Лекционный курс (1134109), страница 7
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. . , XSm − XSm−1 , Xt − XSm }Òðåáóåòñÿ ïðîâåðèòü, ÷òîE(h(Xn )|XS1 , ..., XSn , Xt ) = E(h(Xn )|Xt ), òî åñòü ïî÷åìó âûïîëíåíî ðàâåíñòâîE(h(Xn )|ξ1 , ..., ξm , ξ) = E(h(Xn )|Xt )? Ñî âòîðîãî êóðñà íàì èçâåñòíî: åñëèξ, η - íåçàâèñèìûå ñëó÷àéíûå âåêòîðà â Rk è R1 è g - îãðàíè÷åííàÿ èçìåðèìàÿôóíêöèÿ, g : Rk+1 → R, òî E(g(ξ, η)|η = x) = Eg(ξ, x). ÒàêæåE(Z|Y = x) = φ(x)èE(Z|Y ) = φ(Y ),ãäå φ - áîðåëåâñêàÿ.
Òîãäà â íàøåì ñëó÷àåE(h(Xn )|ξ1 = x1 , ..., ξm = xm , ξ = x) = Eh(Xn − Xt + X1 + ... + Xm+1 ) == ψ(X1 + ... + Xm+1 ),ãäå ψ - áîðåëåâñêàÿ ôóíêöèÿ, à ξ1 = Xs1 , ξ2 = XS2 − XS1 , ..., ξm = XSm −XSm−1 , ξ = Xt − Xm ; Xn = Xn − Xt + (Xt − Xm ) + ... + XS1 . Ýòî íåîáõîäèìîïîÿñíèòü: ñòðîèì àïïðîêñèìàöèþE(h(Xn )|Xt ) = E(E(h(Xn )|XS1 , ..., XSm+1 , Xt )|Xt )Èñïîëüçóÿ òåëåñêîïè÷åñêîå ñâîéñòâî, ïîëó÷èì: åñëè A2 ⊂ A1 , òîE(E(ξ|A1 )|A2 ) = E(ξ|A2 ) =m+1X= E(ψ(ξk )|k=1m+1Xξk ) = ψ(ξ1 + ... + ξm+1 ),k=1÷.ò.ä.•Ëåêöèÿ 10(Xt , F)t∈T X : Ω → St (St , Bt )T ⊂ R, (Ω, F, P )P (C|Ft ) = P (C|Xt ) (∗)C ∈ F≥t = σ{Xu , u ≥ t, u ∈ T } E(IC |A) = P (C|A)FtX - åñòåòñòâåííàÿ ôèëüòðàöèÿ. Åñëè (St , Bt ) - áîðåëåâñêîå ïðîñòðàíñòâî,òî (*) ⇔ E(f (Xt )|Xs1 , . .
. , Xsm , Xs ) = E(f (Xt )|Xs ),ãäå s1 < . . . < sM < s ≥ t f - ëþáàÿ èçìåðèìàÿ : St → R.36Ïðèìåð. Ïóñòü X0 , ξ1 , xi2 , . . . - íåçàâèñèìûå ñë. âåëè÷èíû.X0 : Ω → Rmξk : Ω → Rm , k = 1, 2, . . .Ïîëîæèì Xn+1 = hn+1 (Xn , ξn+1 ), n = 0, 1, . . ., ãäå h(èçì.):Rm × Rq → RmÒîãäà {Xn } - ìàðêîâñêèé ïðîöåññ (Âåðíî è äëÿ íåïðåðûâíîãî âðåìåíè)◦Äîêàçàòåëüñòâî.E(f (Xn+1 |Xn = xn , . . . , X0 = x0 ) =E(ξ|η) = φ(η), E(ξ|η = x) = φ(x)= E(f (hn+1 (Xn , ξn+1 ))|Xn = xn , . . . , X0 = x0 ) =E(g(ξ, η)|η = x) = Eg(ξ, x) åñëè ξ è eta - íåçàâèñèìû.= E(f (hn+1 (xn , ξn+1 ) == E(f (Xn+1 )|Xn = xn )Ñëåäîâàòåëüíî, {Xn } - Ìàðêîâñêèé ïðîöåññ.
•Ðàññìîòðèì ñëó÷àé, êîãäà St ⊂ Sn . S - êîíå÷íîå è ñ÷åòíîå ìíîæåñòâî S ={0, 1, . . . , r} èëè S = {0, 1, . . .}. Ââåäåì â S ìåòðèêó:½0, x = y ;ρ(x, y) =1, x 6= y .Òîãäà S - ïîëüñêîå ïðîñòðàíñòâî.  ýòîì ñëó÷àå {Xt }t∈T - ìàðê. ïðîöåññ.⇔ P (Xt = j|Xs1 , . . . , Xsm = im , Xs = i) = P (Xt = j|Xs = i) äëÿ∀s1 < . . . < sm < s ≥ t (âñå ∈ T ) è ∀i1 , . . . , im , i, ò.÷. P (.
. .) 6= 0.Ââåäåì Ss = {i ∈ S : P (Xs = i) 6= 0}, òîãäà îïðåäåëåíû ôîðìóëûpij (s, t) := P (Xt = j|Xs = i) s ≤ t, s, t ∈ T. i ∈ Ss , j ∈ StÎ÷åâèäíî, ôóíêöèè pij , íàçûâàåìûå ïåðåõîäíûìè âåðîÿòíîñòÿìè, îáëàäàþòñëåäóþùèìè ñâîéñòâàìè:1).pPij (s, t) ≥ 0 ∀i ∈ Ss , j ∈ St s ≥ t2). pij (s, t) = 1j3).pij (s, s) = δPij4).pij (s, t) =pik (s, u)pkj (u, t) ∀s < u < t -óð-å Êîëìîãîðîâà-×åïìåíà.k∈SuÇàìå÷àíèå. Ëåãêî îïðåäåëèòü pij (s, t) ∀i, j ∈ S s ≤ t, òàê ÷òîáû âûïîëíÿëèñü1)-4).
À èìåííî, ïîëîæèì:pij (s; t) = 0, åñëè i ∈ Ss è j ∈ St37pij (s; t) = pi0 (s),j (s; t), j ∈ S , ãäå i0 = i0 (s) ∈ SÏîýòîìó äàëåå áåç îãðàíè÷åíèÿ îáùíîñòè ñ÷èòàåì, ÷òî pij (s; t) çàäàíû äëÿs ≤ t (s, t ∈ T ) è âñåõ i, j ∈ S .Ïîñ÷èòàåì P (Xt1 = j1 , . . . , Xtn = jn ) = P (Xtn = jn |Xtn−1 = jn−1 )P (Xt1 =j1 , . . . , Xtn−1 = jn−1 ) = [ â ñèëó ìàðêîâîñòè ìîæíî âûêèíóòü âñþ ïðåäûñòîðèþ,ïîëó÷èì ] = pjn−1 ,jn (tn−1 , tn )P (Xt1 = j1 , . . . , Xtn−1 = jn−1 ) = [ ýòè ôîðìóëûâåðíû ïðè âåðîÿòíîñòè óñëîâèÿ íå ðàâíîé íóëþ, àíàëîãè÷íûì îáðàçîìïîëó÷àåì] = pjn−1 ,jn (tn−1 , tn )(tn−2 , tn−1 ) · . . . · pj1 ,j2 (t1 ; t2 )P (Xt1 = j1 ) == P (Xt1 ) · pj1 ,j2 (t1 ; t2 ) · .
. . · pjn−1 ,jn (tn−1 ; tn )Ðàññìîòðèì T = [0, ∞) èëè T = Z+PP (Xt1 = j1 ) =[ïî ôîðìóëå ïîëíîé âåðîÿòíîñòè] =pi (0) · pij1 (0; t1 ), ãäåipi (0) = P (X0 = i), i ∈ S - íà÷àëüíîå ðàñïðåäåëåíèå. ÒîãäàXXP ((Xt1 , . . . , Xtn ) ∈ B) =pi (0)pij1 (0; t1 ) · . . . · pjn−1 ,jn (tn−1 ; tn )i(j1 ,...,jn )∈BÒ Å Î Ð Å Ì À Ïóñòü S - äèñêðåòíîå ïðîñòðàíñòâî. Ïóñòü äëÿ ìíâ St ⊂ S, t ∈ T (íåïóñòûõ) çàäàíû ôóíêöèè pij (s; t),Ps ≤ t, i ∈ Ss , j ∈ St ,óäîâëåòâîðÿþùèå óñëîâèÿì 1)-4).
Ïóñòü pi (0) ≥ 0 èpi (0) = 1. Òîãäà íàiíåêîòîðîì ïðîñòðàíñòâå (Ω, F, P )∃ ìàðêîâñêèé ïðîöåññ {Xt }t∈T (t = [0, ∞)èëè T = Z+ ), ò.÷. pi (0) = P (X0 = i), pij (s; t) = P (Xt = j|Xs = i).Èòàê, ìàðêîâñêàÿ öåïü ìîæåò áûòü ïîñòðîåíà ñ ïîìîùüþ ïåðåõîäíûõ âåðîÿòíîñòåé;íåîáõîäèìî òîëüêî âûïîëíåíèå óñëîâèé 1)-4).Ïðèìå÷àíèå. (ïóàññîíîâñêèé ïðîöåññ) Ïóñòü m(B), B ∈ B(R+ ) - ëîêàëüíîêîíå÷íàÿ ìåðà, ò.å. m(B) < ∞ äëÿ îãðàíè÷åííûõ ìíîæåñòâ B S = {0, 1, 2, . . .}.Îïðåäåëèì :(m((s;t])j−i −m((s;t]), j ≥ i;(j−i)! epij (s; t) =0j < i.Ïóñòü pij (s; s) = δijËåãêî âèäåòü, ÷òî 1),2),3) î÷åâèäíî âûïîëíÿþòñÿ (ðàçëîæåíèå ýêñïîíåíòûâ ðÿä).
Ïðîâåðèì4):Ppik (s; u)pkj (u; t), s < u < t. ×òî ïîëó÷àåòñÿ:pij (s; t) =kj−kX m ((s; u])k−im ((u; t])e−m((s;u]) ·e−m((u;t]) =(k − i)!(j − k)!i≤k≤jP=eÈñïîëüçóåì :−m((s,t]) i≤k≤jNPl=0(j−i)!(k−i)!(j−k)!m((s; u])k−i m((u; t])j−k(j − i)!l l N −lCNab= (a + b)NÒåïåðü, åñëè âçÿòü pi (0) = δi0 , òî ýêâèâàëåíòíîå îïðåäåëåíèå ïóàññîíîâñêîãî38ïðîöåññà òàêîå: {Nt , t ≥ 0} - ìàðêîâñêèé ïðîöåññ (öåïü Ìàðêîâà ñî çíà÷åíèÿìèâ S = {0, 1, . . .}, èìåþùèé pi (o) = δi0 è pij (s; t) = {.
. . (m íàçûâàåòñÿâåäóùåé ìåðîé).  ÷àñòíîñòè, ïðè m ((s; t]) = (t − s)λ, λ = const > 0ïîëó÷àåòñÿ ñòàíäàðòíûé ïóàññîíîâñêèé ïðîöåññ èíòåíñèâíîñòè λ.Óïðàæíåíèå. Äîêàçàòü ýêâèâàëåíòíîñòü îïðåäåëåíèé ïóàññîíîâñêîãî ïðîöåññà:1. N0 = 02. N - èìååò íåçàâèñèìûå ïðèðàùåíèÿ.3. Nt − Ns ∼ π(m(s; t]), s ≤ t è òîãî îïðåäåëåíèÿ, êîòîðîå áûëî ðàíüøå.Çàìå÷àíèå. Ïðåäûäóùàÿ òåîðåìà ÿâëÿåòñÿ ñëåäñòâèåì òåîðåìû Êîëìîãîðîâà.Îïð.: Ôóíêöèÿ P (s, x, t, B) íàçûâàåòñÿ ïåðåõîäíîé ôóíêöèåé, åñëè âûïîëíåíûóñëîâèÿ (s, t ∈ T ⊂ R, x ∈ Ss , B ∈ Bt , ò.å.
èìååòñÿ ñåìåéñòâî èçìåðèìûõïðîñòðàíñòâ (St , Bt )t∈T1). P (s, x, t, ·) ÿâëÿåòñÿ ìåðîé íà Bt2). P (s, ·, t, B) ∈ Bs |B(R) ½1, x ∈ B3). P (s, x, s, B) = δx (B) =0, xnotinB4). äëÿ ∀s < u R< t :P (s, x, t, B) = P (s, x, u, dy)P (u, y, t, B) (óðàâíåíèå Êîëìîãîðîâà-×åïìåíà,Suèíòåãðèðóåì ïî âñåì ïðîìåæóòî÷íûì çíà÷åíèÿì y)Îïð.: Ìàðêîâñêèé ïðîöåññ {Xt , t ∈ T } îáëàäàåò ïåðåõîäíîé ôóíêöèåé,åñëè:P (s, x, t, B) = P (Xt ∈ B|Xs = x)èëè òàê P (s, Xs , t, B) = P (Xt ∈ B|Xs )Óïðàæíåíèå. Íàéòè ïåðåõîäíûå ôóíêöèè âèííåðîâñêîãî ïðîöåññà ñîçíà÷åíèÿìè â Rm . äèñêðåòíîì ñëó÷àå P (s, i, t, B) =Pj∈Bpij (s; t).Ò Å Î Ð Å Ì À. (Ýðãîäè÷åñêàÿ).
Ïóñòü ∃j0 ∈ S è h, δ > 0, òàêèå ÷òîpij0 (h) ≥ δ äëÿ ∀i ∈ S . Òîãäà äëÿ ∀i, j ∈ S ñóùåñòâóåò:lim pij (t) = pejt→∞(∗)×òî òàêîå pij0 (h) -?Îïð.: Ìàðêîâñêàÿ öåïü íàçûâàåòñÿ îäíîðîäíîé, åñëè pij (s; t) = pij (t −s) t ≥ s.Ñìûñë îïðåäåëåíèÿ ñîñòîèò â òîì, ÷òî ñèñòåìà îñóùåñòâëÿåò ïåðåõîä èç iâ j çà t-s. Ñìûñë (*) ñîñòîèò â òîì, ÷òî ñèñòåìà, êàê áû, çàáûâàåò èç êàêîãî39ñîñòîÿíèÿ îíà ñòàðòîâàëà.◦Äîêàçàòåëüñòâî. (òåîðåìû)Îáîçíà÷èì mj (t) = inf pij (t), Mj (t) = sup pij (t).
Î÷åâèäíî, ÷òî mj (t) ≤iipij (t) ≤ Mj (t). Äîêàæåì, ÷òî ó íèõ îáùèé ïðåäåë. Çàìåòèì, ÷òî mj (t) íåóáûâàåò è Mj (t) íå âîçðàñòàåò ñ ðîñòîì t. Òîãäà mj (s + t) = inf pij (s + t).iÄëÿ îäíîðîäíîéP öåïè óðàâíåíèå Êîëìîãîðîâà-×åìïåíà äàåò:pij (s + t) =pik (s)Pkj (t)kPPïîýòîìó mj (s + t) = inf pik (s)pkj (t) ≥ mj (t) inf pik (s) = mj (t)iikkÀíàëîãè÷íî äëÿ Mj (t).Îñòàëîñü óáåäèòüñÿ, ÷òî Mj (t) − mj (t) → 0, t → ∞ :Mj (t) − mj (t) = sup paj (t) − inf pbj (t) = sup(pa,j (t) − pb,j (t)) =baa,bÓðàâíåíèå Êîëì.-×åïì. :XX= sup{pak (h)pkj (t − h) −pbk (h) − pkj (t − h)} =a,bKkt>hsupa,bP+X−XX(pak (h) − pbk (h))pkj (t − h) = sup(a+ +)≤a,bkáåðåòñÿ ïî k, ò.÷. pak[P≥ 0 (ýòî ìíîæåñòâî èíäåêñîâ çàâèñèòP(h) − pbk (h)pbk (h) = 1, ïîýòîìópak (h) =îò à è b).
Çàìåòèì, ÷òîkPk−P+(pak (h) − pbk (h)) = 0](pak (h) − pbk (h)) ++−XX≤ sup( (pak (h) − pbk (h))Mj (t − h) +(pak (h) − pbk (h))mj (t − h)) =a,bÏî çàìå÷àíèþ âûøåsupa,b+X(pak (h) − pbk (h))(Mj (t − h) − mj (t − h))Èòàê,+XMj (t) − mj (t) ≤ [Mj (t − h) − mj (t − h)](pak (h) − pbk (h))a,bÅñëè j0 íå ïðèíàäëåæèòJa,b (+XkP+pak (h) − pbk (h) ≤P=Ja,b+X), òîpak (h) ≤ 1 − pkj0 ≤ 1 − δk40P+Åñëè j0 ∈ Ja,b , òî<1−δ :+PP+pak (h) − pbk (h) ≤pak (h) − pbj0 ≤ 1 − δ .
 èòîãåkMj (t) − mj (t) ≤ (1 − δ)[Mj (t − h) − mj (t − h)] ≤u=t−h£t¤ht≤ (1 − δ)[ h ] [Mj (u) − mj (u)]0<u<tÇàìå÷àíèå.  óñëîâèÿõ ýðãîäè÷åñêîé òåîðåìû: |pij (t) − pej | ≤ (1 − δ)[ h ] .Ò.å. ñêîðîñòü ñõîäèìîñòè ýêñïîíåíöèàëüíî áûñòðàÿ.tËåêöèÿ 11pij (s, t) = P (Xt = j|Xs = i) - ïåðåõîäíûå âåðîÿòíîñòè.Îïð.: Öåïü îäíîðîäíàÿ, åñëè Pij (s, t) = pij (t − s)Äëÿ îäíîðîäíûõ öåïåé óðàâíåíèå Êîëìîãîðîâà-×åïìåíà çàïèñûâàåòñÿ ïðîñòî:Xpij (s + t) =pik (s)pkj (t) ∀i, js, t ≥ 0kP (s + t) = P (s)P (t) - ïîëóãðóïïîâîå ñâîéñòâî. P (t) = (pij (t)) - ìàòðèöà.(P (t))t≥0 - Pïîëóãðóïïà (ñòîõàñòè÷åñêàÿ)[pij (t) ≥ 0, pij (t) = 1, pij (0) = δij ⇔ P (O) = I]jpij (t) è pi (0) - ïîçâîëÿþò ñòðîèòü ìàðêîâñêóþ öåïü.P (t) - ñòàíäàðòíàÿ ñòîõàñòè÷åñêàÿ ïîëóãðóïïà ⇒+∃Q = ddt |t=0 P (t) - èòåðàòîð ïîëóãðóïïû.
Ò.å. ∃qij = d+dt |t=0 pij (t)T tf − ftÌàòðèöà Q íàçûâàåòñÿ èíôèíèòèçåìàëüíîé.Af =Ò Å Î Ð Å Ì À. Åñëè ñòîõàñòè÷åñêàÿ ïîëóãðóïïà ñòàíäàðòíà (ò.å.p(t) → I ïðè t → 0+), òî ∀i 6= j ∃ êîíå÷íûå qij ≤ 0 è ∀i ∃qi = qii ∈ [0, ∞]Áåç äîêàçàòåëüñòâà.Ýðãîäè÷åñêàÿ òåîðåìà:(*) pij (t) → pej ∀j ïðè t → ∞ ⇒ P (Xt = j) → pej , t →P∞, ò.ê. pj (t) =pi (0)pij (t)i|pj (t) − pej | = |Xpi (0)(pij (t) − pej )| → 0iâ ñèëó (*).41Ñèñòåìû ìàññîâîãî îáñëóæèâàíèÿ.
Ïîñòóïàåò ïîòîê çàÿâîê íà îáñëóæèâàíèå.n - ïðèáîðîâ, "ñèñòåìà ñ îòêàçîì", çàÿâêè îáðàçóþò ïóàñ. ïîòîê. η ∼ exp(µ)½µe−µz , z ≥ 0;pη (z) =0,z < 0.?Xt -÷èñëî çàíÿòûõ ïðèáîðîâ. P (Xt = j) → pejÔîðìóëû Ýðëàíãà..Îïèñûâàþò â ýòîé ìîäåëè ñòàöèîíàðíîå ðàñïðåäåëåíèå.pej =ρjj!nPk=0ρ=λµρkk!, j = 0, 1, . . . , nÄîêàæåì êðóïíî-áëî÷íî.Óïðàæíåíèå. pe = (ep1 , . . .) ÿâëÿåòñÿ ñîáñòâåííûì âåêòîðîì ìàòðèöû P ∗ (t) ∀t(îòâå÷àþùèé ñîáñòâ. çíà÷åíèþ 1). Ïóñòü âûïîëíåíû óñëîâèÿ ýðãîäè÷åñêîéòåîðåìû, òîãäà ∃epËP Å Ì Ì PÀ. Ïóñòü âûïîëíåíî óñëîâèå ýðãîäè÷åñêîé òåîðåìû, òîãäàpej = 0 èëèpej = 1jÅñëèPjjpej = 1, òî pe íàçûâàåòñÿ ñòàöèîíàðíî ðàñïðåäåëåííûì.Îïð.:Ïðîöåññ {Xt , t ∈ T } íàçûâàåòñÿ ñòàöèîíàðíûì (ñòàöèîíàðíûì âóçêîì ñìûñëå ), åñëè ∀n, ∀t1 , .
. . , tn ∈ T ∀h t1 + h . . . tn + h ∈ TLaw(Xt1 , dots, Xtn ) = Law(Xt1 +h . . . Xtn +h )Ò Å Î Ð Å Ì À.Åñëè ó îäíîðîäíîé ìàðêîâñêîé öåïè {Xt , t ≥ 0} ∃ñòàöèîíàðí. ðàñïðåäåëåíèå pe, òî Ì.Ö. Y − {Yt , t ≥ 0} èìåþùàÿ íà÷àëüíîåðàñïðåäåëåíèå pe è òå æå ïåðåõîäíûå âåðîÿòíîñòè pij , ÷òî è öåïü X, ÿâëÿåòñÿñòàöèîíàðíûì ïðîöåññîì.Ìû äîêàæåì, ÷òî ∃ ïðîöåññ Y◦Äîêàçàòåëüñòâî.?Xipi (0)P ((Yt1 . . .
Ytn ) ∈ B) = P ((Yt1 +h . . . Ytn +h ) ∈ B)XX X?pij1 (t1 ) . . . pjn−1 jn (tn−1 , tn ) =P (Xt1 = i) . . .i(j1 ...jn )∈Bj1 ...jn ∈BÇàìåòèì, ÷òî pij (s, t) = pij (t − s) = pij (s + h, t + h). Åñëè âçÿòü pi (0) =pej = pj (t). Òîãäà ðàâåíñòâî (ïåðâîå) âåðíî. pe - ñîáñòâåííûé âåêòîð p∗ (t) •42.
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