz neravenstva
αe t2 2
+ β = 1 + α
n
n
t
n=
∞
∑ ( )
1
22
!
> 1 +
n
n
t
n=
∞
∑ ( )
1
22α
!
= e t2 2α
, t ∈ R \ { 0 },
v¥tekaet τ ( ζ ) > 2 α = ( )Eζ2 1 2/
. Sledovatel\no, ζ ne qvlqetsq stroho
subhaussovskoj.
Zameçanye 3.2. To, çto summa dvux stroho subhaussovskyx sluçajn¥x vely-
çyn moΩet ne b¥t\ stroho subhaussovskoj, b¥lo otmeçeno ranee v [7] (sm. takΩe
[11, c. 26], prymer 1.2.5).
4. Slabo subhaussovskye sluçajn¥e πlement¥. V dal\nejßem X budet
oboznaçat\ (koneçnomernoe yly beskoneçnomernoe) dejstvytel\noe normyro-
vannoe prostranstvo, X
*
— soprqΩennoe k X banaxovo prostranstvo. Znaçe-
nye lynejnoho funkcyonala x
* ∈ X na πlemente x ∈ X budem oboznaçat\ çerez
〈 x
*, x 〉.
Kak y preΩde, budem sçytat\, çto fyksyrovano nekotoroe, dostatoçno boha-
toe, osnovnoe veroqtnostnoe prostranstvo ( Ω, A, P ). Proyzvol\noe yzmerymoe
separabel\noznaçnoe otobraΩenye ξ : Ω → X budem naz¥vat\ sluçajn¥m πle-
mentom. Esly 0 < p < ∞ , to budem hovoryt\, çto sluçajn¥j πlement ξ v nor-
myrovannom prostranstve X:
ymeet syl\n¥j p-j porqdok, esly E ξ p < ∞;
ymeet slab¥j p-j porqdok, esly E x
p*, ξ < ∞ dlq vsex x
* ∈ X;
centryrovan, esly ξ ymeet slab¥j perv¥j porqdok y E 〈 x
*, ξ 〉 = 0 dlq vsex
x
* ∈ X
*
.
Budem hovoryt\, çto dlq sluçajnoho πlementa ξ so slab¥m vtor¥m porqd-
kom v normyrovannom prostranstve X otobraΩenye R : X
* → X qvlqetsq:
kovaryacyonn¥m operatorom, esly
Rx x1 2
* *, = E x x1 2
* *, ,ξ ξ – E Ex x1 2
* *, ,ξ ξ , x1
* , x2
* ∈ X;
korrelqcyonn¥m operatorom, esly
Rx x1 2
* *, = E x x1 2
* *, ,ξ ξ , x1
* , x2
* ∈ X
*
.
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SLABO SUBHAUSSOVSKYE SLUÇAJNÁE ∏LEMENTÁ V BANAXOVÁX … 1199
Pust\ X — banaxovo prostranstvo. Yzvestno, çto:
a) dlq kaΩdoho sluçajnoho πlementa ξ slabo vtoroho porqdka v X suwest-
vuet kovaryacyonn¥j operator Rξ : X
* → X [14] (teorema 3.1.2);
b) kaΩd¥j lynejn¥j symmetryçn¥j poloΩytel\n¥j operator R : X
* → X s
separabel\noj oblast\g znaçenyj qvlqetsq kovaryacyonn¥m operatorom neko-
toroho sluçajnoho πlementa ξ slabo vtoroho porqdka v X [14] (teorema 3.2.2).
V çastnosty, esly X — separabel\noe hyl\bertovo prostranstvo, to toΩdest-
venn¥j operator I : X
* → X qvlqetsq kovaryacyonn¥m operatorom [15, c. 110];
c) operator R : X
* → X qvlqetsq kovaryacyonn¥m operatorom nekotoroho
sluçajnoho πlementa ξ syl\no vtoroho porqdka v X tohda y tol\ko tohda, koh-
da dlq nekotoroj syl\no 2-summyruemoj posledovatel\nosty ( xk ) v X ymeem
R x
* =
k
k kx x x
=
∞
∑
1
*, , x
* ∈ X
*
[14] (predloΩenye 3.2.3). V çastnosty, esly ξ — sluçajn¥j πlement syl\no
vtoroho porqdka v X, to eho kovaryacyonn¥j operator Rξ : X
* → X qvlqetsq
qdern¥m (y, sledovatel\no, kompaktn¥m) operatorom.
PokaΩem, çto kovaryacyonn¥j operator sluçajnoho πlementa so slab¥m po-
rqdkom r, r > 2, ymeet svojstvo kompaktnosty. ∏to utverΩdenye m¥ v¥vedem
yz sledugweho predloΩenyq, kotoroe ostalos\ nezameçenn¥m v [14].
PredloΩenye 4.1. Pust\ X — banaxovo prostranstvo s soprqΩenn¥m X
*
,
B
X* — zamknut¥j edynyçn¥j ßar v X
*
, Xσ* — prostranstvo X
* s topolo-
hyej σ ( X
*, X ). Pust\, dalee, 0 < p < r < ∞, ξ — sluçajn¥j πlement v X so
slab¥m porqdkom r, T ξ : X
* → L r ( Ω , A , P ) — ynducyrovann¥j operator.
Spravedlyv¥ sledugwye utverΩdenyq:
a) mnoΩestvo x x B
p
X
* *, : *ξ ∈{ } ravnomerno yntehryruemo;
b) Tξ , kak operator yz Xσ* v Lp , qvlqetsq sekvencyal\no neprer¥vn¥m;
s) T B
Xξ *( ) — kompaktnoe mnoΩestvo v Lp ;
d) Tξ , kak operator X
* v Lp , qvlqetsq kompaktn¥m.
Dokazatel\stvo. a) Sohlasno teoreme o zamknutom hrafyke operator Tξ
ohranyçen y poπtomu
E x
p r p
*, ξ( ) / = E x
r*, ξ ≤ || Tξ ||r < ∞, x
* ∈ B
X* .
b) Pust\ xk
* ∈ X
*
, k = 1, 2, … , y x xk
*, → 0 dlq vsex x ∈ X. Nam nuΩno
pokazat\, çto Tξ xk
* → 0 v Lp , t. e. lim ,*
k k
p
xE ξ = 0. Po teoreme Banaxa –
Ítejnxauza ymeem sup *
k k
p
x < ∞. Oçevydno, xk
p*, ( )ξ ω → 0 dlq kaΩdoho
ω ∈ Ω. Uçyt¥vaq utverΩdenye a), sohlasno teoreme Vytaly o sxodymosty
yntehralov zaklgçaem, çto lim ,*
k k
p
xE ξ = 0, y utverΩdenye b) dokazano.
s) Vvydu separabel\noznaçnosty ξ moΩem sçytat\, çto X separabel\no.
Kak yzvestno, B
X* qvlqetsq metryzuem¥m y kompaktn¥m podmnoΩestvom v
Xσ* . Sledovatel\no, s uçetom utverΩdenyq b) poluçaem, çto Tξ , kak otobra-
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1200 N. N. VAXANYQ, V. V. KVARACXELYQ, V. Y. TARYELADZE
Ωenye yz B X X
X* , ( , )*σ( ) v Lp , qvlqetsq neprer¥vn¥m. Otsgda y yz kompakt-
nosty B X X
X* , ( , )*σ( ) sleduet utverΩdenye s). UtverΩdenye d) sleduet yz
utverΩdenyq s).
Sledstvye 4.1. Pust\ X — banaxovo prostranstvo y ξ — sluçajn¥j
πlement v X so slab¥m vtor¥m porqdkom y kovaryacyonn¥m operatorom
Rξ : X
* → X. Esly ξ ymeet slab¥j porqdok r pry nekotorom r , 2 < r < ∞,
to Rξ — kompaktn¥j operator.
Dokazatel\stvo. Rξ moΩno predstavyt\ v vyde Rξ = Tξ
*Tξ . V sylu ut-
verΩdenyq 4.1, d) Tξ , kak operator yz X
*
v L2, qvlqetsq kompaktn¥m. Sledo-
vatel\no, kompakten y Rξ .
VozmoΩn¥e obobwenyq ponqtyq subhaussovosty dlq sluçajn¥x vektorov
(πlementov) v koneçnomern¥x y beskoneçnomern¥x normyrovann¥x prostran-
stvax rassmotren¥ v rabotax [8 – 10].
Opredelenye 4.1. Pust\ X — (koneçnomernoe yly beskoneçnomernoe) dej-
stvytel\noe normyrovannoe prostranstvo. Sluçajn¥j πlement ξ v X
naz¥vaetsq:
haussovskym, esly vse sluçajn¥e velyçyn¥ 〈 x
*, ξ 〉, x
* ∈ X
*
, qvlqgtsq
haussovskymy;
slabo subhaussovskym, esly vse sluçajn¥e velyçyn¥ 〈 x
*, ξ 〉, x
* ∈ X
*
, qvlq-
gtsq subhaussovskymy;
stroho subhaussovskym, esly vse sluçajn¥e velyçyn¥ 〈 x
*, ξ 〉, x
* ∈ X
*
, qvlq-
gtsq stroho subhaussovskymy;
F-subhaussovskym, esly vse sluçajn¥e velyçyn¥ 〈 x
*, ξ 〉, x
* ∈ X
*
, qvlqgtsq
subhaussovskymy y dlq nekotoroho c ≥ 0 y vsex x
* ∈ X
* v¥polneno
neravenstvo
τ ( 〈 x
*, ξ 〉 ) ≤ c xE *, ξ
2 1 2( ) / ;
γ-subhaussovskym, esly suwestvuet separabel\noznaçn¥j centryrovann¥j
haussovskyj sluçajn¥j πlement η v X takoj, çto dlq vsex x
* ∈ X
* v¥pol-
neno neravenstvo
E exp 〈 x
*, ξ 〉 ≤ E exp 〈 x
*, η 〉.
PredloΩenye 4.2. Pust\ X — banaxovo prostranstvo, ξ — slabo subha-
ussovskyj sluçajn¥j πlement v X, Tξ : X
* → S G( Ω, A, P ) — ynducyrovann¥j
operator. Spravedlyv¥ sledugwye utverΩdenyq:
a) Tξ — neprer¥vn¥j lynejn¥j operator yz X
* v S G( Ω );
b) τ w ( ξ ) = sup ,
*
*
*
x B
X
x
∈
( )τ ξ < ∞;
c) kovaryacyonn¥j operator Rξ sluçajnoho πlementa ξ qvlqetsq kom-
paktn¥m y || Rξ || ≤ τ w ( ξ );
d) esly dim ( X ) < ∞, to suwestvuet ε > 0 takoe, çto Eeε ξ 2
< ∞.
Dokazatel\stvo. UtverΩdenye a) sleduet yz teorem¥ o zamknutom hrafy-
ke, utverΩdenye b) — yz utverΩdenyq a). Pervaq çast\ utverΩdenyq c) sleduet
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SLABO SUBHAUSSOVSKYE SLUÇAJNÁE ∏LEMENTÁ V BANAXOVÁX … 1201
yz sledstvyq 4.1, a vtoraq çast\ — yz predloΩenyq 2.1, b4). UtverΩdenye d)
sleduet standartn¥m putem yz predloΩenyq 2.1, b6).
Pryvedem sledugwyj yzvestn¥j rezul\tat.
Teorema 4.1. Pust\ ξ — sluçajn¥j πlement v normyrovannom prostran-
stve X. Tohda:
a) esly ξ — haussovskyj sluçajn¥j πlement, to suwestvuet ε > 0 takoe,
çto Eeε ξ 2
< ∞ ( Fernyk, Landau y Íepp, Skoroxod, sm. [14], sledstvye 2,
b) predloΩenyq 5.5.6);
b ) esly ξ — γ-subhaussovskyj sluçajn¥j πlement, to suwestvuet ε > 0
takoe, çto Eeε ξ 2
< ∞ ([9], teorema 3.4);
s) esly X = Lp , 1 ≤ p < ∞, y ξ — F-subhaussovskyj sluçajn¥j πlement v
X, to suwestvuet ε > 0 takoe, çto Eeε ξ 2
< ∞ ([9], teorema 4.3).
Ysxodq yz πtoj teorem¥ y ymeq v vydu predloΩenye 4.2, d), estestvenno po-
stavyt\ vopros: veren ly rezul\tat, analohyçn¥j teoreme 4.1, dlq slabo subha-
ussovskyx sluçajn¥x πlementov. Okaz¥vaetsq, çto predloΩenye 4.2, d) verno
tol\ko v koneçnomernom sluçae. Dlq formulyrovky sledugweho rezul\tata
vvedem oboznaçenyq:
exp
(
m
)
( x ) = exp ( exp
(
m
–
1
)
( x ) ), m = 2, 3, … ,
y
exp
(
1
)
( x ) = exp ( x ) = e
x
;
ln
(
m
)
( x ) = ln
(
1
)
( ln
(
m
–
1
)
( x ) ), m = 2, 3, … ,
y
ln
(
1
)
( x ) = ln ( max ( 1, x ) ).
Qsno, çto dlq kaΩdoho m funkcyy exp
(
m
)
y ln
(
m
)
opredelen¥ dlq vsex
x ∈ R.
Teorema 4.2 [16]. V proyzvol\nom beskoneçnomernom normyrovannom pros-
transtve X dlq lgboho natural\noho çysla m suwestvuet symmetryçn¥j
sçetnoznaçn¥j slabo subhaussovskyj sluçajn¥j πlement ξ v X, dlq kotoroho
E ln
(
m
)
|| ξ || = ∞.
Dokazatel\stvo. Sohlasno lemme Dvoreckoho (sm. [14], lemma 2.2.1), dlq
kaΩdoho natural\noho j najdutsq πlement¥ x j , 1 , … , x j, j v X takye, çto dlq
vsex dejstvytel\n¥x çysel t1 , … , t j ymeem
k
j
kt
=
∑
/
1
2
1 2
≤
k
j
k j kt x
=
∑
1
, ≤ 2
1
2
1 2
k
j
kt
=
∑
/
. (4.1)
Qsno, çto
k
j
j kx x
=
∑
/
1
2
1 2
*
,( ) = sup ( ) :*
,
k
j
k j k
k
j
kt x x t
= =
∑ ∑ ≤
1 1
2 1 ≤
≤ x t x x t
k
j
k j k
k
j
k
* *
,sup ( ) :
= =
∑ ∑ ≤
1 1
2 1 ≤ 2 || x
* ||, x
* ∈ X
*
. (4.2)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1202 N. N. VAXANYQ, V. V. KVARACXELYQ, V. Y. TARYELADZE
Oboznaçym çerez p0 celug çast\ çysla exp
(
m
+
1
)
( 4 ) y pust\ pk = p0 + k dlq
vsex natural\n¥x k. Oçevydno, çto çysla pk zavysqt takΩe y ot m . Lehko vy-
det\, çto ln
(
m
+
2
)
( pk ) > 1 dlq kaΩdoho natural\noho k y m. Sluçajn¥j πle-
ment ξ postroym po v¥brann¥m πlementam x j, k sledugwym obrazom:
P ξ = ±[ ]ln( ) ,p xk j k =
c
j k2
µ µ , k = 1, 2, … , j = 1, 2, … ,
hde c — normyrugwaq konstanta y
µk =
1
2 1 2 2
p p p p pk k k
m
k
m
kln( ) ln ( ) ln ( ) ln ( )( ) ( ) ( )… [ ]+ +
, k = 1, 2, … .
Lehko zametyt\, çto ξ — symmetryçn¥j sçetnoznaçn¥j sluçajn¥j πlement,
dlq kotoroho P 1< < ∞[ ]ln( )m ξ = 1. Dlq dokazatel\stva slaboj subhausso-
vosty sluçajnoho πlementa ξ vospol\zuemsq teoremoj 2.1, sohlasno kotoroj ξ
qvlqetsq slabo subhaussovskym sluçajn¥m πlementom tohda y tol\ko tohda,
kohda ϑ ξ2 x*,( ) < ∞. Otmetym predvarytel\no, çto
ln( )pk
n
k[ ] µ <
ln( )p
p
k
n
k
[ ] ≤
n
e
n
, n, k = 1, 2, … ,
y poπtomu
E x
n*, ξ
2
= c x x p
j
j
k
j
j k
n
k
n
k
=
∞
=
∑ ∑ [ ]
1 1
2
µ µ*
,, ln( ) ≤
≤ c
n
e
x x
n
j
j
k
j
j k
n
=
∞
=
∑ ∑
1 1
2
µ *
,, .
Teper\, prymenqq neravenstvo (4.2), poluçaem, çto ϑ ξ2 x*,( ) < ∞ dlq kaΩ-
doho x
* ∈ X
*
.
Ostaetsq pokazat\, çto E ln
(
m
)
|| ξ || = ∞. ∏to sleduet yz levoho neravenstva v
(4.1), sohlasno kotoromu || x j, k || ≥ 1 dlq vsex k y j, y yz sootnoßenyq
ln
(
m
)
( ln ( x ) )
1
/
2 ≥ ln ( ln
(
m
)
( x ) )
1
/
2
dlq kaΩdoho natural\noho m y dlq vsex x ≥
≥ exp
(
m
+
1
)
( 4 ), kotoroe lehko v¥vesty yz πlementarnoho neravenstva
ln ( ln ( x ) )
1
/
2 ≥ ( ln
(
2
)
( x ) )
1
/
2
dlq vsex x ≥ ee4
= exp
(
2
)
( 4 ). V samom dele,
E ln
(
m
)
|| ξ || = c p x
j
j
k
j
k
m
k j k
=
∞
=
∑ ∑ ( )( )
/
1 1
1 2µ µ ln ln( )( )
, ≥
≥
c
p
j
j
k
j
k
m
k2 1 1
1
=
∞
=
+∑ ∑
µ µ ln ( )( ) = ∞,
çto y trebovalos\ dokazat\.
Zameçanye 4.1. 1. Lehko proveryt\, çto dlq sluçajnoho πlementa ξ, po-
stroennoho v xode dokazatel\stva teorem¥ 4.2, ymeem E ln
(
m
+
1
)
|| ξ || < ∞.
2. Pust\ H — beskoneçnomernoe separabel\noe hyl\bertovo prostranstvo s
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SLABO SUBHAUSSOVSKYE SLUÇAJNÁE ∏LEMENTÁ V BANAXOVÁX … 1203
ortonormyrovann¥m bazysom ( ek ). Opredelym sluçajn¥j πlement ξ v H so-
otnoßenyqmy
P ξ = ±[ ]p ek k =
c
k2
µ , k = 1, 2, … ,
hde çysla pk , µk y c v¥bran¥ tak, kak pry dokazatel\stve teorem¥ 4.2. Tohda
lehko proveryt\, çto ξ — slabo subhaussovskyj sluçajn¥j πlement y
E ln
(
m
+
1
)
|| ξ || = ∞, E ln
(
m
+
2
)
|| ξ || < ∞.
3. Netrudno pokazat\, çto v lgbom normyrovannom prostranstve (daΩe na
çyslovoj osy) moΩno postroyt\ sluçajn¥j πlement ξ, dlq kotoroho
E ln
(
m
)
|| ξ || = ∞ dlq lgboho m . M¥ ne znaem, spravedlyv ly sledugwyj vary-
ant teorem¥ 4.2: pust\ X — proyzvol\noe beskoneçnomernoe normyrovannoe
prostranstvo. Tohda v X suwestvuet slabo subhaussovskyj sluçajn¥j πlement
ξ, dlq kotoroho pry vsex natural\n¥x m
E ln
(
m
)
|| ξ || = ∞.
Zameçanye 4.2. Teorema 4.2 pokr¥vaet teoremu 2.2.1 yz [14], kotoraq ut-
verΩdaet, çto v lgbom beskoneçnomernom normyrovannom prostranstve X su-
westvuet sçetnoznaçn¥j sluçajn¥j πlement, kotor¥j ymeet slab¥j porqdok r
dlq kaΩdoho r > 0 y ne ymeet syl\noho porqdka p ny dlq kakoho p > 0.
Pust\ X — beskoneçnomernoe separabel\noe banaxovo prostranstvo, 2 ≤ r <
< ∞. Vvedem po analohyy s [15] sledugwye oboznaçenyq:
R ( X ) — klass vsex symmetryçn¥x poloΩytel\n¥x separabel\noznaçn¥x
operatorov R : X
* → X;
CR ( X ) — podklass R ( X ), sostoqwyj yz kompaktn¥x operatorov;
Rr ( X ) — podklass R ( X ), sostoqwyj yz kovaryacyonn¥x operatorov vsex
sluçajn¥x πlementov v X so slab¥m r-m porqdkom;
Rw sub ( X ) — podklass R ( X ), sostoqwyj yz kovaryacyonn¥x operatorov vsex
separabel\noznaçn¥x slabo subhaussovskyx sluçajn¥x πlementov v X;
Rgaus ( X ) — podklass R ( X ), sostoqwyj yz kovaryacyonn¥x operatorov vsex
haussovskyx sluçajn¥x πlementov v X . Proyzvol\n¥j operator Rgaus ( X )
budem naz¥vat\ takΩe haussovskoj kovaryacyej.
Yzvestno, çto R ( X ) = R2 ( X ) [14] (teorema 3.2.2). Sledstvye 4.1 pokaz¥vaet,
çto esly 2 < r < ∞, to
Rgaus ( X ) ⊂ Rw sub ( X ) ⊂ Rr ( X ) ⊂ CR ( X ).
Estestvenno voznykagt sledugwye vopros¥: moΩno ly dat\ v podxodqwyx ter-
mynax vnutrennee opysanye klassov Rr ( X ) pry r > 2 y Rw sub ( X )? Naprymer,
ymegt ly mesto sledugwye ravenstva pry vsex r > 2:
Rw sub ( X ) = Rr ( X ) = R ( X )?
Sluçajn¥j πlement, rassmotrenn¥j v zameçanyy 4.1 (2), ymeet svojstvo
R ξ ∈
2< <∞r
r H∩ S ( ) ,
hde Sr ( H ) pry fyksyrovannom r oboznaçaet klass Íattena (po povodu klas-
sov Íattena sm. [17]). Ymeq v vydu πtot fakt, moΩno postavyt\ vopros: vern¥
ly sledugwye ravenstva:
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1204 N. N. VAXANYQ, V. V. KVARACXELYQ, V. Y. TARYELADZE
Rr ( H ) = Sr ( H ) ∩ R ( H ) dlq vsex r > 2,
Rw sub ( H ) =
2< <∞
r
r H H∩ ∩S R( ) ( )?
PredloΩenye 4.3. Pust\ X — banaxovo prostranstvo, ξ — γ-subhaus-
sovskyj sluçajn¥j πlement v X. Tohda:
a) ξ — slabo subhaussovskyj sluçajn¥j πlement;
b) suwestvuet S ∈ Rgaus ( X ) takoj, çto
R x xξ
* *, ≤ τ ξx*,( ) ≤ Sx x* *, , x
* ∈ X
*
;
s) R ξ ∈ Rgaus ( X ).
Dokazatel\stvo. Po opredelenyg γ -subhaussovskoho sluçajnoho πlemen-
ta suwestvuet centryrovann¥j haussovskyj sluçajn¥j πlement η v X takoj,
çto
Eexp ,*x ξ ≤ Eexp ,*x η , x
* ∈ X
*
.
Sledovatel\no,
Eexp ,*t x ξ( ) ≤ exp
,*t x2 2
2
E η
= exp
,* *t R x x2
2
η
, t ∈ R. (4.3)
Oçevydno, çto utverΩdenye a) v¥tekaet yz (4.3). UtverΩdenye b) takΩe sleduet
yz (4.3) (moΩem vzqt\ S = R η ). DokaΩem utverΩdenye s). Sohlasno utverΩde-
nyg b) suwestvuet S ∈ Rgaus ( X ) takoj, çto R x xξ
* *, ≤ S x x* *, dlq vsex
x
* ∈ X
*
. Yz posledneho sootnoßenyq v¥tekaet R ξ ∈ Rgaus ( X ) ([14], sledstvye 2
predloΩenyq 6.3.4).
PredloΩenye 4.4. Pust\ X — koneçnomernoe banaxovo prostranstvo y ξ
— sluçajn¥j πlement v X. Sledugwye utverΩdenyq πkvyvalentn¥:
i) ξ qvlqetsq slabo subhaussovskym;
ii) ξ qvlqetsq γ-subhaussovskym;
iii) ξ qvlqetsq F-subhaussovskym.
Dokazatel\stvo. Ymplykacyq ii) ⇒ i) sleduet yz predloΩenyq 4.3. Ymp-
lykacyq iii) ⇒ i) oçevydna. PokaΩem, çto i) ⇒ iii). Rassmotrym ynducyrovan-
n¥j operator Tξ : X
* → S G( Ω, A, P ). Vvydu toho, çto Tξ ( X
*
) — koneçnomer-
noe vektornoe prostranstvo, v nem proyzvol\n¥e dve norm¥ πkvyvalentn¥. V
çastnosty, πkvyvalentn¥ norm¥ τ ( ⋅ ) y ⋅ L2
. Sledovatel\no, dlq nekotoroj
konstant¥ b > 0 ymeem
τ ξx*,( ) ≤ b x
L
*, ξ
2
= b R x xξ
* *,
1 2/
, x
* ∈ X
*
,
hde Rξ : X
* → X — kovaryacyonn¥j operator ξ, t. e. ξ qvlqetsq F-subhaussov-
skym. UtverΩdenye ii) sleduet yz utverΩdenyq iii), tak kak v koneçnomernom
prostranstve proyzvol\n¥j kovaryacyonn¥j operator qvlqetsq haussovskym.
Nam ponadobqtsq nekotor¥e vspomohatel\n¥e fakt¥. Napomnym sledug-
wee opredelenye. Pust\ X — normyrovannoe prostranstvo. Oboznaçym çerez
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SLABO SUBHAUSSOVSKYE SLUÇAJNÁE ∏LEMENTÁ V BANAXOVÁX … 1205
l X2
weak ( ) mnoΩestvo vsex slabo 2-summyruem¥x posledovatel\nostej ( xk ) v X,
çerez l X2
weak ( ) vektornoe prostranstvo. Krome toho, kak lehko vydet\,
|| ( xk ) || 2, w = sup ,
*
*
x k
kx x
≤ =
∞
∑
/
1 1
2
1 2
< ∞ dlq vsex ( xk ) ∈ l X2
weak ( ),
y funkcyonal || ⋅ || 2, w qvlqetsq normoj v l X2
weak ( ), a esly X — banaxovo pros-
transtvo, to l X w2 2
weak ( ), ,⋅( ) takΩe qvlqetsq banaxov¥m prostranstvom (sm.,
naprymer, [17]).
Sledugwyj rezul\tat moΩno najty v [14] (teorema 5.4.4 y teorema 5.5.3 (b)).
Teorema 4.3. Pust\ X — proyzvol\noe banaxovo prostranstvo,
( xk ) ∈ l X2
weak ( ) y R : X
* → X — operator, zadann¥j ravenstvom
R x
* =
k
k kx x x
=
∞
∑
1
*, , x
* ∈ X
*
.
Esly R — haussovskaq kovaryacyq y ( )gk k∈N
— posledovatel\nost\ nezavysy-
m¥x standartn¥x haussovskyx sluçajn¥x velyçyn, to rqd
k k kx g∑ sxodytsq
poçty navernoe v prostranstve L2 ( Ω, X ).
Teper\ dokaΩem, çto v beskoneçnomernom sluçae ymplykacyq i) ⇒ ii) v pred-
loΩenyy 4.4 neverna.
Teorema 4.4. Pust\ X — proyzvol\noe beskoneçnomernoe banaxovo prost-
ranstvo. Spravedlyv¥ sledugwye utverΩdenyq:
a) suwestvuet symmetryçn¥j sçetnoznaçn¥j slabo subhaussovskyj sluçaj-
n¥j πlement ξ v X, dlq kotoroho R ξ ∉ Rgaus ( X );
b) suwestvuet symmetryçn¥j sçetnoznaçn¥j slabo subhaussovskyj sluçaj-
n¥j πlement ξ v X, kotor¥j ne qvlqetsq γ - subhaussovskym.
Dokazatel\stvo. a) Dopustym protyvnoe. Rassmotrym sluçajn¥j πlement
ξ v X s raspredelenyem
P ξ = ± +[ ]ln( )k xk1 =
c
k k2 1 2ln( )+[ ]
, k = 1, 2, … ,
hde ( xk ) ∈ l X2
weak ( ) y c — normyrugwaq konstanta. Oçevydno, çto ξ — sym-
metryçn¥j sçetnoznaçn¥j sluçajn¥j πlement.
Dlq dokazatel\stva slaboj subhaussovosty sluçajnoho πlementa ξ vospol\-
zuemsq neravenstvom ln( )k kn+[ ]1 ≤ 2 n e n/( ) , kotoroe spravedlyvo dlq vsex
natural\n¥x k y n. Ymeem,
E x
n*, ξ
2
= c
k
k
x x
k
n
k
n
=
∞ −
∑ +[ ]
1
2 21ln( )
,* ≤
≤ 2
1
2
c
n
e
x x
n
k
k
n
=
∞
∑ *, ≤ 2
1
2
c
n
e
x x
n
k
k
n
=
∞
∑ *, <
< ( ) ,
*x x nk w
n n n
2
2 2
< ∞.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1206 N. N. VAXANYQ, V. V. KVARACXELYQ, V. Y. TARYELADZE
Takym obrazom, ϑ ξ2 x*,( ) < ∞ dlq kaΩdoho x
* ∈ X
*
y, sledovatel\no, so-
hlasno teoreme 2.1 sluçajn¥j πlement ξ qvlqetsq slabo subhaussovskym. Leh-
ko vydet\, çto kovaryacyonn¥j operator sluçajnoho πlementa ξ moΩno pred-
stavyt\ v vyde
R ξ x
* = c
k k
x x x
k
k k∑ +
1
1ln( )
,*
, x
* ∈ X
*
.
Po predpoloΩenyg R ξ ∈ Rgaus ( X ). Tohda sohlasno teoreme 4.3 poluçaem
( xk ) ∈ l X2
weak ( ) ⇒
k
k kk k
x g∑ +
1
1ln( )
∈ L2 ( Ω, X ).
Yz posledneho sootnoßenyq sohlasno teoreme o zamknutom hrafyke sleduet
suwestvovanye konstant¥ C > 0 takoj, çto
k
j
k k
L
k k
x g
=
∑ +1
1
1
2
ln( )
≤ C x x x j w
( , , , )
,1 2 2
… (4.4)
dlq vsex x k ∈ X, k = 1, 2, … , j = 1, 2, … .
Poskol\ku X beskoneçnomerno, sohlasno lemme Dvoreckoho dlq kaΩdoho
natural\noho j suwestvugt πlement¥ x j, 1 , x j, 2 , … , x j, j v X, kotor¥e dlq vsex
dejstvytel\n¥x çysel t1 , … , t j udovletvorqgt sootnoßenyg (4.1). Dlq kaΩ-
doho fyksyrovannoho j v kaçestve sovokupnosty πlementov x 1 , x 2 , … , x j voz\-
mem ukazann¥j v¥ße nabor πlementov x j, 1 , x j, 2 , … , x j, j . Yz (4.1) y (4.4) sleduet,
çto dlq lgboho natural\noho j
k
j
k k=
∑ +
/
1
1 2
1
1ln( )
≤ 2 C.
Poluçyly protyvoreçye, poskol\ku levaq çast\ πtoho neravenstva neohra-
nyçena. UtverΩdenye b) sleduet yz utverΩdenyq a) y yz predloΩenyq 4.3, c).
Zametym, çto teorema 4.4, b) v¥tekaet takΩe yz teorem 4.1, b) y 4.2.
Sledugwaq teorema pokaz¥vaet, çto esly X — beskoneçnomernoe banaxovo
prostranstvo, to ymplykacyq ii) ⇒ iii) v predloΩenyy 4.4 takΩe neverna.
Teorema 4.5. Pust\ X — beskoneçnomernoe banaxovo prostranstvo. Tohda
suwestvuet symmetryçn¥j sçetnoznaçn¥j ohranyçenn¥j γ-subhaussovskyj slu-
çajn¥j πlement ξ, kotor¥j ne qvlqetsq F-subhaussovskym.
Dokazatel\stvo. Pust\ ( xk ) — posledovatel\nost\ v X so sledugwymy
svojstvamy:
k kx∑ < ∞ y dlq nekotoroj posledovatel\nosty xn
*( ) v X
*
ymeet mesto ravenstvo x xk n
*, = δn, k , k, n = 1, 2, … ( δn, k — symvol Kronekera).
Yzvestno, çto takug posledovatel\nost\ moΩno najty v kaΩdom beskoneçno-
mernom banaxovom prostranstve. Pust\ ( αk ) — proyzvol\naq posledovatel\-
nost\ stroho poloΩytel\n¥x çysel,
k k∑ α = 1. Rassmotrym sluçajn¥j πle-
ment v X, kotor¥j prynymaet znaçenyq ± xk s veroqtnostqmy αk / 2 , k = 1,
2, … . Qsno, çto ξ — symmetryçn¥j ohranyçenn¥j sluçajn¥j πlement v X y
E x
s*, ξ =
k
k k
s
x x∑ α *, dlq vsex s > 0 y x
* ∈ X
*
. (4.5)
PokaΩem, çto ξ qvlqetsq γ-subhaussovskym. Lehko vydet\, çto operator
R : X
* → X, opredelqem¥j ravenstvom
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SLABO SUBHAUSSOVSKYE SLUÇAJNÁE ∏LEMENTÁ V BANAXOVÁX … 1207
R x
* =
k
k kx x x∑ *, , x
* ∈ X
*
,
est\ haussovskaq kovaryacyq. V samom dele, esly gk , k = 1, 2, … , — nezavysy-
m¥e standartn¥e haussovskye sluçajn¥e velyçyn¥, to rqd γ =
k k kx g∑ poçty
navernoe absolgtno sxodytsq v X y Rγ = R. Pust\ m — proyzvol\noe natu-
ral\noe çyslo. S uçetom ravenstva (4.5) pry kaΩdom x
* ∈ X
*
ymeem
E x
m m
*, ξ
2 1 2( ) / =
k
k k
m
m
x x∑
/
α *,
2
1 2
≤
≤
k
k
m
kx x∑ /
/
α1 2
1 2
*, ≤ Rx x* *,
1 2/
,
otkuda poluçaem (sm. sledstvye 2.2, a))
τ ξx*,( ) ≤
e
x
2
ϑ ξ*,( ) ≤
e
Rx x
2
1 2* *,
/
.
Poskol\ku e R/2 takΩe est\ haussovskaq kovaryacyq, yz posledneho ra-
venstva sleduet, çto ξ qvlqetsq γ-subhaussovskym. Ostaetsq pokazat\, çto ξ
ne qvlqetsq F-subhaussovskym. S uçetom sledstvyq 2.1, a) y ravenstva (4.5)
ymeem
τ ξ
ξ
x
x
n
n L
*
*
,
,
( )
2
≥
1
4
4
2
β
ξ
ξ
x
x
n L
n L
*
*
,
,
=
1
4
1 4
1 2β
α
α
n
n
/
/ =
1 1
4
1 2β αn
/
dlq kaΩdoho natural\noho n. Poskol\ku 1 / αn → ∞, n → ∞, poluçaem, çto ξ
ne qvlqetsq F-subhaussovskym.
Zameçanye 4.3. V [9] pryvodytsq prymer neohranyçennoho γ-subhaus-
sovskoho sluçajnoho πlementa v hyl\bertovom prostranstve, kotor¥j ne
qvlqetsq F-subhaussovskym. Teorema 4.5 neskol\ko usylyvaet prymer 5.1 v [9]
y ee dokazatel\stvo.
Nakonec, sformulyruem bez dokazatel\stva rezul\tat, kotor¥j pokaz¥vaet,
çto dlq nekotor¥x beskoneçnomern¥x banaxov¥x prostranstv ymplykacyq
iii)<⇒ ii) v predloΩenyy 4.4 ostaetsq spravedlyvoj.
Teorema 4.6. Pust\ X — banaxovo prostranstvo. Rassmotrym sledugwye
utverΩdenyq:
i) kaΩd¥j F-subhaussovskyj sluçajn¥j πlement ξ v X qvlqetsq γ-sub-
haussovskym;
ii) kaΩd¥j stroho subhaussovskyj sluçajn¥j πlement ξ v X qvlqetsq
γ-subhaussovskym;
iii) X ymeet koneçn¥j kotyp;
iv) dlq kaΩdoho F-subhaussovskoho sluçajnoho πlementa ξ v X suwestvu-
et ε > 0 takoj, çto Eeε ξ 2
< ∞.
Tohda ymegt mesta sledugwye utverΩdenyq:
a) i) ⇒ ii) ⇒ iii);
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1208 N. N. VAXANYQ, V. V. KVARACXELYQ, V. Y. TARYELADZE
b) esly, dopolnytel\no, X qvlqetsq G L-prostranstvom ( v çastnosty,
esly X — banaxova reßetka), to iii) ⇒ i);
s) esly, dopolnytel\no, X qvlqetsq G L-prostranstvom (v çastnosty,
esly X — banaxova reßetka), to iii) ⇒ iv).
Zameçanye 4.4. 1. Ymplykacyq i) ⇒ ii) v utverΩdenyy a) teorem¥ 4.6 oçe-
vydna. Ymplykacyq ii) ⇒ iii) v tom Ωe utverΩdenyy faktyçesky dokazana v [9]
(prymer 5.2).
2. Dokazatel\stvo utverΩdenyq b) teorem¥ 4.6 budet opublykovano pozdnee.
Opredelenyq y svojstva LG-prostranstv y prostranstv koneçnoho kotypa moΩ-
no najty, naprymer, v [18].
3. UtverΩdenye s) teorem¥ 4.6 v¥tekaet yz utverΩdenyq b) πtoj Ωe teorem¥
y yz teorem¥ 4.1, b). Zametym, çto teorema 4.1, s) qvlqetsq çastn¥m sluçaem
utverΩdenyq s) teorem¥ 4.6. Nam neyzvestno, spravedlyva ly ymplykacyq
iv)<⇒ iii) teorem¥ 4.6.
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19, # 1. – P. 1 – 25.
2. Paley R. E. A. C., Zygmund A. On some series of functions // Proc. Cambridge Phil. Soc. – 1932. –
26, # 4. – P. 337 – 357; 458 – 474. – 28, # 3. – P. 190 – 205.
3. Jain N. C., Marcus M. B. Central limit theorems for C ( S )-valued random variables // J. Function.
Anal. – 1975. – 19, # 3. – P. 216 – 231.
4. Pisier G. Les inéqualités de Khintchine – Kahane d’aprés C. Borell // Sémin. Géometrié Espaces
Banach, 1977–1978. Ecole Polytechnique. – Paris, 1978. – Exp. 7. – P. 1 – 14.
5. Pisier G. Sur l’espace de Banach des séries de Fourier aléatoires presque sûrement continues //
Sémin. Géométrie Espaces Banach, 1977–1978. Ecole Polytechnique. – Paris, 1978. – Exp. 17 – 18.
– P. 1 – 33.
6. Ledoux M., Talagrand M. Probability in Banach spaces. – Berlin; New York: Springer, 1991.
7. Buld¥hyn V. V., Kozaçenko G. V. O subhaussovskyx sluçajn¥x velyçynax // Ukr. mat. Ωurn.
– 1980. – 32, # 6. – S. 723 – 730.
8. Ostrovskyj E. Y. ∏ksponencyal\n¥e ocenky raspredelenyq maksymuma nehaussovskoho
sluçajnoho polq // Teoryq veroqtnostej y ee prymenenyq. – 1990. – 35, # 3. – S. 723 – 730.
9. Fukuda R. Exponential integrability of sub-Gaussian vectors // Probab. Theory. Relat. Fields. –
1990. – 85, # 4. – P. 505 – 521.
10. Antonini R. G. Subgaussian random variables in Hilbert spaces // Rend. Semin. mat. Univ. Padova.
– 1997. – 98. – P. 89 – 99.
11. Buld¥hyn V. V., Kozaçenko G. V. Metryçeskye xarakterystyky sluçajn¥x velyçyn y
processov. – Kyev: TViMS, 1998.
12. Ergemlidze Z., Shangua A., Tarieladze V. Sample behavior and laws of large numbers for Gaussian
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13. Kahane J. P. Some random series of functions. – Lexington: D. C. Heath and Co., 1968.
14. Vaxanyq N. N., Taryeladze V. Y., Çobanqn S. A. Veroqtnostn¥e raspredelenyq v banaxov¥x
prostranstvax. – M.: Nauka, 1985.
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Poluçeno 17.06.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
|
| id | umjimathkievua-article-3678 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:46:58Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b4/6a1f3e3be39514e65208815834ec57b4.pdf |
| spelling | umjimathkievua-article-36782020-03-18T20:02:00Z Weakly Sub-Gaussian Random Elements in Banach Spaces Слабо субгауссовские случайные элементы в банаховых пространствах Vakhaniya, N. N. Kvaratskheliya, V. V. Tarieladze, V. I. Вахания, Н. Н. Кварацхелия, В. В. Тариеладзе, В. И. Вахания, Н. Н. Кварацхелия, В. В. Тариеладзе, В. И. We give a survey of properties of weakly sub-Gaussian random elements in infinite-dimensional spaces. Some new results and examples are also given. Наведено огляд властивостей слабко субгауссових випадкових елементів у нескінченновимірних просторах, а також декілька нових результатів та прикладів. Institute of Mathematics, NAS of Ukraine 2005-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3678 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 9 (2005); 1187–1208 Український математичний журнал; Том 57 № 9 (2005); 1187–1208 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3678/4082 https://umj.imath.kiev.ua/index.php/umj/article/view/3678/4083 Copyright (c) 2005 Vakhaniya N. N.; Kvaratskheliya V. V.; Tarieladze V. I. |
| spellingShingle | Vakhaniya, N. N. Kvaratskheliya, V. V. Tarieladze, V. I. Вахания, Н. Н. Кварацхелия, В. В. Тариеладзе, В. И. Вахания, Н. Н. Кварацхелия, В. В. Тариеладзе, В. И. Weakly Sub-Gaussian Random Elements in Banach Spaces |
| title | Weakly Sub-Gaussian Random Elements in Banach Spaces |
| title_alt | Слабо субгауссовские случайные элементы в банаховых пространствах |
| title_full | Weakly Sub-Gaussian Random Elements in Banach Spaces |
| title_fullStr | Weakly Sub-Gaussian Random Elements in Banach Spaces |
| title_full_unstemmed | Weakly Sub-Gaussian Random Elements in Banach Spaces |
| title_short | Weakly Sub-Gaussian Random Elements in Banach Spaces |
| title_sort | weakly sub-gaussian random elements in banach spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3678 |
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