А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 24
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= , . . EjfbN () ; f ()j2, N , , 208 . +" fbNW , . . bfNW () =ZWN ( ; )fbN ( ) d; WN (), n 2 N, , a) WN () 0QRb) WN () d = 1Q;c) Ejf^NW () ; f ()j2 ! 0 N ! 1 2 (; ].(, F%% ) WN () = aN B (aN ), sin 2 21BN () = 2 BN (0) = 21 2 aN % 1, aN =N ! 0 (N ! 1).' " ) . + L2- X = fX (t) t 2 T Rg. @ X (t) kk L2(E), X s < t. D , , - Fs = fX (u) u 6 s u 2 T g, , inf fkX (t) ; yk : y 2 L2(E Fs P )g = kX (t) ; E(X (t)jFs)k:=, -Fs , , .
J %%,, % '% %, %, & Xu u 6 s,u 2 T ( % %% L2(E)), % ) % . + " " , . Hs (X ) s 2 T , X (u), u 6 s, u 2 T , H;1 (X ) := \s2T Hs (X ) H (X ) = L2OX ] L2OX ] { L2(E) Xu , u 2 T . =, H (X ) { L2(E), ) Hs (X ), s 2 T .*, X (t) = % s %%, = &' X (t) ( L2(E))-% %% Hs (X ). *. ^(s t) := inf fkX (t) ; gk : g 2 Hs (X )g:(11.29)( ( ), L { H ( k k, ) x 2 H , inf fkx ; gk : g 2 Lg " h = PL x, PL { H L.@ , Ps X (t), Ps { H (X ) Hs (X ).209+ X , H;1 (X ) = H (X ),.. Hs (X ) = Ht (X ) s t 2 T .
+ X , H;1 (X ) = 0. ', -% %%,< % , '% T = R T = Z. 4< & %%, X%. @ EY = 0 Y 2 H (X ), Y - L2(E) " X , ) .+, ^(s t) = ^(s + u t + u) s t u 2 T:(11.30)C , y = c1X (s1)+: : :+cn X (sn) 2 Hs (X ), ck 2 C , sk 2 T ,sk 6 s z = c1X (s1+u)+: : :+cn X (sn +u) 2 Hs+u (X ) kX (t);yk2 = kX (t+u);zk2( ).
J (11.30) (t) := ^(s s + t) t 2 T:=) (11.29), , (t) = 0 t 6 0 (t 2 T ) (v) 6 (u) v 6 u (v u 2 T ):(11.31)( (11.31) , , (11.29) , .A 11.12. (% % X 1) , (t0) = 0 t0 > 0,t0 2 T ( (t) = 0 t 2 T ),2) , (t) ! R(0) t ! 1(11.32) R { % $% X .2 1) J X , X (t) 2 Ht(X ) = Hs (X ) , ,^(s t) = 0 s t 2 T .B&%. + (t0) = 0 t0 > 0 (t0 2 T ).
@ X (s+t0) 2 Hs (X ) s 2 T . (11.31) X (t) 2 Hs (X ) t 6 s + t0. = , Ht(X ) = Hs (X ) t 6 s + t0. s , X.2) %, (11.32). C s t 2 TR(0) = kX (t)k2 = kPs X (t)k2 + kX (t) ; Ps X (t)k2 = kPs X (t)k2 + (t ; s): (11.33)+" kPs X (t)k ! 0 s ! ;1 t 2 T . @ , L1 L2 { H L1 L2, PL1 x 6 PL2 x x 2 H . @ , kP;1 X (t)k 6 kPsX (t)k s t 2 T .0, P;1 X (t) = 0, .. X (t)?H;1 (X ) t 2 T , H (X )?H;1 (X ).+ H;1 (X ) H (X ), , H;1 = 0.B&%.
+ H;1 = 0. 9 (11.33), , 11.13. Ht, t 2 T { #) H , .. Hs Ht s 6 t, s t 2 T . " \s2T Hs = 0, Psx ! 0 s ! ;1 x 2 H , Ps { H Hs .2102 + kPs xk 6 kPtxk x 2 H s 6 t, , kPtk xk ! 0 k ! 1, tk+1 < tk , k 2 Z+ limk!;1 tk = ;1.= Lk := Htk " Htk+1 , .. Htk+1 # Lk = Htk , k 2 N.
@ x 2 H m > k mX;1Ptk x = Ptm x + Qj x(Qj { H Lj ). =j =kkPtk xk = kPtm xk22+Xm;1j =kP kQ xk2 6 kP xk2 6 kxk2.0, 1tj =0 jmX;1220kPtk x ; Ptm xk =j =kkQj xk2:@ , m > k ,kQj xk ! 0 k m ! 1(, k > m). H ) " y = klimP x.!1 tk+ Ptk x 2 Htm k > m, y 2 Htm m 2 N. +, y 2 \1m=0 Htm , \1m=0Htm = \t2T Ht Ht . +"y = 0. 2@ , 11.12 . 2A 11.14 (4,). L % ( . )X = fX (t) t 2 T g, T = R T = Z X (t) = M (t) + N (t) t 2 T(11.34) M = fM (t) t 2 T g { , N = fN (t) t 2 T g { %, M ?N , .. M (s)?N (t) s t 2 T .
> , % M N -%% %. J % X , M (t) 2 Ht (X ) ( N (t) 2 Ht (X )) t 2 T .2 + M (t) = P;1 X (t), N (t) = X (t) ; M (t), t 2 T , (11.34). + M N M (t) 2 H;1 (X ) N (t)?H;1 (X ) t 2 T .0, M ?N . * (11.34) , X (t) 2 Ht (M ) # Ht (N ). +"Ht(X ) Ht(M ) # Ht(N ).
0 , M (t) 2 H;1 (X ) Ht(X ). =Ht(M ) Ht(X ) (11.34) Ht(N ) Ht(X ) t 2 T . @, Ht(M ) # Ht (N ) Ht (X ) Ht(X ) = Ht(M ) # Ht(N ) t 2 T:(11.35)@ , Ht(N )?H;1 (X ) t 2 T , H;1 (N )?H;1 (X ). Ht (N ) Ht(X ), " H;1 (N ) H;1 (X ). *, H;1 (N ) = 0,.. N { . 4 , Ht (M ) H;1 (X ), t 2 T . * (11.35), H;1 (X ) Ht (M ) # Ht(N ). 9, Ht (N )?H;1 (X ), H;1 (X ) Ht (M ), t 2 T . 0, Ht(M ) = H;1 (X ), t 2 T , M { .9', % M N { % . C " H (X )u 2 T , Su ,Su(c1X (t1) + : : : + cnX (tn )) = c1X (t1 + u) + : : : + cnX (tn + u)211 ckP2 C , tk 2 T , k =1 : : : n, n 2 N. 2 . C,Pn k=1 ck X (tk ) = mj=1 dj X (sj ), dj 2 C , sj 2 T , j = 1 : : : m, m 2 N, X u 2 T0=knXk=1ck X (tk ) ;mXj =1dj X (sj )k = k2nXk=1ck X (tk + u) ;mXj =1dj X (sj + u)k2:K , u 2 T(Sux Suy) = (x y) x y 2 Lin(Xt t 2 T )..
Su { Xt t 2 T , kSuxk = kxk x 2 Lin(X (t) t 2 T ), u 2 T . 0, Su H (X ), Su.3 ( Lin(X (t) t 2 T )), (Su )u2T { H (X ), ..Su+v = SuSv u v 2 T S0 = I { . *, %( ) % (Su )u2T H (X ). 11.15. 4 # u v 2 TSuHv (X ) = Hu+v (X ) Su H;1 (X ) = H;1 (X )(11.36)Pu+v Su = SuPv P;1 Su = Su P;1 (11.37) Pt { H (X ) Ht (X ), t 2 T f;1g.2 C u v 2 T , , Su (Lin(X (t) t 6 v t 2 T )) 2 Hu+v (X ), "SuHv (X ) Hu+v (X ). 9, Su S;u = I , Hu+v (X ) = SuS;u Hu+v (X ) SuHv (X ).+ (11.36) . D x 2 H;1 (X ), x 2 Ht (X ) t 2 T .+ Su x 2 Ht+u (X ) t 2 T .
0, Sux 2 H;1 (X ),.. Su H;1 (X ) H;1 (X ). +" H;1 (X ) = SuS;u H;1 (X ) SuH;1 (X ), (11.36). Su, Pv , u v 2 T , X (t), t 2 T , , ) (11.37). C u v t 2 T Pu+v Su X (t) = Pu+v X (t + u). 0 , X (t) = yv (t) + zv (t), yv (t) = Pv X (t) 2 Hv (X ) zv (t)?Hv (X ). C, X (t + u) = SuX (t) = Su yv (t) + Su zv (t), Su yv (t) 2 Hu+v (X ) h 2 Hu+v (X ) (11.36) h = Sug, g 2 Hv (X ).
+"(h Suzv (t)) = (g zv (t)) = 0:D H = L # L? (L? { L), " x 2 H x = y + z, y 2 L, z 2 L? ( y = PLx, PL { H L). +", SuPv X (t) = Suyv (t) = Pu+v X (t + u). + (11.37). K, P;1 SuX (t) = P;1 X (t + u), X (t) = y;1 (t) + z;1 (t), y;1 2 H;1 (X ) z;1(t)?H;1 (X ). @ X (t + u) = Su X (t) = Su y;1(t) + Su z;1(t).+ Su y;1(t) 2 H;1 (X ) Suz;1(t)?H;1 (X ) ( (11.36)) Su P1 X (t) = Su y;1(t) = P;1 X (t + u). 2212' %,% % 11.14. + M (t) 2 H;1 (X ) H (X ), EM (t) = 0, , , EN (t) = 0, t 2 T .
@ , M N { % . C t u v 2 T , 11.15 Su, (M (v + u) M (t + u)) = (P;1 X (v + u) P;1 X (t + u)) = (P;1 Su X (v) P;1 SuX (t)) == (Su P;1 X (v) SuP;1 X (t)) = (P;1 X (v) P;1 X (t)) = (M (v) M (t)):K,(N (v + u) N (t + u)) = (X (v + u) ; M (v + u) X (t + u) ; M (t + u)) == (X (v) X (t)) ; (P;1 Su X (v) SuX (t)) ; (SuX (v) P;1 SuX (t)) + (M (v) M (t)) == (X (v) X (t)) ; (M (v) X (t)) ; (X (v) M (t)) + (M (v) M (t)) == (X (v) ; M (v) X (t) ; M (t)) = (N (v) N (t)):9' %%, ' (11.34). K , X (t) = U (t)+V (t), U = fU (t) t 2 T g, V = fV (t) t 2 T g { , U ?V U (t) V (t) 2 Ht(X ), t 2 T .
@ Ht(X ) = Ht (U ) + Ht(V ) = H (U ) # Ht (V ) t 2 T:0,H;1 (X ) = \t2T (H (U ) # Ht(V )) = H (U ) # \t2T Ht (V ) = H (U ) + 0 = H (U ):@ ,M (t) = PH;1(X )X (t) = PH (U )X (t) = PH (U )U (t)+PH (U )V (t) = PH (U )U (t)+0 = U (t) t 2 T:@ . 2@ 11.16. M (11.34) .
- , L2- ( ) X , T R, , , { . 9 (. 11.14) . 2 - 11.14.= , 11.15 (11.30):^(s + u t + u) = kX (t + u) ; Ps+u X (t + u)k = kSu X (t) ; Ps+u Su X (t)k == kSuX (t) ; SuPs X (t)k == kX (t) ; Ps X (t)k = ^(s t):1 " = f"n n 2 Zg #) % L2- X = fXn n 2 Zg, Hn (X ) = Hn (") n 2 Z.= 213A 11.17.
4 , % % % X = fXn n 2 Zg , , . #) % " = f"n n 2 Zg 2 fcn g1n=0 2 l , , ..Xn =1Xk=0ck "n;k n 2 Z(11.38) .2 %, X { . 3 , m 2 Z Hm (X ) L2(E) " hm;1 + cX (m), hm 2 Hm;1 (X ), c 2 C . + X (m) = gm;1 + zm, gm;1 2 Hm;1 (X ) zm?Hm;1 (X ), Hm;1 L2(E) hm;1 + czm;1, , ", hm;1 2 Hm;1 (X ),c 2 C (, ). ', zm 6= 0 (0 { L2(E), 0 ..). *Hm;1(X ) = Hm (X ) (11.36) Hn;1 (X ) = Hn (X ) n 2 Z, . n = zn=kznk - , 11.13. K , n 2 Z tk = n ; k, k 2 Z+. + " Lk := Htk " Htk+1 n;k (k 2 Z+). +kPtk X (n)k ! 0, k ! 1, (11.38), 1Xk=0jck j2 6 kXn k2 < 1:@ , (11.38) n 2 Z, "" ck , k 2 Z, ) , n. C , , 1XXn = c(kn)n;k k=0= (Xn n;k ), n k 2 Z. +) H (X ). K , "k = Sk 0, k 2 Z (-, Sk { H (X )). 9 (11.36), , "k 2 Hk (X ) "k ?Hk;1 (X ).
-, kvarepsilonkk = kxi0k = 1, k 2 Z. + Hk (X ) " Hk;1 (X ) k , , , "k = k k , k 2 C ,k 2 Z f"k g { ) X . L2(E) c(kn)X0 =1Xk=0c(0)k 0;k "0;k=1Xk=0ck ";k : Sn (n 2 Z), ) ,Xn = Sn X0 =1Xk=0ck Sn";k =1Xk=0ck Sn S;k 0 =1Xk=0ck Sn;k 0 =1Xk=0ck "n;k :B&%. C " = f"n n 2 Zg { (.. - ) X ), (11.38) , X { , Hn (X ) Hn ("), n 2 Z. = H;1 (X ) Hn "214 n 2 Z. ( "n+1?Hn ("), n 2 Z, " zn?H;1 (") n 2 Z. "n { H (X ). 0, H;1 = 0. 2@ , (11.38) %, % %- % % %,% % % & =. C X = fXn n 2 Ng Xn = Mn + Nn = Mn +1Xk=0ck "n;k n 2 Z(11.39) Mn { ), f"n n 2 Ng { ) Nn, (n 2 Z), fck g1k=1 2 l2.
9, PHm(X )Mn+m = Mn+m Hm (X ) = Hm (") m n 2 Z, n2 = EjXn+m ; PHm(X )Xn+m j2 = EjNn+m ; PHm (X )Nn+m j2 == EjNn+m ; PHm (")Nn+m j2 == Ejn;1Xk=0ck "n+m;k j2 =n;1Xk=0jck j2: (11.40)J (11.40) ) (. (11.32)), X = fXn n 2 Ng, .. ) fNn n 2 Ng, n ! 1n2!1Xk=0jck j2 = EjN0j2 = EjX0j2 = R(0) R { X .* 11.17 11.15 , , .. (11.38), 12X1;ikf () = p ck e 2 O; ]:2 k=0*) A 11.18 (0). (% % %X = fXn n 2 Zg , f () #Z;log f ()d > ;1:C " ] H2 (. O?, .
]). .% & %, , % = . ( .215E. 11.1. + X = fX (t) = ei(t+) t 2 Rg, , O0 2] ( ), E2 < 1. 9, X (11.13), G = E2F , F { .E. 11.2. ) Y = fY (t) = cos(t + ) t 2 Rg:+, Y { . D X = fX (t) t 2 Rg { , , fRe X (t) t 2 Rg fIm X (t) t 2 Rg { ?E. 11.3. O ] + fNt t > 0g { , ) > 0. + fNt t > 0g P ( = ;1) = P ( = 1) = 1=2. , $ , X (t !) = (!)(;1)Nt(!) t > 0 ! 2 E:(11.41)( " . ( EX (t) cov(X (s) X (t)),s t > 0 ( 3.19).E. 11.4.
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