On One Property of the Wiener Integral and its Statistical Application

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1 saqatvelos eceebata eovl aaes oabe # 9 BUETIN OF THE GEORGIAN NATIONA AADEM OF SIENES vol o 9 Maheacs O Oe Pope o he Wee Ieal a s Sascal Applcao Pee Babla* Elzba Naaaa** Mzeva Pasasa & Gol Sohaze # * I Javahshvl Tbls Sae Uves ** Acae Mebe I Javahshvl Tbls Sae Uves & Soh Uves # A Tseeel Sae Uves as ABSTRAT Fo he Wee eal oe pope o veso s esablshe Ths pope s se o cosco o opaaec sascal esao o he o loahc evave o sbo ao pocesses hch s obseve Wee ose 9 Bll Geo Nal Aca Sc e os: Wee eal loahc evave opaaec esao e he s o epee ao pocesses ξ η hee s a saa Wee pocess be e obsevao Us he obseve vales (aecoes ξ ξ ξ e ae o cosc a esae o he o loahc evave o he sbo o he pocess η Fo ao vales (he η a o o epe o hs poble s cosee [] he oe-esoal case To solve he poble e appl he echqe o opaaec esao o Wee eals I he s pa o hs o e ve oe veso heoe o he Wee eal Ths heoe eealzes he ell-o esl o Rooe [] o he oe-esoal case I he Wee eal heo hs esl s o epee ees a eables s o cosc he seco pa o he o e-esoal appoaos o esao o he o loahc evave e-esoal space We ll cose he space o coos cos o [ ] he -aleba o Boel sbses a he Wee ease μ o e he ( a be pos o [ ] We ll eoe b ( he F be soe coal o I s asse ha boe le h veces a he pos ( ( ( F ( F ( ( e I s esoo ha o e have F ( F he sese o so coveece I s ell o ([] ha o a coos boe coal F ebese eal h espec o a Wee ease ( s hs eal ha s calle he Wee eal ca be ee b he eqal F μ ( l π 9 Bll Geo Nal Aca Sc

2 O Oe Pope o he Wee Ieal a s Sascal Applcao hee F ep E ( E s a -esoal Eclea space A aaloos ola also hols o lple Wee eals le s cose he aso hch e call he Wee Fo a coos boe coal aso o ( μ ( (ea he e he coseao o he s o epee ao elees e ae a oce he ollo B E e he he ea e E be a sepaable lea opolocal space h a Boel -aleba o s sbses epee ao elees ξ a η h vales E a h pobabl sbos ( A P{ ω ξ A} ν ( A P{ ω : η A} especvel o B ( E be ee o he pobabl space { I P} ζ ξ ca a sbo ( A P{ ω : ζ A} he covolo o ( A μ( A ν ( A B( E E μ : a Ω Fo he s η e (* hee μ ( A eoes he sh o μ oo he veco E Iee b he eealze ola o oal pobabl e oba { η A} P{ ξ A η } P{ η } P{ ξ A } ν ( P ξ E Whch s eqvale o (* I π ( A s a ease sch ha slaeosl boh a μ ae absolel coos h espec o π (o π 5 μ he (* ca be ee he o eaple μ ρ ρ μ ( ν ( hee ρ a ρ μ π π E hch s sall apple he e-esoal case o sbo eses h espec o a ebese ease The ole hch hs epesso plas vaos pobles o aalss a pacla he heo o paabolc eeal eqaos s ell-o (see o eaple [] We ae eese he qeso o veso o aso ( Fo he sae o splc e ve he esl o [] hch has bee eoe above e ξ be a saa oall sbe ao vale h paaees ( e he R be a co sch ha he covolo ϕ hee ϕ ep π s ee Theoe (Rooe [] Ue he above asspos ( τ l τ We ea eael ha he co a all s evaves ae as a hole he pass hs s o he l e ca e ( Sch sple vaa o Rooe s heoe as eealzao E ae boe he Bll Geo Nal Aca Sc vol o 9

3 Pee Babla Elzba Naaaa Mzeva Pasasa Gol Sohaze Bll Geo Nal Aca Sc vol o 9 Theoe e he co have evaves o all oes a he ae oehe h s evaves as a hole be boe The he eal eqao ep π ( has a solo he class o boe cos A solo o ( ca be epesee as hee a s s s s ϕ ϕ s a oal evave o s-h oe o he co ϕ ϕ hee s s a o-eave ee be Poo The o co ll be soh oall o e he eal as a Macla sees a he po : Sbs hs epesso ( a a o acco ha o > o o ep π e oba ( ( Ae eleea asoaos e ca e ( We eeae hs eqal o es a lpl boh pas o he esl eqal b he cosa Ae ha e eeae ( o es a lpl boh pas o he esl eqal b he cosa a so o I he eeal case e eeae eqal ( es a lpl boh pas o he esl eqal b a so o I ao o ( e oba a be o eqales: 6 6 ( (6

4 O Oe Pope o he Wee Ieal a s Sascal Applcao Bll Geo Nal Aca Sc vol o 9 No o ( e sbac (5 a (6 sbac he e eqal a he e eqal a so o We o so a o acco he cobaoal e The e all oba (7 hch coces h he eqal e ae o pove All hese oal asoaos ae se becase as ca be easl vee b he coo o he heoe he absole vale o he eeal e o he oal sees (7 vashes a a scel qc ae a hee (7 covees absolel a ol The heoe s pove oolla I he coos o Theoe e cose he eqal ep π (8 The (9 Iee e oce he oao ϕ ψ he (8 aes he o ep ψ π ϕ o hch Theoe s applcable Ae sbso a asoao e oba (9 e s o o ( Isea o ( e cose he eqal ep ψ π E ep ( Ioc he oao e e ( he o

5 Pee Babla Elzba Naaaa Mzeva Pasasa Gol Sohaze Bll Geo Nal Aca Sc vol o 9 π ep ( e s appl Theoe o eqal ( o Fo hs e ee hs eqal as ep e π Us oolla e oba ep ( Fo he se o oolla e ee ( he o π ep e B ola (9 e oba ep Pocee so sep b sep -es aloehe e all oba

6 O Oe Pope o he Wee Ieal a s Sascal Applcao 5 Bll Geo Nal Aca Sc vol o 9 ( Ths e have sho ha ( ca be vee a he vese s e o ( Noe ha ola ( e asse a sep-b-sep applcao o eeal eqaos o he cos The poe o he ae a he paal ae s calclae as sal b he Neo ola o hhe evaves e s asse ha s ae o he class o boe coals oehe h he evaves The e ca asse ha he sees ( covee ol a absolel Iee as see o eqal ( hee es all evaves a he ae boe The o he eeal e ( e oba a esae M (I he epesso o he le s asse o be he ce cosa hle ae he vaables o he eeal e o he sees Ths esae ( ves he ese poo Sce he sees ( covee absolel a ol e ca pass o he l e he sao s The e oba he l he vese o he Wee aso (: l ( Theeoe he ollo saee s e Theoe The vese aso o he Wee aso ( ess he class o coals oehe h he evaves o all oes hch ae boe ae as a hole a he vese aso ca be calclae b ola ( hee s he ohooal poeco o he coal Rea Feqel sea o ( he ollo eo o a clcal Wee ease s se: ep l E F F π μ I ha case ola ( has he o

7 6 Pee Babla Elzba Naaaa Mzeva Pasasa Gol Sohaze Bll Geo Nal Aca Sc vol o 9 (5 hle ( s e as l (6 Rea I s ees o oe ha (6 (le ( ca be se o calclae soe Wee eals We llsae hs b seveal eaples: e The o (6 e oba Theeoe μ Hece e have E μ e The o (6 e oba Theeoe ( μ Hece a o acco he pece eaple e have E μ B aaloos calclaos o < e sho ha { } E μ I e ae e he ep a ae ca o soe calclaos e oba e Ee e μ 5 e s calclae ( ep μ Fo hs e ae ep hee Fo e have ep I s eas o calclae ha

8 O Oe Pope o he Wee Ieal a s Sascal Applcao 7 Bll Geo Nal Aca Sc vol o 9 ep ep Theeoe ep ep ep l ep ep l ( ep l ep μ Ths ( ( ( μ ep ep Ths ples ha ( ep ep μ e s pocee o he applcao o he esls obae Asse ha e have he s o ao elees W hee W s a saa Wee pocess a oes o epe o W e be epee obsevaos (aecoes o a elee We ae o esae he loahc evave h l o he sbo μ o he ao pocess alo h We cose he pos < < < e a as We oce he oao W W W W a cose he eqal W Ths s he eqal he e-esoal space R a e asse he esece o sbo eses e ca e ep E p p π B ola (5 e have p p (7 Fo bev e oce he oao

9 8 Pee Babla Elzba Naaaa Mzeva Pasasa Gol Sohaze The (7 aes he o Bll Geo Nal Aca Sc vol o 9 ( ( ϕ D p ~ D p ( ~ ( ϕ (8 I he e-esoal case he loahc evave o a sbo h es p calclae b he ola R s ( ap h l h (9 p hee ( eoes he scala poc o he vecos R a h s he ohooal poeco o he veco h R O he vales o he obseve cos a he selece pos e copose he a o obsevaos ( Ma ( cosss o epee a eqall sbe ao vales Us hs a e cosc he esae o he es (acall o p a he esae p b ola (8 a he soh loahc evave b ola (9 e se he sasc Fo esa he o es hee ( p pˆ ( a Fo a he scsso e ae eqal The Fo (8 e oba pˆ apˆ ap No o he loahc evave l ( l ( h ( ( R ae eve posve boe cos h I o case ( ( a ( ( h D ap h ϕ( h ( ap h p e ca e a esae he o D a D ( ( h ϕ( ( ( ϕ(

10 O Oe Pope o he Wee Ieal a s Sascal Applcao 9 So he ollo saee s e Theoe I he space [ ] e have he obsevao o he ealzao o he s o epee ao pocesses W hee W s a saa Wee pocess he he esae o he loahc evave o he sbo o he ao pocess s ve b he ola l D a ( ( hϕ( l ( h l l D ϕ hee ( ( s a eve posve sooh es co < < [ ] a as ae chose so ha < as he pos ateaa ves eals et Tvsebsa a s sas aoeebs Sesaeb pee babla* elzba aaaa** zeva acaca & ol soaze # * avasvls sa Tblss saelo vese ** aaes ev avasvls sa Tblss saelo vese & sos vese # a eetls saelo vese qtas asos aela ves eals Sebebs et Tvseba oelc aoeeba cob loatl aoebls aapaael sas Seasebs asaeba set SeTvevT pocess aalebsatvs oelzec avveba eba ves SeSoTebs pobebs REFERENES E Naaaa (989 Nopaaec esao o pobabl eses a eesso cves le Acaec Pblshes Gop Doech P Rooe (957 aa J Mah 9: 5- I M ovalch (96 UMN (Avaces o Maheacal Sceces VIII (9: 97- ( Rssa Dales (967 UMN (Avaces o Maheacal Sceces : -5 ( Rssa Receve Decebe 8 Bll Geo Nal Aca Sc vol o 9

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