А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 21
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ak (n) % ak < 1 n ! 1 k 2 Z+ (..ak (n) 6 ak (n + 1) k n 2 Z+ limn!1 ak (n) = ak k 2 Z+). Xkak = nlim!1Xkak (n):(9.52)* " , (9.50) (9.51) , P (t) { %%% ' t > 0.- , s t > 0 i j n 2 Z+(Pn (s + t))ij =Xk(Pn (s))ik (Pn (t))kj , ak (n) = (Pn (s))ik (Pn (t))kj ( s > 0,i j 2 Z+), C9.10 , % P (t), t > 0 &<%, .. (8.19).9' %,, % P (t), t > 0 %% &% 0, P (t) = I +P (t) = I +Zt0Zt0QP (s)ds(9.53)P (s)Qds:(9.54)C (9.51), ..QP (s) = P (s)Q s > 0:(9.55)177C, ( ) i j 2 X s > 0(QnPn (s))ij = qii(Pn (s))ij +1Xk=0ak (n) ak (n) = qik (Pn (s))kj (1 ; ik ), k n 2 Z+.
+ C9.10, lim Q P (s) = QP (s). K , Pn (s)Qn ! P (s)Q n ! 1.n!1 n n= , 1 k k+1XQnPn (s) = s Qk!n = Pn (s)Qn n 2 Z+ s > 0:k=0C " () " Pn (t) = I +.. i j 2 X , t > 0(Pn (t))ij = ij + qiiZt0Zt0QnPn (s)ds(Pn (s))ij ds +Z tX10 k=0qik (Pn (s))kj (1 ; ik )ds:0 n ! 1 1. 3 (9.53). 2A 99.12. % Q, ) , # (9.41). > (9.16) % ., #) Q-%, , (9.23).2 0 X , " X = Z+. + , = (P (t) t > 0), % %,% % 99.10, &% % ,%, .. (Pe(t) t > 0) {- (9.16), ..Pe0(t) = QPe(t) t > 0 Pe(0) = I(Pe(t))ij > (P (t))ij i j 2 X t > 0:(9.56)J j 2 X (x(t))i = (Pe(t))ij , (x(0))i = ij , i 2 X .
@ i 2 X ,t>0nXX0(x(t))i = qik (x(t))k + qik(x(t))k k=0..k>nx0n(t) = Qn xn(t) + Rn (t)P - xn(t) = ((x(t))0 : : : (x(t))n) (Rn(t))i = k>n qik (x(t))k, i = 0 : : : n.K (9.49) xn(t) = Pn (t) +178Zt0Pn (t ; s)Rn (s)ds n 2 Z+ t > 0.. i 6 n, j 6 n(xn(t))i = (Pn (t))ij +Z tXn0 k=0(Pn (t ; s))ik (Rn (s))k ds:9, (Pn (u))ik > 0, (Rn(s))k > 0 i k = 0 : : : n s u > 0, (9.56).%, = &% % 0 % % %% % &%. @ (P (t) t > 0) -. D (Pe(t) t > 0) P{ - P , j peij (t) 6 1 i 2 X .
*(9.56) j pij (t) = 1, i 2 X , , peij (t) = pij (t) i j 2 X ,t > 0. K , , ( ) -, ) . 2E. 9.13. + Q (9.41) . C, (. C9.11) (P (t) t > 0) , Q-. D (P (t) t > 0) { , ) Q-, .E. 9.14. ( C8.29). + Q (9.41) . @ ) ) Q-.1. C > 0 (Q ; I )x = 0(9.57) x = 0, .. x = (0 0 : : : ), , supi jxij < 1.2. C > 0 (9.57) x = 0.E. 9.15. (. 9.5).
C, supi qi < 1, - .* , & % %% -= 0 % % %% , A 99.16 (. O?]). qi < 1. , - >, , #) : X (s ! ) ! 1 s ! t , X (s ! ) ! 1 s ! t .J%%, %%, = 0& %%, ' &. = (X (t) t > 0) X = Z+ - % , Q qii;1 = i qii = ;(i + i) qii+1 = i qij = 0 ji ; j j > 1(9.58)179 0 = 0, i > 0, i 2 N i > 0, i 2 Z+. D i = 0 i 2 Z+, % , i = 0, i 2 Z+, { % .E. 9.17. + , Pi ;i 1 < 1.
C, - , { .E. 9.18. C, , Pi ;i 1 < 1, , { , .T ) , . +n;10 = 1 n = 01 : :: : n 2 N(9.59)1 2 : :nR=1Xn1n;11XX1 X1 T =X1 ):S=(inn+n nn=1 n n i=1n=2i=0 i in=0A 99.19 (M%). Q { %, -#) #(9.43). 1. " R = 1, ) .Q-%, -2. " R < 1 S = 1, ) Q-%. , %.3. " R < 1 S < 1, # T < 1, ) Q-%, #) .+ % .=, R = 1 Q- C O?].C ' %% % (, ..). C 4 " , , O?], O?], O?].D P (X (0) = i) 6= 0, t > 0 Gii (t) := P (X (t1) 6= i X (t2) = i 0 < t1 < t2 6 tj X (0) = i).. " , , i, ( ) " t.
0 i , limt!1 Gii (t) = 1, . 0 i R , ) 01 tdGii (t).E. 9.20. C, ) .1. 3 , .1802. + ( ) , Xn(n n);1 = 1:3. + ( ) , XnXn = 1 n(n n);1 < 1:n (9.57).E. 9.21. C, " , Xnn < 1 Xn(n n);1 = 1 { , Xnn = 1 Xn(n n);1 = 1:181 10. 4$ $* . . "# - . F # , . # , A2 . > % # .B& ' &;% & %()) % ' % ' . + " .
C L2- X = fX (t) t 2 Oa b]g, (E F P ), X (t) = m(t) +nXk=1k (t)zk (10.1) m(t) = EX (t), 1(t) : : : n(t) , z1 : : : zn { L2(E F P ). @ X (t), t 2 Oa b], . + X " ). @, r(s t) = cov(X (t) X (s)) =nXk=1k k (s)k (t) s t 2 Oa b] k { " zk , k = 1 : : : n.K (10.1), , ) . (, 3.6.+, L2(E F P ) { , " z1 z2 : : : , X (t) 2 L2(E F P ) J " 1(t) 2(t) : : : .
= , ) , " " "". = , "" z1 z2 : : : , 1 2 : : : (, L2Oa b]). 1, ) (10.1) . J -. - . 0. +.C % % %, .+ K | S. + K - . 4 - A = fKg( S), " M = fB 2 A : (B ) < 1g -.+ B 2 K Z (B ) 2 L2(E) == L2(E F P ), . . Z (B ) = Z (B !) EjZ (B )j2 < 1, (Z (B ) Z (C )) = (B \ C ) 8 B C 2 K(10.2)182 ( ) = EZ 2 L2(E). @ Z () .* (10.2) , B C 2 K B \ C = ?, Z (B ) ? Z (C ), . .(Z (B ) Z (C )) = (?) = 0.*, Z , , Z .A 10.1. * Z -1S L2 (E) $% K, .
. B =Bk , B B1 : : : 2 K, Bn \ Bm = ?k=1 n 6= m, Z (B ) =1Xk=1Z (Bk )(10.3) .2 (10.2) (, "", .. (x y) = (x y) (x y) = (x y) x y 2 L2(E) 2 C ) kZ (B ) ;nXk=1Z (Bk )k2 = (Z (B ) ;nXk=1Z (Bk ) Z (B ) ;nXk=1Z (Bk )) == (B ) ;nXk=1(Bk ) ! 0 n ! 1: 2', Z (?) = 0 .., " (10.3) , Z .. - K. + , '% % ! 2 E, % (10.3), % % %%, 1, % %% B B1 : : : 2 K. 10.2. + S = O0 1), K = fOa b) 0 6 a 6 b < 1g (Oa a) = ?).+ Z (Oa b)) = W (b) ; W (a), W | . 9 ) , , Z | K, ) 3.@ ' Z K M.D | - K, ) Sn 2 K, n 2 N, , 1SS = Sn, Sn \ Sm = ? n 6= m (Sn ) < 1, n 2 N ( { n=1, S = S1).
= Kn = K \ Sn, . . Kn = fB 2 K : B Sng, An = fKng | - Sn. @A 2 A = fKg () A = "(A) =1n=11Xn=1An An 2 Ann (An)(10.4)(10.5) n | jKn Kn An. ( , A S Sn 2 K, n 2 N, " (10.5) .183F %%, , % (S) < 1 (. . %& S Sn, n 2 N).= A , ) m 7 )S , ) K (S 2 K). C B = Bi, Bi 2 K, i = 1 : : : m,i=1Bi \ Bj = ? (i 6= j ), 'Z (B ) =mXi=1Z (Bi):(10.6)Sr Dj , Dj 2K, j =1 : : : r,j =1mPPm Pr Z (B \ D ) Z (B ) =2 %. C, B =Dj \ Dl = ? (j 6= l). @ (10.3)iiji=1i=1 j =1rr PmPP Z (Dj ) =Z (Dj \ Bi), < % %,j =1%..j=1 i=1( f : S ! C,f=mXci 1 Bi i=1(10.7)Sm ci 2 C , Bi 2 A, i = 1 : : : m, Bi = S Bi \ Bj = ? i 6= j , .
. Bi S. i=1= (10.7) Jf =mXi=1ciZ (Bi):(10.8) 10.3. * (10.8) .Pr2 + (10.7) f f = dj 1 Dj ,j =1Sr dj 2 C , Dj 2 A, j = 1 : : : rQ Dj = S, Di \ Dj = ? (i 6= j ). ( , j =1(Bi \ Dj ) 6= 0, ci = dj . 0, , Z (?) = 0, rXj =1dj Z (Dj ) ==rmXXdjZ (Dj \ Bi) =mXXdj Z (Bi \ Dj ) =i=1 j : (Bi\Dj )6=0m rmciZ (Bi \ Dj ) =ciZ (Bi \ Dj ) = ciZ (Bi):i=1 j : (Bi \Dj )6=0i=1 j =1i=1Xmj =1Xi=1XXX 10.4.
f g | $%. (Jf Jg) = hf gi h i | L2 (S) = L2(S A ).1842 (10.9)Pr2 + f (10.7), g = dj 1 Dj , D1 : : : Dr j =1 S. @(Jf Jg) ===Xmm XrXi=1 j =1ZXm i=1i=1ciZ (Bi)ZrXj =1 Xm Xrdj Z (Dj ) =cidZj 1 Bi\Dj ()(d) =ci1 BirXj =1dj 1 Dj (d) =Zi=1 j =1m XrXi=1 j =1cidZj (Bi \ Dj ) =ZcidZj 1 Bi ()1 Dj ()(d) =f ()g()(d) = hf gi: 2% 10.5. J | $% L2(E).2 D f g | , , , f + g, 2 C , | . 10.4(J (f + g) ; J (f ) ; J (g) J (f + g) ; J (f ) ; J (g)) == hf + g f + gi ; hf f + gi ; hg f + gi ;; Zhf + g f i + Z hf f i + Zhg f i ; Zhf + g gi + Zhf gi + Zhg gi = 0:0,J (f + g) = Jf + Jg: 2(10.10) 10.6. $% L2(S).2 + f 2 L2(S).
C " > 0 H = H (") > 0 , Zf 2()1 fjf ()j>H g(d) < ":@ (7.3) f ()1 fjf ()j<H g n ;12P hn =rnk 1 Dnk , Dnk = f : rnk < f () 6 rnk +k=0+ H 2;n+1 g 2 A, k = 0 : : : 2n ; 1. + "sup jf ()1 fjf ()j<H g ; hn ()j 6 H 2;n+1 n 2 N ,f 1 fjf j<H g ; hn 6 H 2;n+1 ((S))1=2 | L2(S). + 4.2 ( ) Bnk 2 A, k = 0 : : : 2n ; 1, n 2 N, " k n(Dnk 4 Bnk ) < "H ;1 2;n;1 :@X2n ;1k=0rnk 1 Dnk ;X2n ;1k=0rnk 1 Bnk 6 HX2n ;1k=01 Dnk ; 1 Bnk = HX2n ;1k=0(Dnk 4 Bnk ) < ":185PP2n ;1mn= rnk 1 Bnk ci1 Bi , B1 : : : Bmn 2 A i=1k=0 S, . 2@ J L2(S).
C f 2 L2(S) 'Jf = nlimJf ( L2(E)),(10.11)!1 n() ffng | , fn L;!f n ! 1. 10.7. * (10.11) .2()f 2 + 10.6 ) fn L;!2n ! 1. + 10.5 10.4, kJfn ; Jfmk = kJ (fn ; fm)k = fn ; fm ! 0 n m ! 1, . . ) ffng .+ L2(E) , , ) Jfn L2(E). +, " ) L2 () . + gn ;! f , gn . @ gn ; fn 6 gn ; f + fn ; f ! 0 n ! 1. 0,kJfn ; Jgn k = kJ (fn ; gn )k = fn ; gn ! 0 n ! 1:()()()( Jfn L;! Jgn L;! , Jfn ; Jgn L;! ; .
@ , kJfn ; Jgnk!! k ; k = 0, . . = .. 2% 10.8. 4 # f g 2 L2(S)2J22| L2 () ) hn ;! f . (Jf Jg) = hf gi(10.12)L2(S) L2(E). hn 2 L2(S) (()Jhn L;!Jf n ! 1:(10.13)(10.12) fn gn , n 2 N,2 C L2 ()L2 () fn ;! f , gn ;! g (. 10.6), 10.4 . 3 J , 10.5. + J (10.12), kJf ; Jhn k = kJ (f ; hn)k = f ; hn ! 0 n ! 1: 2B&% %, %< Jf f 2 L2(S A ), | -2 .M S Sn 2 K (Sn) < 1, n 2 N. @fn := f jn 2 L2(Sn An n) n 2 N Z1 ZXf 2 L2(S A ) ()2f ()(d) =f 2()n (d) < 1186n=1 nn(10.14) An = fK \ Sn g, n | jKn Kn An. + " (10.14) S Sn,n 2 N.C f 2 L2(S) 'Jf =1Xn=1Jn fn(10.15) L2(E), fn (10.14), Jn L2(Sn) . 10.9.
* (10.15) .2 D f 2 L2(S), (10.14) fn 2 L2(Sn ) n 2 N. @, Jnfn ? Jmfm n 6= m. D fn fm | , ", . . (Z (B ) Z (D)) = 0 B 2 An D 2 Am (An | , Kn , n 2 N). C fn fm, ) , . 0, X2 MXMN_NX Jnfn ; Jnfn =kJn fnk2 ! 0n=1n=1n=M ^N N M ! 1 Jn (10.14).@, Jf S. + S =1S= ;n , ;n 2 K, ;n \ ;m = ? (n 6= m), (;n ) < 1, n 2 N. = hnm = f jn \;m ,n=11PPN(n )m n 2 N. @ fn = f jn f jn \;m , n 2 N.
@ , hnm L;!fnm=1m=1 N ! 1 , , (10.13)Jn fn =1Xm=12Jnhnm L2(E). + Jnm h 2 L2(Sn \ ;m ) L2(Sn) Jnmh = Jn h, n m 2 N. 9, 7 , 1Xn=1Jn fn =1 X1Xn=1 m=1Jnm hnm:K, gm = f j;m Jem { , L2(;m ) , Jn L2(Sn ), m n 2 N, 1Xem=1Jmgm =1 X1Xm=1 n=1Jnmhnm :(10.16)= , Jnm hnm ? Jklhkl, (n m) 6= (k l), , , 11 XXn=1 m=1kJnmhnm k =211 XXn=1 m=1kkhnm 2L2(n\;m)=Zf 2()(d) < 1: 2187- , J L2(S A ) Jf =Zf ()Z (d):(10.17)A 10.10.
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