Лекционный курс (1134109), страница 3
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Áóäåì ñòðîèòü áðîóíîâñêîåäâèæåíèå íà [0,1]. Ðàññìîòðèì ïîñëåäîâàòåëüíîñòü íåçàâèñèìûõ ãàóññîâñêèõâåëè÷èí íà (Ω, F, P ), ðàñïðåäåëåííûå ïî N (0, 1) ξkω . Ââåäåì íåñëó÷àéíûå∞RtPôóíêöèè Øàóäåðà Sk (t) = Hk (u)du k = 1, 2, . . .. Ïóñòü WT (ω) :=ξk (ω)Sk (t), t ∈k=a0[0, 1] - ÿâíàÿ êîíñòðóêöèÿ áðîóíîâñêîãî äâèæåíèÿH1 (t) ≡ 1, t ∈ [0, 1]H2 (t)...Hk (t)6H2 (t)and so on112¾n(t)−2n/2 I( k−1H2n +k (t) = 2n/2 I[ k−1k−1k , 1 ≤ k ≤ 2112n , 2n + 2n+1 ]2n + 2n+1 , 2n ]Ëèðè÷åñêîå îòñòóïëåíèå. Ìîëîäîé àðõèòåêòîð ñäàåò ïðîåêò, íó è ïîíÿòíî,âîëíóåòñÿ. Áîëåå îïûòíûé ñîâåòóåò åìó óñòàíîâèòü íà ôàñàäå ñîáà÷êó...èâñå îáñóæäåíèÿ ñâåäóòüñÿ ê òîìó, ÷òîáû óáðàòü ñîáà÷êó.
Òàê âîò ìîæíîáûëî îïðåäåëèòü è íå ÷åðåç ôóíêöèè Øàóäåðà2{Hk }∞k=1 îáðàçóþò ïîëíóþ îðòîíîðìèðîâàííóþ ñèñòåìó â ïðîñòðàíñòâå L [0, 1]1Óïðàæíåíèå. Ïðîâåðèòü îðòîíîðìèðîâàííîñòü è ïîëíîòó (ä-òü, ÷òî èíäèêàòîðûïðîìåæóòêîâ ìîãóò áûòü àïïðîêñèìèðîâàíû â ìåòðèêå ôóíêöèÿìè Õààðà).Âñïîìíèì ñëåäñòâèå èç ðàâåíñòâà Ïàðñåâàëÿ:< f, g >=∞X< f, Hk >< g, Hk >k=1Ë Å Ì Ì À 1.
Ïóñòü ak = O(k ε ), ãäå ε < 12 . Òîãäà∞Pk=1ak Sk (t), t ∈ [0, 1]ÿâëÿåòñÿ íåïðåðûâíîé ôóíêöèåé íà îòðåçêå [0,1].Äîêàçàòåëüñòâî.Pn→∞◦ Äîñòàòî÷íî óáåäèòüñÿ, ÷òî sup|ak |Sk (t) −→ 0. Îöåíèì: (∗) =nt∈[0,1] k>2Pn1|ak |Sk (t) ≤ 2(n+1)ε 2− 2 −1 = c0 2−n( 2 −ε) , ε < 12 .2n <k≤2n+113nSk (t) ≤ 2− 2 −1 , íîñèòåëè ýòèõ ôóíêöèé: 2n < k ≤ 2n+1 , íîñèòåëè íå ïåðåñåêàþòñÿ.P −m( 1 −ε) n→∞P2(∗) ≤2−→ 0. Ïî òåîðåìå Âåéåðøòðàññàak Sk (t) ÿâëÿåòñÿn≤mkíåïðåðûâíîé ôóíêöèåé (ðÿä ñõîäèòñÿ ðàâíîìåðíî), ÷.ò.ä.
•Ë Å Ì Ì À 2. Ïóñòü ξ1 , ξ2 ,... - ïîñëåäîâàòåëüíîñòü ñòàíäàððòíûõãàóññîâñêèõ √âåëè÷èí, ò.å. ξk ∼ N (0, 1), k = 1, 2, ...(íåçàâèñèìîñòüíå ïðåäïîëàãàåòñÿ).√Òîãäà ∀c > 2 è ï.â. ω ∈ Ω ∃ N = N (ω, c) : |ξk (ω)| ≤ a logk, ∀k ≥ N, ãäåa=const.Äîêàçàòåëüñòâî.◦ Íà÷èíàÿñ√íåêîòîðîãî N = N (ω, c), âñå ξk (w) ëåæàò âíóòðè ïîëîñû Y ∈√[−a logk, a logk]. •.Äîêàçàòåëüñòâî (ëåììû 2).◦ Ïóñòü ξ ∼ N (0, 1). P(ξ > x) =2x√1 e− 2x 2π−√12πR∞x2e−u /2u2 du≤√12πR∞e−x2x√1 e− 2x 2π2− x2u22du=/ïî ÷àñòÿì/= √−12π, òàê êàêR∞x2e−u /2u2 d uR∞ 1xu d(e2− u2)=≥ 0 ïðè x > 0.ïðè x → ∞ ôóíêöèÿ x√12π e- ÿâëÿåòñÿ àñèìïòîòè÷åñêè ýêâèâàëåíòíîéèñõîäíîìó èíòåãðàëó (äëÿ ïðîâåðêè íàäî åù¼ ðàç ïðîèíòåãðèðîâàòü ïî÷àñòÿì).
√P(|ξk | > c logk áåñê. ÷àñòî ïî k)PP(Ak ) < ∞ ïî Áîðåëþ-Êàíòåëëè.Âñïîìíèì 2-îé êóðñ: P(Ak .) = 0, åñëèkT SAn ).P(Ak áåñê. ÷àñòî)=P(n k≥nÒàêèì îáðàçîì,√PPc2 logkc2 u√ 1 k 2 < ∞, åñëè√1 √1 e− 2 ≤constP(|ξk | > c logk) ≤2πlogkk≥N c logkk≥N√c > 2.√Ñëåäîâàòåëüíî, ïî ëåììå Áîðåëÿ-Êàíòåëëè äëÿ ï.â. ω ∈ Ω |ξk (ω)| ≤ c logk, k ≥N (c, ω). •Ëèðè÷åñêîå îòñòóïëåíèå (åùå ïàðà ñîâåòîâ). Ìû äîëæíû ïðèâûêíóòü, ÷òîèäåé íå òàê ìíîãî, â èíòåãðèðîâàíèè - äâå ãëóáêèõ èäåè: ïî ÷àñòÿì èñâåäåíèå êðàòíûõ ê ïîâòîðíûì (åñòü, ïðàâäà, è òðåòüÿ - ïåðåíîñ ìåðûñ îäíîãî ïðîñòðàíñòâà íà äðóãîå (çàìåíà òî áèøü - çàìå÷.ðåä.)). Åñëèíóæíî èìåòü îöåíêó õâîñòîâ, ïîíÿòíîå äåëî - èíòåãðèðóåì ïî ÷àñòÿì.Ò Å Î Ð Å Ì À. Ïðîöåññ Wt (ω) =∞Pk=1ξk (ω)Sk (t) (*) ÿâëÿåòñÿ âèííåðîâñêèìïðîöåññîì íà [0,1].Äîêàçàòåëüñòâî.√◦ Ïî ëåììå 2 |ξk (ω)| ≤ c logk ≤ c0 k ε , ε ≤ 12 , k ≥ N =⇒ ïî ëåììå 1 ðÿä(*) ñõîäèòñÿ ðàâíîìåðíî íà îòðåçêå [0,1] äëÿ ï.â.
ω ∈ Ω =⇒ ïî òåîðåìåÂåéåðøòðàññà ýòî íåïðåðûâíàÿ ôóíêöèÿ. Ñëåäîâàòåëüíî, òðàåêòîðèè W14íåïðåðûâíû ñ âåð 1. Ïðîâåðèì ïóíêòû 1-3 èç Îïð.∗ .∞P1)Ðÿäξk (ω)Sk (t) ñõîäèòñÿ â ñðåäíå-êâàäðàòè÷åñêîì ê âåëè÷èíå Wt ∀t ∈k=1[0, 1]. (Íóæíî ïðîâåðèòü, ÷òî ðÿä ñõîäèòñÿ, ò.å. ïðîâåðèòü ôóíäàìåíòàëüíîñòü,NPò.å. E |ξk Sk |2 −→ 0, M, N −→ 0) [Ïîòîìó, ÷òî è òà è äðóãàÿ ñõîäèìîñòèMâëåêóò ñõîäèìîñòü ïî âåðîÿòíîñòè, à ðàç ïîòî÷å÷íî ðÿä ñõîäèòñÿ ê W (t), òîñõîäèòñÿ â ñðåäíåêâàäðàòè÷íîì]. Òîãäà EW (t) = 0: |EW (t)| = |E(W (t) −Wnp(t))| [ò.ê. EWn (t) = 0] ≤ [íåðàâåíñòâî Êîøè-Áóíÿêîâñêîãî] ≤≤ |E(W (t) − Wn (t))|2 → 0.
Ò.ê. Wn (t) → W (t) â ñðåäíåì êâàäð.Óïðàæíåíèå. cov(W (s), W (t)) = min(s, t) - ðàâåíñòâî ÏàðñåâàëÿÓïðàæíåíèå. ζ = (ζ1 , . . . , ζn ) - ãàóññîâñêèé ⇔ ∀c1 , . . . , ck ∈ RPãàóññîâñêèé ⇒ck Wk - ãàóññîâñêàÿ âåëè÷èíà •Pkck ζk -( ëåêöèÿõ ýòîò êóñîê äîêàçàòåëüñòâà äåéñòâèòåëüíî ñêîìêàí - ïðèì.ðåä.)Çàäà÷è.1.-...0Wt :−nPk2T2|W (tk,n ) − W (tk−1,n )| −→ T ï.í. - äîêàçàòü2. N = {Nt , t ≥ 0} - ïóàññîíîâñêèé ïðîöåññ èíòåíñèâíîñòè λNt ?−→ ? t → ∞tËåêöèÿ 4Ò Å Î Ð Å Ì À.Ò Å Î Ð Å Ì À.
(Âèííåð-Çèãìóíä-Ïýëè). Ñ âåðîÿòíîñòüþ åäèíèöàòðàåêòîðèè áðîóíîâñêîãî äâèæåíèÿ. W = {Wt , t ≥ 0} íå äèôôåðåíöèðóåìûíè â îäíîé òî÷êå ïîëóîñè [0, +∞).Äîêàçàòåëüñòâî.◦ Ðàññìîòðèì ïðîìåæóòîê [k, k + 1), ãäå k ∈ {0, 1, 2, . . .}. Åñëè Wt (ω)äèôôåðåíöèðóåìû â òî÷êå s ∈ [k, k + 1), òî |Wt − Ws | ≤ l|t − s|, äëÿíåêîòîðîãî l ∈ N è t ∈ [s; s + 1q ) q ∈ N. Èç äèôôåðåíöèðóåìîñòè ñëåäóåòäèôôåðåíöèðóåìîñòü ñïðàâà. À l - çàâèñèò îò s è ω è q çàâèñèò îò ω , s è15j+1n )l.
Ðàññìîòðèì ñîâîêóïíîñòü Al,n,i = {|W (k +i + 1, i + 2, i + 3. Ïóñòü n>4q. Íàéäåì i=i(s,n)k1ns1n1ns+− W (k +1n )|≤7ln}j =k+11qÅñëè Wt (ω) äèôôåðåíöèðóåìà â òî÷êå s ∈ [k, k + 1), òî¯ µ¶µ¶¯ ¯µ µ¶¶ µ µ¶¶¯¯ ¯¯¯¯W k + j + 1 − W k + j ¯ = ¯ W k + j + 1 − W (s) − W k + j − W (s) ¯ ≤¯¯¯¯nnnn¯ µ¯ ¯ µ¯¶¶¯¯ ¯¯j+1j437l≤ ¯¯W k +− W (s)¯¯ + ¯¯W k +− W (s)¯¯ ≤ l + l =nnnn4Ïóñòü Dk - ìíîæåñòâî òàêèõ òî÷åê íà [k; k+1), äëÿ êîòîðûõ Wt - äèôôåðåíöèðóåìàÒîãäà∞ [∞ \ \n[Dk ⊂Al,n,il=1 q=1 n>4q i=1Äëÿ êàæäûõ l, q ∈ N P (Ñëåäîâàòåëüíî, P (TTnSn>4q k=lnSn>4q i=1Al,n,i ), P (∞Tn=1Al,n,i ) ≤ lim inf P (nBn ) ≤ lim inf P (Bn ).nnSi=1Al,n,i ) ≤ lim infnP (Al,n,i ) =[ò.ê. ïðèðàùåíèÿ íåçàâèñèìû]=¢D¯ ¡¢¡¢¯¡¯ ¡ ¢¯j+1¯− W k + nj ¯= P 3 ¯W n1 ¯ < 7ln = W k+ n¶µ³´7l|W ( n1 )|7ln√<=P(|ξ|<= [ξ ∼ N (0, 1)] =P11√√nn14l√1 √2π nn√12πnPi=17l√nRP (Al,n,i )e−x22dx ≤− √7ln= c √ln èòîã嵶3l√P(Al,n,i ) ≤ lim inf n c=0n>4qnn>4q i=1n\ [Ò.å.PÃ∞ ∞∞[[ \ [!Al,n,i=0l=1 q=1 n>4q i=1Âñåãäà ìîæíî ñ÷èòàòü, ÷òî âåðîÿòíîñòíîå ïðîñòðàíñòâî (Ω, F, P ) ïîïîëíåíî.Ò.å.
åñëè P (A) = 0, à B ⊂ A, òî P (B) = 0, èìååì (Ω, F, P )∞SÒ.î. Dk ⊂ A, P (A) = 0, ò.ê. ïðîñòðàíñòâî ïîëíîå P (Dk ) = 0, ò.î. P (Pk ) =k=1160. •Ñëåäóùàÿ òåîðåìà íàçûâàåòñÿ ÒÅÎÐÅÌÎÉ Êàêóòàíè - ëåãêî çàïîìíèòü"êàê ó Òàíè".Ò Å Î Ð Å Ì À. (Ìàðêîâñêîå ñâîéñòâî). ∀ ôèêñèðîâàííîãî a > 0 ïðîöåññYt = Wt+a − Wa , t > 0 ÿâëÿåòñÿ áðîóíîâñêèì äâèæåíèåì, ïðè÷åì {Yt , t ≥ 0}è σ{Ws , s ∈ [0; a]} íåçàâèñèìû.Äîêàçàòåëüñòâî.◦ßñíî, ÷òî Y0 = 0, Yt èìååò íåçàâèñèìûå ïðèðàùåíèÿ, Yt −Ys ∼ N (0, t−s)è òðàåêòîðèè íåïðåðûâíû.Äîñòàòî÷íî óáåäèòüñÿ, ÷òî (Ws1 , ..., Wsn ) è (Yt1 , ..., Ytn ) íåçàâèñèìû, 0 ≤t1 < ... < tn .
Âåêòîð (Ws1 , ..., Wsn ) ïîëó÷àåòñÿ èç âåêòîðà (Ws1 , Ws2 −Ws1 , ..., Wsn −Wsn−1 ) ëèíåéíûì ïðåîáðàçîâàíèåì (äîìíîæåíèåì íà ìàòðèöó).Âåêòîð (Yt1 , ..., Ytn ) ïîëó÷àåòñÿ ëèíåéíûì ïðåîáðàçîâàíèåì èç âåêòîðà (Yt1 , Yt2 −Yt1 , ..., Ytn − Ytn−1 ).Òî åñòü äîñòàòî÷íî ïðîâåðèòü íåçàâèñèìîñòü âåêòîðîâ (Ws1 , Ws2 −Ws1 , ..., Wsn −Wsn−1 ) è (Yt1 , Yt2 − Yt1 , ..., Ytn − Ytn−1 ), ÷òî î÷åâèäíî.
•Ìîæíî ëè â óòâåðæäåíèè ïîñëåäíåé òåîðåìû âìåñòî êîíñòàíòû a èñïîëüçîâàòüñëó÷àéíóþ âåëè÷èíó τ ? Òîãäà Xt = W (t + τ ) − W (τ ), t ≥ 0.Îïð.: Ïîòîê σ -àëãåáð F=(Ft )t∈T , T ⊂ R1 , íàçûâàåòñÿ ôèëüòðàöèåé, åñëèFs ⊂ Ft ∀s < t, s, t ∈ T.Ïðèìåð: X = Xt , t ∈ T , FX = (FtX )t∈T - åñòåñòâåííàÿ ôèëüòðàöèÿ, åñëèFtX = σ{Xs , s ≤ t}, s ∈ T.Îïð.: τ : Ω → T ∪ {∞} íàçûâàåòñÿ ìàðêîâñêèì ìîìåíòîì îòíîñèòåëüíîôèëüòðàöèè (Ft )t∈T , åñëè {τ ≤ t} ∈ Ft ∀t ∈ T. Åñëè τ < ∞ ï.í., òî τíàçûâàåòñÿ ìîìåíòîì îñòàíîâêè.Ïðèìåð: {Xn , n ∈ N } - ïîñëåäîâàòåëüíîñòü äåéñòâèòåëüíûõ ñëó÷àéíûõâåëè÷èí, B - áîðåëåâñêîå ìíîæåñòâî â R1 , τ = inf {n : Xn ∈ B }, (τ = ∞,åñëè Xn ∈ B ∀n). äèñêðåòíîì ñëó÷àå τ - ìàðêîâñêèé ìîìåíò ⇔ {τ = n} ∈ Fn , {τ = n} ={X1 B, X2 B, ..., Xn−1 B, Xn ∈B} ∈ Fn .Çàäà÷à íà 5+Ïóñòü X = {Xt , t ≥ 0} - ïðîöåññ ñ ï.í.
íåïðåðûâíûìè òðàåêòîðèÿìè, ïðèíèìàþùèé∀t çíà÷åíèÿ â ìåòðè÷åñêîì ïðîñòðàíñòâå (S, ρ). Îïðåäåëèì τ = inf {t ≥ 0 :Xt ∈F}, ãäå F - çàâêíóòîå ïîäìíîæåñòâî S . Òîãäà τ - ìàðêîâñêèé ìîìåíò17îñòàíîâêè îòíîñèòåëüíî F. ÷àñòíîñòè äëÿ âèííåðîâñêîãî ïðîöåññà W = {Wt , t ≥ 0} è ∀a > 0 τa =inf {t ≥ 0 : Wt = a} - ìàðêîâñêèé ìîìåíò, ò.ê. {a} - çàìêíóòî.Óïðàæíåíèå: äîêàçàòü, ÷òî τa - ìîìåíò îñòàíîâêè.Ëåêöèÿ 5Ò Å Î Ð Å Ì À. (Ñòðîãî ìàðêîâñêîå ñâîéñòâî áðîóíîâñêîãî äâèæåíèÿ).Ïóñòü W = {Wt , t ≥ 0} - áðîóíîâñêîå äâèæåíèå. Ïóñòü τ - ìîìåíò îñòàíîâêèîòíîñèòåëüíî åñòåñòâåííîé ôèëüòðàöèè FW = (FtW )t≥0 .
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