On the Skitovich-Darmois theorem and Heyde theorem in a Banach space
According to the well-known Skitovich-Darmois theorem, the independence of two linear forms of independent random variables with nonzero coefficients implies that the random variables are Gaussian variables. This result was generalized by Krakowiak for random variables with values in a Banach space...
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| Дата: | 2008 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3239 |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509291760320512 |
|---|---|
| author | Myronyuk, M. V. Миронюк, М. В. |
| author_facet | Myronyuk, M. V. Миронюк, М. В. |
| author_sort | Myronyuk, M. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:48:57Z |
| description | According to the well-known Skitovich-Darmois theorem, the independence of two linear forms of independent random variables with nonzero coefficients implies that the random variables are Gaussian variables. This result was generalized by Krakowiak for random variables with values in a Banach space in the case where the coefficients of forms are continuous invertible operators. In the first part of the paper, we give a new proof of the Skitovich-Darmois theorem in a Banach space. Heyde proved another characterization theorem similar to the Skitovich-Darmois theorem, in which, instead of the independence of linear forms, it is supposed that the conditional distribution of one linear form is symmetric if the other form is fixed. In the second part of the paper, we prove an analog of the Heyde theorem in a Banach space. |
| first_indexed | 2026-03-24T02:38:46Z |
| format | Article |
| fulltext |
UDK 519.2
M. V. Myrongk (Fiz.-texn. in-t nyz\kyx temperatur NAN Ukra]ny, Xarkiv)
DO TEOREM SKYTOVYÇA – DARMUA TA XEJDE
U BANAXOVOMU PROSTORI
By the well-known Skitovich – Darmois theorem, the independence of two linear forms of independent
random variables with nonzero coefficients implies that the random variables are Gaussian variables.
This result was generalized by Krakowiak to the case of random variables with values in a Banach
space, where coefficients of the forms are continuous invertible operators. In the first part of the paper,
we give a new proof of the Skitovich – Darmois theorem for a Banach space.
Heyde proved another characterization theorem of a Gaussian distribution similar to the Skitovich –
Darmois theorem, where, instead of the independence of linear forms it is assumed that the conditional
distribution of one of linear forms is symmetrical if another form is fixed. In the second part of the
paper, we prove an analog of the Heyde theorem for a Banach space.
Yzvestnaq teorema Skytovyça – Darmua utverΩdaet, çto yz nezavysymosty dvux lynejn¥x form
ot nezavysym¥x sluçajn¥x velyçyn s nenulev¥my koπffycyentamy sleduet, çto sluçajn¥e ve-
lyçyn¥ qvlqgtsq haussov¥my. ∏tot rezul\tat b¥l obobwen Krakovqkom dlq sluçajn¥x vely-
çyn so znaçenyqmy v banaxovom prostranstve, kohda koπffycyentamy form qvlqgtsq nepre-
r¥vn¥e oborotn¥e operator¥. V pervoj çasty rabot¥ pryvedeno novoe dokazatel\stvo teorem¥
Skytovyça – Darmua v banaxovom prostranstve.
Xejde dokazal blyzkug k teoreme Skytovyça – Darmua xarakteryzacyonnug teoremu, v ko-
toroj vmesto nezavysymosty lynejn¥x form predpolahalos\, çto uslovnoe raspredelenye od-
noj lynejnoj form¥ pry fyksyrovannoj druhoj qvlqetsq symmetryçn¥m. Vo vtoroj çasty ra-
bot¥ dokazan analoh teorem¥ Xejde v banaxovom prostranstve.
1. Vstup. U 1953 r. V. P. Skytovyç ta H. Darmua dovely nastupnu teoremu, qka
xarakteryzu[ haussiv rozpodil na dijsnij prqmij.
Teorema Skytovyça – Darmua [1, s. 3] (§ 3.1). Nexaj ξ1, … , ξn — neza-
leΩni vypadkovi velyçyny. Qkwo linijni formy L1 = α ξ1 1 + … + α ξn n ta
L2 = β ξ1 1 + … + β ξn n , de koefici[nty α j , βj [ vidminnymy vid nulq, nezaleΩ-
ni, to vypadkovi velyçyny ξj [ haussovymy.
Perenesenng ci[] xarakteryzacijno] teoremy na riznomanitni alhebra]çni
struktury prysvqçeno velyku kil\kist\ doslidΩen\ (dyv., napryklad, [2 – 12]).
Pry c\omu koefici[ntamy form [ topolohiçni avtomorfizmy vidpovidno] alheb-
ra]çno] struktury.
U 1962 r. S. Hur’[ ta I. Olkin uzahal\nyly teoremu Skytovyça – Darmua dlq
vypadkovyx velyçyn, wo nabuvagt\ znaçen\ u Rn
, ta nevyrodΩenyx matryc\ u
qkosti koefici[ntiv form [13]. Potim ]x dovedennq sprostyv A. Zinher (dyv. [1],
§ 3.2). Teorema Skytovyça – Darmua uzahal\ngvalas\ takoΩ u vypadku, koly
vypadkovi velyçyny nabuvagt\ znaçen\ u hil\bertovomu prostori. Pry c\omu
koefici[ntamy form buly deqki linijni neperervni operatory (dyv. [8, 10]). U
1985 r. V. Krakovqk uzahal\nyv teoremu Skytovyça – Darmua dlq vypadkovyx
velyçyn, wo nabuvagt\ znaçen\ u dovil\nomu separabel\nomu banaxovomu pros-
tori X, koly koefici[ntamy form [ oborotni linijni neperervni operatory Aj ,
Bj [9]. Zaznaçymo, wo dovedennq teoremy Skytovyça – Darmua ta ]] uzahal\-
nen\ zvodyt\sq do vyvçennq rozv’qzkiv rivnqnnq v xarakterystyçnyx funkciona-
lax
ˆ
j
n
j j jA u B
=
∗ ∗∏ +( )
1
µ v =
ˆ ˆ
j
n
j j
j
n
j jA u B
=
∗
=
∗∏ ∏( ) ( )
1 1
µ µ v , u, v ∈ ∗X , (1)
de X ∗
— prostir, sprqΩenyj do prostoru X, a Aj
∗
, Bj
∗
— operatory, sprqΩe-
© M. V. MYRONGK, 2008
1234 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
DO TEOREM SKYTOVYÇA – DARMUA TA XEJDE U BANAXOVOMU PROSTORI 1235
ni do operatoriv Aj , Bj .
U cij statti my navedemo nove dovedennq teoremy Skytovyça – Darmua u se-
parabel\nomu banaxovomu prostori. Zaznaçymo, wo ce dovedennq pryncypovo
vidriznq[t\sq vid dovedennq, navedenoho u [9]. U roboti [9] dovedennq teoremy
SkytovyçaK– Darmua u banaxovomu prostori provodylos\ za nastupnog sxemog.
Avtor pokazu[, wo xarakterystyçni funkcionaly rozpodiliv vypadkovyx vely-
çyn ne dorivnggt\ nulg, ta loharyfmu[ rivnqnnq (1). Potim special\nym çy-
nom obyra[ skinçennovymirnyj pidprostir, na qkomu intehru[ otrymane rivnqn-
nq, pomnoΩene na wil\nist\ deqkoho haussovoho rozpodilu. Dali, vykorysto-
vugçy ocinky rostu vynykagçyx pry dovedenni cilyx funkcij, pokazu[, wo roz-
v’qzkamy rivnqnnq (1) [ xarakterystyçni funkcionaly haussovyx rozpodiliv.
Blyz\ku ideg raniße vykorystav A. Zinher pry dovedenni uzahal\nennq teoremy
Skytovyça – Darmua u skinçennovymirnomu prostori (dyv., napryklad, [1], § 3.2).
Dovedennq teoremy Skytovyça – Darmua, navedene u cij statti, ©runtu[t\sq,
qk i klasyçne dovedennq ci[] teoremy, na vykorystanni metodu skinçennyx riz-
nyc\. Ves\ kompleksnyj analiz pry c\omu „pryxovano” u teoremax Kramera ta
Marcynkevyça, qki, oçevydno, magt\ misce i v banaxovomu prostori. Ideg c\oho
dovedennq raniße vykorystav H. Fel\dman pry dovedenni teoremy Skytovyça –
Darmua dlq lokal\no kompaktnyx abelevyx hrup (dyv. [3] ).
TakoΩ u cij statti dovedeno blyz\ku do teoremy Skytovyça – Darmua teore-
mu Xejde u banaxovomu prostori, de zamist\ nezaleΩnosti linijnyx form pered-
baça[t\sq, wo umovnyj rozpodil odni[] linijno] formy pry fiksovanij inßij [
symetryçnym.
My budemo vykorystovuvaty standartni fakty z teori] jmovirnisnyx rozpodi-
liv u banaxovomu prostori (dyv., napryklad, [14]). Nahada[mo deqki poznaçennq
ta oznaçennq. Nexaj X — separabel\nyj banaxiv prostir, X ∗
— sprqΩenyj do
n\oho prostir. Poznaçymo çerez Aut ( )X mnoΩynu usix oborotnyx neperervnyx
operatoriv, çerez 〈 x, f 〉 znaçennq funkcionala f X∈ ∗
na elementi x X∈ . Piv-
hrupu vidnosno zhortky jmovirnisnyx rozpodiliv na X poznaçymo çerez M
1( )X .
Dlq µ ∈ M
1( )X çerez µ poznaçymo rozpodil, wo vyznaça[t\sq formulog
µ( )E = µ(– )E dlq bud\-qko] borelevo] mnoΩyny E X⊂ . Xarakterystyçnyj
funkcional imovirnisnoho rozpodilu µ vypadkovo] velyçyny ξ zi znaçennqmy u
X vyznaça[t\sq formulog
ˆ ( )µ f = E e i f〈 〉[ ]ξ, =
X
i x fe d x∫ 〈 〉, ( )µ , f X∈ ∗
.
Zaznaçymo, wo ˆ ( )µ f = ˆ ( )µ f .
Rozpodil µ ∈ M 1( )R budemo nazyvaty haussovym, qkwo abo µ — vyrodΩe-
nyj rozpodil, abo µ [ absolgtno neperervnym vidnosno miry Lebeha u R ta ma[
wil\nist\ ρ( )t =
1
2πσ
e
t m
–
( – )2
2σ
, t ∈R, m ∈R , σ > 0. Xarakterystyçna funk-
ciq haussovoho rozpodilu ma[ vyhlqd ˆ ( )µ t = e
imt t– 1
2
2σ
, t ∈R, m ∈R , σ ≥ 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
1236 M. V. MYRONGK
Oznaçennq 1. Vypadkova velyçyna ξ nazyva[t\sq haussovog, qkwo dlq
bud\-qkoho elementa f X∈ ∗
haussovog [ vypadkova velyçyna 〈 〉ξ, f . Inßymy
slovamy, rozpodil µ na X nazyva[t\sq haussovym, qkwo dlq bud\-qkoho ele-
menta f X∈ ∗
joho odnovymirnyj obraz µ f = µ � f –1
[ haussovym rozpodi-
lom, a otΩe, isnugt\ dijsni çysla m f ta σ f ≥ 0 , dlq qkyx ˆ ( )µ f t =
= e
im t tf f– 1
2
2σ
, t ∈R.
Zaznaçymo, wo u banaxovomu prostori ma[ misce nastupne tverdΩennq (dyv.
[14, s. 4], §. 2.4).
TverdΩennq 1. Nexaj X — banaxovyj prostir. Xarakterystyçnyj funk-
cional haussovoho rozpodilu µ ∈ M
1( )X ma[ vyhlqd
ˆ ( )µ f = e
i m f Rf f〈 〉 〈 〉, – ,1
2
, f X∈ ∗
, (2)
de m X∈ , a R : X∗ → X — symetryçnyj nevid’[mnyj operator. Element m
nazyva[t\sq serednim rozpodilu µ, a R — joho kovariacijnym operatorom.
Navpaky, qkwo µ ∈ M
1( )X z xarakterystyçnym funkcionalom vyhlq-
duK(2), de m X∈ — deqkyj element, a R : X∗ → X — deqkyj symetryçnyj
nevid’[mnyj operator, to µ — haussiv rozpodil u X z serednim m ta kova-
riacijnym operatorom R .
Vraxovugçy te, wo ˆ ( )µ t f = ˆ ( )µ f t dlq bud\-qkyx f X∈ ∗
, t ∈R, otrymu[mo
〈 〉m f, = mf , 〈 〉R f f, = σ f . (3)
2. Teoremy Kaca – Bernßtejna, Kramera ta Marcynkevyça u banaxovo-
mu prostori. Z oznaçennq 1 bezposeredn\o vyplyva[ nastupne tverdΩennq.
TverdΩennq 2. Rozpodil µ na X haussiv todi i lyße todi, koly dlq
bud\-qkoho f X∈ ∗
funkciq ˆ ( )µ f t [ xarakterystyçnog funkci[g deqkoho ha-
ussovoho rozpodilu na R.
Za dopomohog tverdΩennq 2 lehko otrymaty analohy dlq dijsnoho separa-
bel\noho banaxovoho prostoru takyx vidomyx klasyçnyx teorem, qk teorema Ka-
ca – Bernßtejna pro xarakteryzacig haussovoho rozpodilu nezaleΩnistg sumy
ta riznyci nezaleΩnyx vypadkovyx velyçyn, teorema Kramera pro rozklad haus-
sovoho rozpodilu [15] (§ 4.1), teorema Marcynkevyça [15] (§ 3.13). Dovedennq
cyx teorem zvodyt\sq do opysu rozv’qzkiv vidpovidnyx funkcional\nyx rivnqn\,
qki prypuskagt\ zvuΩennq na koΩnyj odnovymirnyj pidprostir. Proilgstru[-
mo vykladene vywe na prykladi teoremy Kaca – Bernßtejna.
Teorema A (analoh teoremy Kaca – Bernßtejna). Nexaj ξ 1 ta ξ2 — ne-
zaleΩni vypadkovi velyçyny zi znaçennqmy u dijsnomu separabel\nomu banaxovo-
mu prostori X ta z rozpodilamy µ ta ν vidpovidno. Qkwo ξ1 + ξ2 ta
ξ1 – ξ2 [ nezaleΩnymy, to µ ta ν — haussovi rozpodily z odnakovymy kova-
riacijnymy operatoramy.
Dovedennq. Qk i v klasyçnomu vypadku (dyv., napryklad, [1], §. 3.1), lehko
baçyty, wo umova nezaleΩnosti ξ1 + ξ2 ta ξ1 – ξ2 rivnosyl\na tomu, wo xarak-
terystyçni funkcionaly ˆ ( )µ f ta ˆ ( )ν f zadovol\nqgt\ rivnqnnq
ˆ ( ) ˆ ( – )µ νf g f g+ = ˆ ( ) ˆ ( ) ˆ ( )ˆ (– )µ µ ν νf g f g , f, g X∈ ∗
. (4)
Zafiksu[mo f X∈ ∗
ta rozhlqnemo zvuΩennq rivnqnnq (4) na odnovymirnyj pid-
prostir L = t f{ , t ∈ }R . Z teoremy Kaca – Bernßtejna vyplyva[, wo ˆ ( )µ t f ta
ˆ ( )ν t f — xarakterystyçni funkci] deqkyx haussovyx rozpodiliv. Oskil\ky ele-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
DO TEOREM SKYTOVYÇA – DARMUA TA XEJDE U BANAXOVOMU PROSTORI 1237
ment f [ dovil\nym, to dlq bud\-qkoho f X∈ ∗
funkci] ˆ ( )µ f t ta ˆ ( )ν f t [ xa-
rakterystyçnymy funkciqmy deqkyx haussovyx rozpodiliv. Z tverdΩennq 2
vyplyva[, wo ˆ ( )µ f ta ˆ( )ν f — xarakterystyçni funkcionaly haussovyx rozpo-
diliv z kovariacijnymy operatoramy R
1
ta R
2
vidpovidno. Zaznaçymo, wo z teo-
remy Kaca – Bernßtejna vyplyva[, wo dlq bud\-qkoho f X∈ ∗
ˆ ( )µ f t = e
im t tf f– σ 2
, ˆ ( )ν f t = e
in t tf f– σ 2
,
de m f , nf ∈R , σ f ≥ 0. Zvidsy ta z (3) vyplyva[, wo R
1
= R
2
.
Teoremu dovedeno.
ZauvaΩennq 1. U roboti [2] A. Ruxin doviv analoh teoremy Kaca – Bern-
ßtejna dlq lokal\no kompaktnyx abelevyx hrup z odnoznaçnym dilennqm na 2.
Vin zaznaçyv, wo analohiçnymy mirkuvannqmy moΩna dovesty teoremy Kaca –
Bernßtejna dlq vypadkovyx velyçyn, wo nabuvagt\ znaçen\ u hil\bertovomu
prostori. Pizniße teoremu A u hil\bertovomu prostori peredoviv R. Laxa [10].
Qk my baçymo, teoremaKA ne potrebu[ special\noho dovedennq, a bezposeredn\o
vyplyva[ z teoremy Kaca – Bernßtejna.
Nexaj P f( ) — funkciq na X∗
, h — dovil\nyj element X∗
. Poznaçymo çe-
rez ∆h operator skinçenno] riznyci
∆hP f( ) = P f h P f( ) – ( )+ .
Mnohoçlenom na prostori X∗
nazyva[t\sq neperervna funkciq P f( ) taka, wo
pry deqkomu n
∆h
n P f+1 ( ) = 0
dlq vsix f, h X∈ ∗
.
Qk i u vypadku teoremyKA, z spravedlyvosti teorem Kramera ta Marcynkevy-
ça na dijsnij prqmij vyplyva[ ]x spravedlyvist\ u banaxovomu prostori.
Teorema V (analoh teoremy Kramera). Qkwo µ — haussiv rozpodil na dij-
snomu separabel\nomu banaxovomu prostori X ta µ = µ µ1 2∗ , to µ1 ta µ2
takoΩ haussovi rozpodily na X.
Teorema S (analoh teoremy Marcynkevyça). Nexaj X — dijsnyj separa-
bel\nyj banaxiv prostir, µ ∈ M 1( )X . Qkwo
ˆ ( )µ f = eP f( )
, (5)
de P f( ) — mnohoçlen, to µ — haussiv rozpodil na X.
Zaznaçymo, wo z uraxuvannqm (2) dovedennq teoremyKV zvodyt\sq do opysu
rozv’qzkiv rivnqnnq
e
i m f Rf f〈 〉 〈 〉, – ,1
2 = w f w f1 2( ) ( ) , f X∈ ∗
, (6)
u klasi xarakterystyçnyx funkcionaliv imovirnisnyx rozpodiliv na X.
ZauvaΩennq. 2. TeoremuKS moΩna posylyty. A same, qkwo (5) ma[ misce v
deqkomu okoli nulq, to, zastosovugçy tverdΩennq 2 ta rezul\taty rozdilu 2 v
[16], otrymu[mo, wo µ — haussiv rozpodil na X.
3. Mirkuvannq, wo dovodqt\ teoremuKA, rivnosyl\ni teoretyko-jmovirnisno-
mu mirkuvanng, tobto bez perexodu do rivnqn\ u klasi xarakterystyçnyx funk-
cionaliv. Dijsno, nexaj f X∈ ∗
. Todi skalqrni vypadkovi velyçyny 〈 〉ξ1, f ta
〈 〉ξ2, f [ nezaleΩnymy i 〈 〉ξ1, f + 〈 〉ξ2, f ta 〈 〉ξ1, f – 〈 〉ξ2, f takoΩ nezaleΩni.
Z teoremy Kaca – Bernßtejna vyplyva[, wo vypadkovi velyçyny 〈 〉ξ1, f ta
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
1238 M. V. MYRONGK
〈 〉ξ2, f [ haussovymy. Oskil\ky f — dovil\nyj element X∗
, to vypadkovi vely-
çyny ξ
1
ta ξ
2
takoΩ haussovi. Analohiçne dovedennq teoremyKV [ v [17]
(§ 6.3.4) dlq hil\bertovoho prostoru.
4. Vidomog [ teorema Hiri – Lukaça – Laha, v qkij haussiv rozpodil na dijsnij
prqmij xarakteryzu[t\sq nezaleΩnistg linijno] ta kvadratyçno] form vid neza-
leΩnyx vypadkovyx velyçyn (dyv. [1], § 4.2, [15], § 8.3). U roboti [18] cej re-
zul\tat bulo uzahal\neno na banaxovi prostory. Qk v zauvaΩenni 3, tak i pry do-
vedenni neobxidnosti osnovno] teoremy roboty [18] vykorystovugt\sq ti Ω sami
mirkuvannq.
3. Teorema Skytovyça – Darmua v banaxovomu prostori. Dovedemo analoh
teoremy Skytovyça – Darmua v banaxovomu prostori.
Teorema 1 (analoh teoremy Skytovyça – Darmua). Nexaj ξ
1
, … , ξn , n ≥ 2,
— nezaleΩni vypadkovi velyçyny zi znaçennqmy u dijsnomu separabel\nomu bana-
xovomu prostori X ta z rozpodilamy µ j . Nexaj Aj , Bj ∈ K Aut ( )X . Qkwo
linijni formy L1 = A1 1ξ + … + An nξ ta L2 = B1 1ξ + … + Bn nξ [ nezaleΩnymy,
to µ j — haussovi rozpodily.
Dovedennq. Pokladagçy ′ξ j = Aj jξ , zvodymo dovedennq do vypadku, koly
linijni formy magt\ vyhlqd L1 = ξ
1
+ … + ξn ta L2 = C1 1ξ + … + Cn nξ , de
Cj ∈K Aut ( )X . Qk i u klasyçnomu vypadku (dyv., napryklad, [1], § 3.1), lehko ba-
çyty, wo umova nezaleΩnosti L1 ta L 2 rivnosyl\na tomu, wo xarakterystyçni
funkcionaly ˆ ( )µ j f zadovol\nqgt\ rivnqnnq
j
n
j jf C g
=
∗∏ +( )
1
µ̂ =
j
n
j
j
n
j jf C g
= =
∗∏ ∏ ( )
1 1
ˆ ( ) ˆµ µ , f, g X∈ ∗
, (7)
de Cj
∗
— operator, sprqΩenyj do Cj . Takym çynom, dovedennq teoremyK1 zvo-
dyt\sq do opysu rozv’qzkiv funkcional\noho rivnqnnq (7) u klasi xarakterys-
tyçnyx funkcionaliv imovirnisnyx rozpodiliv na X∗
. Zaznaçymo, wo na vidminu
vid teorem A – S my ne moΩemo skorystatysq tverdΩennqm 2 ta zvesty dove-
dennq teoremyK1 do klasyçno] teoremy Skytovyça – Darmua. Sprava v tomu,
wo,Kvzahali kaΩuçy, na vidminu vid rivnqn\, do qkyx zvodqt\sq dovedennq teo-
remKA – S, rivnqnnq (7) ne dopuska[ zvuΩennq na koΩnyj odnovymirnyj pidpros-
tir uKK X∗
.
Poklademo ν j = µ µj j∗ . Todi ˆ ( )ν j f = ˆ ( )µ j f
2
≥ 0. Oçevydno, wo xarakte-
rystyçni funkcionaly ˆ ( )ν j f takoΩ zadovol\nqgt\ rivnqnnq (7). Qkwo my do-
vedemo, wo ˆ ( )ν j f — xarakterystyçni funkcionaly haussovyx rozpodiliv, to z
teoremyKV bude vyplyvaty, wo ˆ ( )µ j f takoΩ xarakterystyçni funkcionaly
haussovyx rozpodiliv. Ce dozvolq[ s samoho poçatku vvaΩaty, wo ˆ ( )µ j f ≥ 0.
Perevirymo spoçatku, wo ˆ ( )µ j f > 0. Cej fakt moΩna dovesty tak samo, qk i
u vypadku X = Rn
(dyv. [1], § 3.2). Dlq povnoty vykladu navedemo tut inße do-
vedennq (dyv. [19]). Zaznaçymo, wo z c\oho dovedennq vyplyva[, wo na bud\-qkij
zv’qznij abelevij hrupi vsi vidminni vid nulq v nuli hrupy neperervni rozv’qzky
rivnqnnq (7), de Cj
∗
— topolohiçni avtomorfizmy hrupy, ne dorivnggt\ nulg.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
DO TEOREM SKYTOVYÇA – DARMUA TA XEJDE U BANAXOVOMU PROSTORI 1239
Poznaçymo Lj = f X∈{ ∗
: ˆ ( )µ j f ≠ }0 , L =
j
n
jL=1∩ , M = g X∈{ ∗
:
j
n
j jC g=
∗∏ ( )1
µ̂ ≠ 0} , N =
j
n
jC M=
∗
1∩ ( ) . Z ohlqdu na (7) lehko baçyty, wo L +
+ N � L. Poznaçymo ( )k N = f X∈{ ∗ : f = f1 + … + fk , f Nj ∈ }. Todi z vklgçennq
L + N � L vyplyva[, wo L +
k
k N=
∞
1∪ ( ) � L . Oskil\ky 0 ∈N , to L +
+
k
k N=
∞
1∪ ( ) = L. Oskil\ky N — vidkryta mnoΩyna, wo mistyt\ nul\, to vnasli-
dok zv’qznosti X∗
ma[mo X∗ =
k
k N=
∞
1∪ ( ) . OtΩe, X∗ = L, tobto ˆ ( )µ j f > 0
dlq vsix f X∈ ∗
, j = 1, 2, … , n.
PokaΩemo, wo ˆ ( )µ j f — xarakterystyçnyj funkcional haussovoho rozpodi-
lu. Poklademo ψ j f( ) = –ln ˆ ( )µ j f . Z (7) vyplyva[
j
n
j jf C g
=
∗∑ +( )
1
ψ = P f( ) + Q g( ) , f, g X∈ ∗
, (8)
de
P f( ) =
j
n
j f
=
∑
1
ψ ( ), Q g( ) =
j
n
j jC g
=
∗∑ ( )
1
ψ .
Dlq rozv’qzannq (8) skorysta[mosq metodom skinçennyx riznyc\, vykorystav-
ßy sxemu dovedennq teoremy Skytovyça – Darmua dlq lokal\no kompaktnyx
abelevyx hrup (dyv. [3]). Nexaj hn — dovil\nyj element X∗
. Poklademo kn =
= –
–
Cn
∗( ) 1
hn . Todi hn + Cn
∗ kn = 0. Nadamo v (8) zminnym f ta g pryrosty hn ta
kn vidpovidno. Vidnimemo vid otrymanoho rivnqnnq (8) ta znajdemo
j
n
l j jn j
f C g
=
∗∑ +( )
1
1–
,
∆ ψ = ∆hn
P f( ) + ∆kn
Q g( ) , f, g X∈ ∗
, (9)
de ln j, = hn + C kj n
∗ = C C kj n n
∗ ∗( )– , j = 1, 2, … , n – 1. Zaznaçymo, wo liva çastyna
rivnqnnq (9) ne mistyt\ funkcig ψn . Nexaj hn –1 — dovil\nyj element X∗
.
Poklademo kn –1 = – –
–
Cn 1
1∗( ) hn –1. Todi hn –1 + Cn –1
∗ kn –1 = 0. Nadamo v (9) zmin-
nym f ta g pryrosty hn –1 ta kn –1 vidpovidno. Vidnimagçy vid otrymanoho
rivnqnnq (9), oderΩu[mo
j
n
l l j jn j n j
f C g
=
∗∑ +( )
1
2
1
–
– , ,
∆ ∆ ψ = ∆ ∆h h P f
2 1
( ) + ∆ ∆k k Q g
2 1
( ) , f, g X∈ ∗
, (10)
de ln j– ,1 = hn –1 + C kj n
∗
–1 = C Cj n
∗ ∗( )– –1 kn –1, j = 1, 2, … , n – 2. Liva çastyna riv-
nqnnq (10) ne mistyt\ funkcij ψn ta ψn –1. Mirkugçy analohiçno, pryxodymo
do rivnqnnq
∆ ∆l l2 1 3 1, ,
K… ∆l n
f C g
,1 1 1ψ +( )∗ =
= ∆ ∆h h2 3
… ∆hn
P f( ) + ∆ ∆k k2 3
… ∆k n
Q g( ) , f, g X∈ ∗ , (11)
de hm — dovil\nyj element X∗
, km = –
–
Cm
∗( ) 1
hm , lm,1 = hm + C km1
∗ =
= C Cm1
∗ ∗( )– km , m = 2, 3, … , n. Nexaj h1 — dovil\nyj element X∗
. Poklademo
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
1240 M. V. MYRONGK
k1 = –
–
C h1
1
1
∗( ) . Todi h1 + C k1 1
∗ = 0. Nadamo v (9) zminnym f ta g pryrosty h1
ta k1 vidpovidno. Vidnimagçy vid otrymanoho rivnqnnq (11), znaxodymo
∆ ∆h h1 2
… ∆hn
P f( ) + ∆ ∆k k1 2
… ∆k n
Q g( ) = 0, f, g X∈ ∗
. (12)
Nexaj h — dovil\nyj element X∗
. Nadamo v (12) zminnij f pryrist h. Vidni-
magçy vid otrymanoho rivnqnnq (12), oderΩu[mo
∆ ∆ ∆h h h1 2
… ∆hn
P f( ) = 0, f X∈ ∗
. (13)
Zaznaçymo, wo f, h ta hm , m = 1, … , n, — dovil\ni elementy X∗
. Poklade-
mo v (13) h1 = … = hn = h. Todi
∆h
n P f+1 ( ) = 0, f, h X∈ ∗
. (14)
OtΩe, P f( ) — mnohoçlen na X∗
. Poznaçymo γ = µ1 ∗ … ∗ µn . Todi ˆ( )γ f =
=
j
n
j f=∏ 1
ˆ ( )µ . OtΩe, ˆ( )γ f = e P f– ( )
. Z teoremyKS vyplyva[, wo ˆ( )γ f — xarak-
terystyçnyj funkcional haussovoho rozpodilu. Todi z teoremyKV otrymu[mo,
wo i ˆ ( )µ j f — xarakterystyçnyj funkcional haussovoho rozpodilu.
TeoremuK1 dovedeno.
4. Teorema Xejde v banaxovomu prostori. Blyz\kyj do teoremy Skytovyça
– Darmua rezul\tat doviv Xejde, qkyj zamist\ nezaleΩnosti linijnyx form
peredbaçav, wo umovnyj rozpodil odni[] linijno] formy pry fiksovanij inßij [
symetryçnym [1] (§ 13.4.1).
Teorema Xejde. Nexaj ξ1, … , ξn — nezaleΩni vypadkovi velyçyny, L1 =
= α ξ1 1 + … + α ξn n ta L 2 = β ξ1 1 + … + β ξn n — linijni formy, v qkyx koefici-
[nty α j , β j — vidminni vid nulq konstanty taki, wo β αi i
–1 ± β αj j
–1 ≠ 0
dlq vsix i ≠ j. Qkwo umovnyj rozpodil L2 pry fiksovanij L1 symetryçnyj,
to vsi vypadkovi velyçyny ξ j [ haussovymy.
Analohiçnyj rezul\tat ma[ misce i u dijsnomu separabel\nomu banaxovomu
prostori.
Teorema 2 (analoh teoremy Xejde). Nexaj ξ1, … , ξn , n ≥ 2, — nezaleΩni
vypadkovi velyçyny zi znaçennqmy u dijsnomu separabel\nomu banaxovomu pros-
tori X ta z rozpodilamy µ j , Aj , B Xj ∈Aut ( ), pryçomu B Ai i
–1 ± B Aj j
–1
K∈
∈ K Aut ( )X dlq vsix i ≠ j. Qkwo umovnyj rozpodil L2 = B1 1ξ + … + Bn nξ pry
fiksovanij L1 = A1 1ξ + … + An nξ [ symetryçnym, to µ j — haussiv rozpodil.
Dovedennq. Qk i pry dovedenni teoremyK1, pokazu[mo, wo dovedennq teore-
myK2 zvodyt\sq do vypadku, koly linijni formy magt\ vyhlqd L 1 = ξ1 + … + ξn
ta L2 = C1 1ξ + … + Cn nξ , de C Xj ∈Aut ( ) taki, wo C Ci j± ∈ Aut ( )X dlq vsix
i ≠ j, a xarakterystyçni funkcionaly ˆ ( )µ j f ≥ 0. Qk i v roboti [4], pokazu[mo,
wo umova symetri] umovnoho rozpodilu L2 pry fiksovanij L1 rivnosyl\na to-
mu, wo xarakterystyçni funkcionaly ˆ ( )µ j f zadovol\nqgt\ rivnqnnq
j
n
j jf C g
=
∗∏ +( )
1
µ̂ =
j
n
j jf C g
=
∗∏ ( )
1
ˆ –µ , f, g X∈ ∗
, (15)
de Cj
∗
— operatory, sprqΩeni do Cj . Oskil\ky ˆ ( )µ j 0 = 1, to isnu[ takyj okil
nulq U, wo vsi ˆ ( )µ j f > 0 pry f U∈ . Vyberemo v U takyj symetryçnyj okil
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
DO TEOREM SKYTOVYÇA – DARMUA TA XEJDE U BANAXOVOMU PROSTORI 1241
nulq V, wob
j
n
j V U
=
∑ ⊂
1
4
λ ( ) , λ j ∈ I C Cn, , ,1
∗ ∗…{ } .
Loharyfmugçy v okoli nulq V rivnqnnq (15), otrymu[mo
j
n
j jf C g
=
∗∑ +( )
1
ψ =
j
n
j jf C g
=
∗∑ ( )
1
ψ – , f, g V∈ , (16)
de ψ j f( ) = –ln ˆ ( )µ j f . Budemo rozv’qzuvaty otrymane rivnqnnq metodom skin-
çennyx riznyc\ tak samo, qk i u klasyçnomu vypadku (dyv. [1], § 13.4.1).
Nexaj hn — dovil\nyj element V. Nadamo v (16) zminnym f ta g pryrosty
C hn n
∗
ta hn vidpovidno. Vidnimagçy vid otrymanoho rivnqnnq (16), ma[mo
j
n
l j jn j
f C g
=
∗∑ +( )
1
∆
,
ψ =
j
n
m j jn j
f C g
=
−
∗∑ ( )
1
1
∆
,
–ψ , f, g V∈ , (17)
de ln j, = C Cn j
∗ ∗+( ) hn , mn j, = C Cn j
∗ ∗( )– hn . Zaznaçymo, wo prava çastyna rivnqn-
nq (17) ne mistyt\ funkcig ψn . Nexaj hn –1 — dovil\nyj element V. Nadamo v
(17) zminnym f ta g pryrosty Cn –1
∗ hn –1 ta hn –1 vidpovidno. Vidnimagçy vid
otrymanoho rivnqnnq (17), oderΩu[mo
j
n
l l j jn j n j
f C g
=
∗∑ +( )
1
1
∆ ∆
– , ,
ψ =
j
n
m m j jn j n j
f C g
=
∗∑ ( )
1
2
1
–
– , ,
–∆ ∆ ψ , f, g V∈ , (18)
de ln j– ,1 = C Cn j–1
∗ ∗+( ) hn –1, mn j, = C Cn j– –1
∗ ∗( ) hn –1. Prava çastyna rivnqnnq
(18) ne mistyt\ funkcij ψn ta ψn –1. Mirkugçy analohiçno, pryxodymo do
rivnqnnq
j
n
l l l j jj j n j
f C g
=
∗∑ … +( )
1
1 2
∆ ∆ ∆
, , ,
ψ = 0, f, g V∈ , (19)
de hi — dovil\nyj element V, li j, = C Ci j
∗ ∗+( ) hi , i = 1, 2, … , n.
Nexaj kn — dovil\nyj element V. Nadamo v (19) zminnym f ta g pryrosty
C hn n
∗
ta –hn vidpovidno. Vidnimagçy vid otrymanoho rivnqnnq (19), ma[mo
j
n
b l l l j jn j j j n j
f C g
=
∗∑ … +( )
1
1
1 2
–
, , , ,
∆ ∆ ∆ ∆ ψ = 0, f, g V∈ , (20)
de bn j, = C Cn j
∗ ∗( )– kn . Zaznaçymo, wo liva çastyna rivnqnnq (20) ne mistyt\
funkcig ψn . Nexaj kn –1 — dovil\nyj element V. Nadamo v (20) zminnym f ta
g pryrosty C kn n– –1 1
∗
ta – –kn 1 vidpovidno. Vidnimagçy vid otrymanoho rivnqn-
nq (20), oderΩu[mo
j
n
b b l l j jn j n j j n j
f C g
=
∗∑ … +( )
1
2
1 1
–
– , , , ,
∆ ∆ ∆ ∆ ψ = 0, f, g V∈ , (21)
de bn j−1, = C Cn j– –1
∗ ∗( ) kn –1. Prava çastyna rivnqnnq (21) ne mistyt\ funkcij
ψn i ψn –1. Mirkugçy analohiçno, pryxodymo do rivnqnnq
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
1242 M. V. MYRONGK
∆ ∆ ∆ ∆ ∆ ∆b b b l l ln n
f C g
2 1 3 1 1 1 1 2 1 1 1 1, , , , , ,
… … +( )∗ψ = 0, f, g V∈ , (22)
de ki — dovil\nyj element V, bi,1 = C Ci
∗ ∗( )– 1 ki , i = 2, 3, … , n.
Z toho, wo ki , hi — dovil\ni elementy V, C Ci j
∗ ∗± ∈ Aut ( )X∗
dlq usix i ≠ j,
a li j, = C Ci j
∗ ∗+( ) hi , bi,1 = C Ci
∗ ∗( )– 1 ki , otrymu[mo, wo v deqkomu okoli nulq W
∆h
n f2 1
1
– ( )ψ = 0, f, h W∈ . (23)
OtΩe, v okoli nulq W ˆ ( )µ1 f = e P f– ( )
, de P f( ) — mnohoçlen. Z zauvaΩennq 2
vyplyva[, wo µ1 — haussovi rozpodily. Analohiçno otrymu[mo, wo vsi µ j —
haussovi rozpodily.
TeoremuK2 dovedeno.
ZauvaΩennq 5. Teoremy Skytovyça – Darmua ta Xejde analohiçnym çynom
moΩna dovesty i v separabel\nomu prostori Freße.
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2. Ruxyn A. L. Ob odnoj teoreme S. N. Bernßtejna // Mat. zametky. – 1969. – 6, # 3. – S. 307 –
310.
3. Feldman G. M. A characterization of the Gaussian distribution on Abelian groups // Probab. Theo-
ry and Relat. Fields. – 2003. – 126. – P. 91 – 102.
4. Feldman G. M. On a characterization theorem for locally compact Abelian groups // Ibid. – 2005. –
133. – P. 345 – 357.
5. Feldman G. M. Arithmetic of probability distributions and characterization problems on Abelian
groups // AMS Transl. Math. Monogr. – Providence, RI, 1993.
6. Feldman G. M., Graczyk P. On the Skitovich – Darmois theorem on compact Abelian groups // J.
Theor. Probab. – 2000. – 13, # 3. – P. 859 – 869.
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bab. and Math. Statistics. – 2000. – 20, # 1. – P. 141 – 149.
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9. Krakowiak W. The theorem of Darmois – Skitovič for Banach valued random variables // Ann.
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P. 85 – 93.
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tist. – 1962. – 33. – P. 533 – 541.
14. Vaxanyq N. N., Taryeladze V. Y., Çobanqn S. A. Veroqtnostn¥e raspredelenyq v banaxov¥x
prostranstvax. – M.: Nauka, 1985. – 368 s.
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1972. – 480 s.
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pislq doopracgvannq — 29.02.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
|
| id | umjimathkievua-article-3239 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:38:46Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b5/338cc20522dc12dd3a8dae68b1c2acb5.pdf |
| spelling | umjimathkievua-article-32392020-03-18T19:48:57Z On the Skitovich-Darmois theorem and Heyde theorem in a Banach space До теорем Скитовича - Дармуа та Хейде у банаховому просторі Myronyuk, M. V. Миронюк, М. В. According to the well-known Skitovich-Darmois theorem, the independence of two linear forms of independent random variables with nonzero coefficients implies that the random variables are Gaussian variables. This result was generalized by Krakowiak for random variables with values in a Banach space in the case where the coefficients of forms are continuous invertible operators. In the first part of the paper, we give a new proof of the Skitovich-Darmois theorem in a Banach space. Heyde proved another characterization theorem similar to the Skitovich-Darmois theorem, in which, instead of the independence of linear forms, it is supposed that the conditional distribution of one linear form is symmetric if the other form is fixed. In the second part of the paper, we prove an analog of the Heyde theorem in a Banach space. Известная теорема Скитовича - Дармуа утверждает, что из независимости двух линейных форм от независимых случайных величин с ненулевыми коэффициентами следует, что случайные величины являются гауссовыми. Этот результат был обобщен Краковяком для случайных величин со значениями в банаховом пространстве, когда коэффициентами форм являются непрерывные оборотные операторы. В первой части работы приведено новое доказательство теоремы Скитовича - Дармуа в банаховом пространстве. Хейде доказал близкую к теореме Скитовича - Дармуа характеризационную теорему, в которой вместо независимости линейных форм предполагалось, что условное распределение одной линейной формы при фиксированной другой является симметричным. Во второй части работы доказан аналог теоремы Хейде в банаховом пространстве. Institute of Mathematics, NAS of Ukraine 2008-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3239 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 9 (2008); 1234–1242 Український математичний журнал; Том 60 № 9 (2008); 1234–1242 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3239/3224 https://umj.imath.kiev.ua/index.php/umj/article/view/3239/3225 Copyright (c) 2008 Myronyuk M. V. |
| spellingShingle | Myronyuk, M. V. Миронюк, М. В. On the Skitovich-Darmois theorem and Heyde theorem in a Banach space |
| title | On the Skitovich-Darmois theorem and Heyde theorem in a Banach space |
| title_alt | До теорем Скитовича - Дармуа та Хейде у банаховому просторі |
| title_full | On the Skitovich-Darmois theorem and Heyde theorem in a Banach space |
| title_fullStr | On the Skitovich-Darmois theorem and Heyde theorem in a Banach space |
| title_full_unstemmed | On the Skitovich-Darmois theorem and Heyde theorem in a Banach space |
| title_short | On the Skitovich-Darmois theorem and Heyde theorem in a Banach space |
| title_sort | on the skitovich-darmois theorem and heyde theorem in a banach space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3239 |
| work_keys_str_mv | AT myronyukmv ontheskitovichdarmoistheoremandheydetheoreminabanachspace AT mironûkmv ontheskitovichdarmoistheoremandheydetheoreminabanachspace AT myronyukmv doteoremskitovičadarmuatahejdeubanahovomuprostorí AT mironûkmv doteoremskitovičadarmuatahejdeubanahovomuprostorí |