А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 17
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2@ , ' & (7.34) Y sx & =, &% (2.10) . B,Ytsx(!) = !(t) t 2 Ts ! 2 E:* , Y (t !) = !(t),t 2 T , ! 2 E = XT , % % s T t > s138(t 2 T ). 2%% , Ts (E F>s Psx ) &% , % s % % x Psx -.. (x 2 Xs). ', F>s . ??, - F>s Tt XT t 2 Ts. C ,F>s = fYt t > s t 2 Tsg. @ Y sx (Yt Psx).+ F>s BT s 2 T , Psx F = BT BT , F>s. *, % '%, % ' (E F ), Psx s 2 T , x 2 Xs, "%," % s % x (Psx- 1), -% % fYtsx t 2 Tsg . + (7.35) u = s, %, %:Psx (Ytsx 2 B ) = P (s x t B ) Psx ; .., s t 2 T (s 6 t) x 2 Xs B 2 Bt:4 , P ( 2 B j) = P ( 2 B ) .., = c .., c = const.', T { R, - C7.1 @ C1.2, Xt t 2 R.
=, " , , Bs s 2 T .* , % ' .-.. % %, , 97.5. + X = R, B = B(R), T = O0 1). Q = 0 ( C, 0). C x 2 R, t > 0,B 2 B(R) P (x t B ) = 1 B (x) Pe(x t B ) = 1 B (x + sign x) sign x = ;1 x < 0, sign 0 = 0, sign x = 1 x > 0.2 3 , P Pe 1') { 4'), .??, , , . + (7.24),, P Pe .-.. , . 2+ ) - < ', %& %, %, % 7.E. 7.6. C, (7.1) , P (AjF6t) = P (AjXt) ..
A 2 F>t t 2 T .( . ?? .. , (E F P ), ) Ejj < 1, , ... E(jA), A { - - (A F ). +, ... && , E. 7.7. + H = L2(E A P ), A F , A {-. C 2 L2(E F P ) PrH H .C, PrH = E(jA) ..139E. 7.8. ( . 7.7). PrH { F P ). + L2(E F P ) L1(E F P ), PrH L1(E F P ). C, E(jA) L2(E(..) " .E. 7.9. + (Xt t 2 T ), T R { . =7, (Xt t 2 U ), U T , . , (Xt t > 0) { , ^ > 0 (Xk k = 0 1 : : : ) . ?E.
7.10. + (Xt t > 0) { ( t) (X B). + (Y E ) { ht : X ! Y , ht 2 BjE , t > 0. C, ht t > 0 { - , (ht(Xt ) t > 0) .E. 7.11. + , ), ) ht, .E. 7.12. + (Xt t > 0) { X R.+ Yt = OXt], O] { . 4 , Yt { ?E. 7.13. ( . 7.10, 7.11). + W (t) = (W1(t) : : : Wm(t)) t > 01 =2Pm2 m- . + Xm (t) =, "k=1 Wk (t) % A. 1 (Xm (t) t > 0) ? , m = 1 X1(t) = jW1(t)j.E. 7.14.
+ fXt t > 0g fYt t > 0g { . 1 fXt + Yt t > 0g fXtYt t > 0g ? T , Yt = c(t), c(t) { ?E. 7.15. + fXk k = 0 1 : : : g { . + Xt = (t ; k)Xk + (k + 1 ; t)Xk+1 t 2 Ok k + 1), k = 0 1 : : : , .. O0 1) (k Xk ). 1 fXt t > 0g? 1 Yt = Xt] t > 0, O] { ?E. 7.16. + 1 2 : : : { ...
, ) 1 ;1 1=2. + S0 = 0, Sn = 1 + : : : + n , n 2 N, Xn = max06k6n Sk .C, fXn n > 0g .E. 7.17. + 1 2 : : : { , 1 2 : : : O0 1], { .. F(x).+ Sn = 1, n 6 , Sn = ;1, n > (n = 1 2 : : : ). 1 fSn n > 1g ?E. 7.18. C, fXt t > 0g , 7.5 G(x) x,x 2 R.E. 7.19. C, %%, fXt t 2 T g,T R &% % %, %, <& t1 < t2 < t3(t1 t2 t3 2 T ) %r(t1 t3)r(t2 t2) = r(t1 t2)r(t2 t3)140(7.36) r(s t) = cov(Xs Xt), s t 2 T .E.
7.20. C, = { 9 Xt = e;tW (e2t), t 2 R, W () { , .E. 7.21. + P (X0 = 1) = P (X0 = ;1) = 1=2 X0 fNt t > 0g > 0. + fXt t > 0g (" $ ). 1 fXt t > 0g ? ( .E. 7.22. C x > 0 x = inf ft > 0 : W (t) = xg, W () { . C, fx x > 0g { ), , 7.7.E. 7.23. + fXt t > 0g { . C s > 0 Y = fXs;t t 2 O0 s]g.
1 Y ? = Y , X ?E. 7.24. C, { " .E. 7.25. + X { O0 1), Y = fYt = (Xt t) t > 0g.C, X Y . C, Y { , . - X Y ?E. 7.26. + fXt t > 0g { X . C, ) hs : X O0 1] ! X , s > 0 (.. hs 2 B(X ) B(O0 1])jB(X ), s > 0) ts fXu 0 6 u 6 tg t s > 0 , Xt+s = hs (Xt ts) .. t s > 0.E. 7.27. + Rm P (x t B ) = P (x + y t B + y) x y 2 Rm, t > 0, B 2 B(Rm).
C, ).B&% && %. O?] , , .. ( , ). O?] " " (. . ??), .. ,) , . 4 X = fXt t 2 T Rg, , % %-& Ft t 2 T (Fs Ft F s 6 t, s t 2 T ). K , (Xt Ft)t2T { ( t (X B)), % X (Ft)t2T (.. Xt 2 Ft jB t 2 T ) s t 2 T , s 6 t B 2 BP (Xt 2 B jFs) = P (Xt 2 B jXs) ..(7.37)3 , , .. Ft = F6t = fXs s 2 T \(;1t 2 T .F ' & % % % . =7 , (. O?, .
2]).141+ (E F P ) - A1 A2 E F :, E) A1 A2, P (A1A2jE ) = P (A1jE )P (A2jE ) Ak 2 Ak k = 1 2:(7.38)@ , (7.1) , fXtg ) F6t F>t t 2 T .E. 7.28. C, (7.38) , E(F1F2jE ) = E(F1jE )E(F2jE ) Fk 2 Mk , Mk { L2(E Ak P ), k = 1 2.E. 7.29. D E ) A1 A2, E ) A1 _ E A2 _ E , A _ E -, ) A E .E. 7.30. (. . 7.6). 9 (7.38) , P (AjA1 _ E ) = P (AjE ) A 2 A2:E. 7.31. + - E A1 E ) A1 A2 ( 2 I ). @\2I E ) A1 A2.E. 7.32. D (7.38), A1 \ A2 E . , (7.1) )) , fXtg F6t \ F>t.', - A1 A2 , ) - E = f Eg.D (Xt t 2 T ) { , -A(U ) = fXt t 2 U g, U T . , T R, , ), ) "" ")". +" , - A(U1) A(U2) U1 U2 T - ).
0 (7.1) ) , , , ) (., , O?]),) C01(T ) ( ), T Rn, ( ) , ) supp U (U T ), " ", (7.1). @ , S - T , S 2 S ) : S1 = S , ;, S2 = T n OS ], OS ] = S @S , ; { ,) @S ( S ). ( - A(U ), U T (, A(U1 U2) = A(U1) _ A(U2)) .# S, A(;" ) ) A(S1) A(S2) " > 0 ( ;" "- ;). C , Zd, ) (. 1).* , , '% %, (. . 7.25), % % %%, =< , , < '<< % %, % %, %, . @, O?, . 91] , , ) " " , ,) .142 8.
. (. # A . & 4. E # # . & 4. . F .+ X %,.. , . @ xi 2 X i, " , i f0 1 : : : rg, N f0g. + B X . + (X B) , , (x y) = 1 x 6= y (x x) = 0, x y 2 X . ' , g : X ! R BjB(R)-. 4 %# ;.M - A, E Aj j 2 J , J { (j2J Aj = E, Ai \ Aj = i 6= j ). C , ) Ejj < 18 E1 A<jE(jA) = : P (Aj ) P (Aj ) 6= 0(8.1)0 P (Aj ) = 0 Aj ( E(jA) .. ).C X = (Xt t 2 T R) t1 : : : tn 2 T- fXt1 : : : Xtn g E fXt1 = j1 : : : Xtn = jn g, j1 : : : jn 2 X .
+" (7.12) (8.1) X = (Xt t 2 T )% ,< ) % %, %, <& m 2 N, s1 < : : : < sm < s 6 t ( T ) <& i j i1 : : : im 2 X %P (Xt = j jXs1 = i1 : : : Xsm = im Xs = i) = P (Xt = j jXs = i)(8.2) P (Xs = i1 : : : Xsm = im Xs = i) 6= 0. ' P (AjB ) = P (AB )=P (B ) P (B ) =6 0 ( P (Aj = x) = '(x), P (Aj) = '()).+ Xs = fi 2 X : P (Xs = i) =6 0g, s 2 X .
C s 6 t (s t 2 T ), i 2 Xs, j 2 X1 pij (s t) = P (Xt = j jXs = i)(8.3)* (8.3) , pij (s t) :1) pPij (s t) > 0 i 2 Xs , j 2 Xt, s 6 t, s t 2 T ,2) pij (s t) = 1 i 2 Xs s 6 t, s t 2 T ,j 2Xt3) pij (s s) = ij i j 2 Xs, j 2 Xt s 2 T , ij | -,4) i 2 Xs, j 2 Xt s 6 u 6 t (s u t 2 T )pij (s t) =Xk2Xupik (s u)pkj (u t):(8.4)1430 (8.4) ;.9 1){3) , 4). 9 (8.2) (8.3), j Xs = i) = X P (Xt = j Xu = k Xs = i) =P (Xt = j j Xs = i) = P (XPt =(Xs = i)P (Xs = i)k=Xk : P (Xu =k Xs=i)6=0=Xk2Xuk Xs = i) =P (Xt = j j Xu = k Xs = i) P (XPu =(X = i)sP (Xt = j j Xu = k)P (Xu = k j Xs = i) =Xk2Xupik (s u)pkj (u t): (8.5)0 (8.4) { " - { T.4 (7.16), .??, , "PXs -.." " i 2 Xs".
9 4 : i j s t , u k 2 Xu , i k, k j .r rrk6isuq>7 j-t. 8.1 Xs X , s 2 T , X { . pi (0), i 2 X0, pi (0) > 0,pi(0) = 1. $% pij (s t) s 6 t (s t 2 T ), i 2 Xs, j 2 Xt, i2X0 . 1){4). (E F P )) % X = fXt t 2 T g Xt t 2 T (.. Xt (! ) 2 Xt t ! 2 E), , pi(0) = P (X0 = i) pij (s t) = P (Xt = j j Xs = i) s 6 t (s t 2 T ), i 2 Xs,j 2 Xt .PA 8.1.
0 2 T O0 1),2 D X ), 0 6 t1 < : : : < tn, n 2 N, Bk Xtk , k = 1 : : : nX XXP (Xt1 2 B1 : : : Xtn 2 Bn) = pi (0)pij1 (0 t1) : : :pjn;1 jn (tn;1 tn): (8.6)ij1 2B1jn 2Bn+" Pt1 :::tn (B1 : : : Bn ) ) (8.6) (. 2.8) 3 . ??. @ (Xt t 2 T ), .-.. . 4 , C7.1 ). 2144=, P (s i t B ) =Xj 2Bpij (s t)(8.7) s 6 t (s t 2 T0), i 2 Xs, B X . ??. +, % - %,< % () %% % '% % '%.@ , % 8.1 & %%.@ 8.2.
B%, % pij (s t), s 6 t(s t 2 T ), i 2 Xs, j 2 Xt, ' %, ', s 6 t (s t 2 T ) i j 2 X . C pij (s t) = 0 i 2 Xs, j 2= Xt pij (s t) = pi j (s t) i 2= Xs, j 2 X , i0 = i0(s) 2 Xs (Xs =6 ). + " , 1) { 4) (%, ' %%, Xt = X t 2 T ). .A 8.3. % N = fNt t > 0g , N | % ; X = f0 1 2 : : : g, pi (0) = i0 0 6 s < t, i j 2 X08 (m((s t]))j;i<;m((st]) j > iepij (s t) = : (j ; i)!0j < i(8.8)m { - B(O0 1)). ( pij (s s) = ij s > 0 i j 2 X .2 + N | , .. N0 = 0 .., ) Nt ; Ns m((st]) t > s > 0, .
. ) + m((s t]). @ ) 2.14. 7.7 N | , " .. Nt = Nt ; N0 m((0t]), t > 0Q X = f0 1 : : : g. = , pi (0) = P (N0 = i) = i0, i 2 X .C 0 6 s < t j ; i Ns = i) = P (N ; N = j ; i)pij (s t) = P (Nt = j j Ns = i) = P (Nt ; NPs(=tsNs = i) (8.8).
+ s > 0 i j 2 X pij (s s) == P (Ns = j j Ns = i) = ij .B&%. + 8.1, fNt t > 0g (8.8) , . =, N0 = 0 .., . . pi (0) = i0, i 2 X . C s < t k > 0 , (8.6) (8.8),P (Nt ; Ns = k) =1Xl=0P (Nt ; Ns = k Ns = l) =1Xl=0P (Nt = k + l Ns = l) =145==1 XXl=0 ipi(0)pil (0 s)plk+l (s t) =1X(m((0 s]))ll!l=01Xl=0p0l (0 s)plk+l (s t) =e;m((0s]) (m((ks! t])) e;m((st]) = (m((ks! t])) e;m((st]):kk(8.9)( , t > 0 Nt X = f0 1 : : : g.1P* (8.9) , P (Nt ; Ns = k) = 1, " P (Nt ; Ns = k) = 0 k < 0.k=0*, Nt ; Ns m((st]), 0 6 s 6 t.C n 2 N 0 = t0 6 t1 < : : : < tn, 0 6 k1 : : : kn P (Nt1 = k1 Nt2 ; Nt1 = k2 : : : Ntn ; Ntn;1 = kn ) == P (Nt1 = k1 Nt2 = k1 + k2 : : : Ntn = k1 + : : : + kn) =X= pi(0)pik1 (0 t1)pk1 k1+k2 (t1 t2) : : : pk1 +:::+kn;1 k1+:::+kn (tn;1 tn) =ik1k2kn= (m((0k t!1])) e;m((0t1]) (m((tk1 !t2])) e;m((t1t2]) : : : (m((tn;k1! tn])) e;m((tn;1tn]) ==Ynm=11n2P (Ntm ; Ntm;1 = km):(8.10)+ (8.10) (8.6) (8.9).
( ) . 2+ m() { - B(O0 1)) m(O0 1)) = 1. = M (t) = m(O0 t)) t > 0 ) M ;1 (t) = inf fu > 0 : M (u) > tg t 2 O0 1):A 8.4. fN (t) t > 0g { % ) m(). f (t) = N (M ;1(t)) t > 0g, = 1. 2 , % f (t) t > 0g = 1 - m() B (O0 1)) % N (t) = (M (t)), t > 0, #) )# m().2 0 , m((s t]) = M (t);M (s) 0 6 s < t M (M ;1 (t)) t t > 0. 2@ , %% '% &%,% %,% -, .A 8.5 ( % ). 1 : : : n : : :| % , . .
#) (;xpi (x) = e x > 00 x < 0: 0 (! ) = 0, t > 0 t(!) = max k :146Xi6k(8.11)i(!) 6 t (8.12)P = 0, . . (!) = 0, (!) > t. f t > 0g | t1? % # .t@ 8.5 , , .. 8.2.63210ot(! )-oo|{z1 (! )-}| 1 {z-12 (! )}|{z}3 (! )1t. 8.22 + 0(!) = 0. + t > 0.
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