А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 23
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X (t + ) = X (t) .. t 2 R. ( - { 3" X .=, %, ,-<% % (., ., O?], . VI, x4 O?], . 1, x15-18). + ) O?], O?], O?].= . , , , ) . *) x174 O?]. " % ', < %% & < , % .+ H { , ) , T . J K : T T ! C ) H , 1. K (t ) 2 H t 2 T ,2. hf K (t )i = f (t) f 2 H , h i { H .A 910.24 (S=).
4 , $% K -) , , % .2 %, K { . C n 2 N, t1 : : : tn 2 T ,z1 : : : zn 2 C 1Xjr=1zj zrhK (tj ) K (tr )i = k kuk2 = hu ui, u 2 H .198nXj =1zj K (tj )k2B&%. %, K { %%, T T .M Lin(K ) { K ( t), t 2 T . C f g 2 Lin(K ),.. f=nXj =1aj K (tj ) g =mXr=1br K (sr )(10.49) aj br 2 C , tj sr 2 T (j = 1 : : : n, r = 1 : : : m), hf gi :=n XmXj =1 r=1aj br K (tj sr ):(10.50)E. 10.25. +, (10.50) , .. , ) " aj , br tj , sr f g 2 Lin(K ).* (10.50) , br = ir , i r 2 f1 : : : mg, hf K (si )i =nXj =1aj K (tj si) = f (si):+ si { , 2. K h i, (10.50), Lin(K )Lin(K ) ().C f 2 Lin(K ) kf k2 = hf f i.
@ Lin(K ) k k. D fn{ Lin(K ), t 2 T , - { 1 { j, jfN (t) ; fM (t)j = jhfN ; fM K (t )ij 6 kfN ; fM k(K (t t))1=2:(10.51)4 , kK (t )k2 = hK (t ) K (t )i = K (t t). @ , ffN g f T . = H Lin(K ). Cf g 2 H h i . C E. 10.26. +, H -, ) K ) . 20 3.3 . +" <&< < <r(s t), s t 2 T ' %%, % ,&% %% H h i ( C10.24 , H ). 0,r(s t) = hr(s ) r(t )i:A 910.27 ().Ff(10.52)2 g X = X (t) tT %2L -% (E P ) % $% r(s t), s t T .
H { ) r . (E P ) ) % L2-% Y = Y (h) h H, h iF2fX (t) = Y (r(t )).. t 2 T2 g(10.53)199(Y (h) Y (g)) = hh gi # h g 2 H(10.54) ( ) = E 2 L2 (E F P ). L2 OX ] H , L2 OX ] { Lin(X ) (.. X ).P2 C h = nk=1 ck r(tk ) 2 H , ck 2 C , tk 2 T , k = 1 : : : n, n 2 N, nXY (h) :=k=1ck X (tk ):3 , " , H . 2)' % 910.27 10.13 % % ,, %E. 10.28.
C, r - (10.21), H ) r h(t) =Zg()f (t )(d) g 2 L2Of ](10.55) L2Of ] { L2(S A ) f (t ), t 2 T (, g t, , f (t )). 0 H Zhh1 h2i = g1()g2()(d)(10.56) hk gk , k = 1 2 (10.55). + Z { B 2 M(E F P ), ) (. 10.10), L2Of ] = L2(S A ). =7, X = fX (t) t 2 T g (10.22), Y , ) C10.27, Y (h) =Zg()Z (d)(10.57) h g (10.55).E. 10.29. C, () H ) r(s t) = minfs tg, s t 2 R+ (.. r { ), h(t) =Zt0g()d t 2 R+Z10g2()d < 1(10.58) d { 3.
=7, L2OW ], .. L2(E F P ) W (t), t 2 R+, (10.57), Z { B 2 M(E F P ), (. 10.2), g (10.58).@ 910.30. j j (. C4.11 q = 1) { " H O0 1] ) r(s t) = minfs tg, s t 2 O0 1]. ( - q > 1.200 11. $* $$ C .> @. 5# . > 3{1. # . &# L2(*). #2 . # . . = . - . -! ?#. - , 2 #. % % .2% , , %, %< % ( ) , % &% , % % 0. C" .- g(t) t 2 T , T ( { ), % , R(s t) = g(s ; t), s t 2 T , ..
(3.6). -L2 { fX (t) t 2 T g (.. EjX (t)j2 < 1, t 2 T ), T { , % . , EX (t) = a t 2 T(11.1)r(s t) = cov(X (s) X (t)) = r(s ; t 0) =: R(s ; t) s t 2 T(11.2)* 3.3 , %%, fR(t) t 2 T g, T { , % % () fX (t) t 2 T g.+ fX (t) t 2 T g, T { , % , n 2 N u t1 : : : tn 2 T(X (t1 + u) : : : X (tn + u)) =D (X (t1) : : : X (tn)).. . 3 , L2- . C " .B %%, Z % < %.A 11.1 ("%). 1% R(n) n 2 N, % , R(n) =Z;einQ(d) n 2 Z(11.3)R Q { % B (O; ]).
H ; O; ].2012 =, (11.3) , n 2 N, t1 : : : tn 2 Z z1 : : : zn 2 CnXkq=1Z Xn2zk zq R(tk ; tq ) = zk eitk Q(d) > 0:; k=1(11.4)B&%. C N > 1 2 O; ] ( R()) N XN1 XfN () = 2NR(k ; q)e;ikeiq = 21k=1 q=1Xjmj<N(1 ; jmj=N )R(m)e;im :(11.5)4 , N ;jmj (k q), k;q = m ( k q 2 f1 : : : N g,jmj < N ).
= B(O; ]) QN fN 3, ..ZQN (B ) = fN ()d B 2 B(O; ]):B@, (2.10), N > 1 Z;einQN (d) =Z;einf (1 ; jnj=N )R(n) jnj < NN ()d =jnj > N:0(11.6) % ( 4), K = O; ] , QN (O; ]) = R(0) < 1 N ( n = 0 (11.6)), fNk g N , QNk ) Q, Q { O; ]. @, (11.6), n 2 ZZ;ZeinQ(d) = klimeinQNk (d) = R(n): 2!1;- ) .A 11.2. fX (t) t 2 Zg { % % . %, (E F P ). ) Z (), B (O; ]), , X (t) =Z;eitZ (d) t 2 Z:2 + : s t 2 Zr(s t) = cov(X (s) X (t)) =Z;ei(s;t)Q(d) =(11.7)Z;eiseitQ(d)(11.8) Q { B(O; ]).
', - ( 10.13), f (t ) = eit, 2 O; ], t 2 Z. + " 202 (10.23). C, L2 = L2(O; ] B(O; ]) Q) L2 , ) ; , J. 24 Q, ) (11.3), ( ), "" Q(f;g) ; , "" Q(f;g)+Q(fg).+ " (11.3) , e;in = ein n 2 Z. E % =, %, %& %%, % '% (; ] % '%. D , Z (f;g) = 0 .. 10.12, " (11.7) (; ]. M, O; ).4 % ' %, % (11.7) % % &,= .A 11.3.
11.2. ;11 NXL2 ()Xk ;! Z (f0g)N k=0N ! 1:(11.9)2 (11.7). Z 1 NX;1Z;11 NXikN k=0 Xk = N k=0 e Z (d) = UN ()Z (d);;8 1 (1 ; eiN)< 6= 0UN () = : N (1 ; ei)1 = 0:0, 10.10 1 NX;1Z2N k=0 Xk ; Z f0gL () = jN ()j G(d)2;(11.10) N () = UN () ; 1 f0g(), G() | , ) Z (). 9 1 NP;1 ik , N e 6 1, 2 R, , N (0) = 0 6= 0, 2 (; ]k=0jN ()j 6 N j1 ;2 eij ! 0 N ! 1 3 (11.10) (11.9). 2@ 11.4.
D fX (t) t 2 Zg | EXt = a, t 2 Z, Xt ; a =Z;eitZ (d)203NP;1NP;12 () N1 (Xk ; a) = N1Xk ; a L;!Z (f0g). 0,k=0k=0;11 NXL2 ()Xk ;! a N ! 1N k=0(11.11)% %, %, EjZ (f0g)j2 = 0, % , < G(f0g) = 0,. . ( ) .0 (11.11) # L2(E).M (E F P ) L2- fXt t 2 Rg. +X () t 2 R, kX (s) ; X (t)k ! 0 s ! t(11.12) kk = (Ejj2)1=2, 2 L2(E F P ). ( - " .A 11.5. (% .
% fX (t) t 2 Rg , %$% R(), (11.2), .2 + X (t) X (t) ; a, a = const, , ( ) - t 2 R. +" - ) ) r(s t) = EX (s)X (t) = R(s ; t) s t 2 R. J fX (t) t 2 Rg % R, s t u v 2 Rjr(u v) ; r(s t)j 6 kX (u) ; X (s)kkX (v)k + kX (s)kkX (v) ; X (t)k ! 0 u ! s v ! t.
4 - { 1 {()j , X (v) L;!X (t), kX (v)k ! kX (t)k v ! t. =, R(t) ; R(0) = r(t 0) ; r(0 0).B&%.kX (s) ; X (t)k2 = ;(R(t ; s) ; R(0)) ; (R(s ; t) ; R(0)) ! 0 s ! t, R() 0. 2% 11.6. 1% R(), % R, R , .2 + R() R . +2 3.3 () fX (t) t 2 Rg r(s t) = EX (s)X (t) = R(s ; t),s t 2 R. 11.5 " R. @jR(s) ; R(t)j = jEX (t)X (0) ; EX (s)X (0)j 6 kX (t) ; X (s)kkX (0)k ! 0 s ! t. @ , R() . 2B % ( ), % R, '%204A 11.7 (F{K). R(t), t 2 R, | % $%. # t 2 RR(t) =Z1;1eitG(d)(11.13)G | % B(R).=, (11.13) .
0 (11.5) , . @ , % 11.7 %, %%, &. C 1 { ] ) 4.@ 11.8. D G, ) (11.13), X (t), t 2 R, f () 3, . . t 2 RR(t) =Z1;1eitf () d f # . * 11.7, 4, , R() 2 L1(R), ) , (??).A 11.9 (0 { 0). (E F P ) % % . % fX (t) t 2 Rg ( ). ) Z (), B (R), X (t) =Z1;1eitZ (d) t 2 R:(11.14)2 11.5 R(t) = EX (t)X (0) ( R) 3.3 .
+ 1{] (11.13). 0,r(s t) = cov(X (s) X (t)) = R(s ; t) =Z1;1ei(s;t)G(d) =Z1;1eiseitG(d) G() | - B(R). @ , - ( 10.13), f (t ) = eit (t 2 R) (10.23). , L2 = L2(R B(R) G) f ()1 fjj6ag, L2 , )ina a ;a, e , n = 0 1 : : : . 2@ &% < %, %% % . 2 7, , , , ) . = .205z ( L2(E)) " = f"(n) n 2 Zg .. 3 , " , ) f () = 1=(2), 2 O; ]. * , " ", " ( ) .A 11.10. (% . % %X = fX (t) t 2 Zg # , 1 fck k 2 Zg 2 l2 (. .jck j2 < 1) .k=;12^" = f"(n) n 2 Zg, L (E F P ) ( .
(E F P )),, PXt =1Xk=;1ck "t;k t 2 Z:(11.15)+, ) (11.15), , % ) . : , X ) $, ". J $ ), ck = 0 k < 0. 2 , X t "k k 6 t (..X (t) t ")" ) ").2 %, (11.15), fck g f"k g .M L2(E^ F P ) ( "k , k 2 Z, " ), . .2 ()PN ct;k "k ; Pn ct;k "k L;!PN0 N n ! 1 M m ! ;1. ', E ct;k "k = 0,k=Mk=mk =ML2 ()N M 2 Z(M 6 N ). C, MN ;! , EMN = 0, jE ;EMN j 6 Ej ;MN j 66 k ; MN k ! 0 (N ! 1, M ! ;1), , E = 0.
*, EXt = 0 t 2 Z. + +r(s t) =Xkcs;k "k Xl Xct;l"l =kcs;k cZt;k =Xjcj cZ;j;(s;t) = R(s ; t) s t 2 Z(11.16). . fX (t) t 2 Zg | . 1X1Y() = pc eik:(11.17)2 k=;1 ;kM (11.17) L2O; ] = L2(O; ] BO; ] mes), mes { 3, fck g 2 l2 Z;(eiseitd = 2 s = t0 s 6= t (s t 2 Z):@ L2O; ]11XX11i(k+s)iscs;j eij:c;k e=pY()e = p2 k=;12 j=;1206(11.18)(11.19)+ + (11.19) (11.18) (11.16)Z;ei(s;t)jY()j2 d =Z;Y()eisY()eitd =Xjcs;j cZt;j = r(s t):(11.20)( Y() 2 L2O; ], .
. jY()j2 2 L1O; ]. *, (11.20) , f () = jY()j2, 2 O; ], fX (t) t 2 Zg.B&%. + f () | fX (t) t 2 Zg. @f () 2 L1OR; ] f () > 0 .. 3p ( G G(B ) = f () d). = Y() = f () (Y() = 0, f () < 0). @BY() 2 L2O; ] , , , f(2);1=2eis 2 O; ]g, s 2 Zg, , 1Xc;k eikY() = p12 k=;1(11.21) fck g 2 l2 (11.21) L2O; ]. @, +, r(s t) = R(s ; t) =Z;ei(s;t)f () d=Z;eisY()eitY()d =Xkcs;k cZt;k : (11.22) S = Z, | ) - B Z(. .- , (fkg) = 1, k 2 Z).
+ f (t ) = ct;, t 2 Z,fck g 2 L2(S B ). @ (11.22) (10.21), , 10.13 , , (E^ F P )) Z (), Xt =Zf (t )Z (d) =1Xk=;1ct;k "k (11.23) "k = Z (fkg), (11.23) L2(E). + " k , k 2 Z, , , . .Ej"k j2 = (fkg) = 1, k 2 Z. 2@ 11.11. * 11.10 , ) , , ) (11.15): 1 X21ikf () = pc e 2 O; ]:2 k=;1 ;k @ %%% %, %%.+ fX (t) t 2 Zg | , R(n) = EXn+k X k , k n 2 Z. J (11.6) , fN (), (11.5), %, %% f () ( )).207J%% ' (11.5) fN %%, % R(m) %%%< , %< &<X0 : : : XN ;1.
8m;1>1 N ;X>>< N ; m k=0 Xm+k X k 0 6 m 6 N ; 1RbN (m) = >Rb(m) ; (N ; 1) 6 m < 0>>:0 m 2 Z (11.24)X bmj :RN (m)e;im 1 ; jNfbN () = 21jmj<N+ NX;1 ;ik 21^fN () = 2N Xk e :k=0(11.25)2 . 3 , EfbN () = fN ()(11.26) ERbN (m) = R(m) m, jmj 6 N ; 1. @ , fN ) :NX;1 NX;1NX;1 NX;1 Z i(k;l)11;i(k;l)fN () = 2NR(k ; l)e= 2Nef ( ) de;i(k;l) =k=0 l=0k=0 l=0==Z;Z;;;1 NX;11 NXi( ;)k e;i( ;)l f ( ) d =2N k=0 l=0 eZ;121 NXi(;)k f ( ) d = YN ( ; )f ( ) d2N k=0 e(11.27); JNX;1 ik2 1 sin(N=2) 21YN () = 2N e = 2N sin(=2) :k=09, f () 2 L1O; ], .. ( 3) ZYN ( ; )f ( ) d ! f () N ! 1:(11.28)* (11.26) (11.28) , % %% %, %%.
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