А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 22
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( (10.17) L2 (S A ) ( | - ) L2Z L2(E F P ). Z -%M = fB 2 A : (B ) < 1g $Z (B ) = J 1 B :(10.18)S12 @ . + S = Sn,n=1 S1 S2 : : : | S (Sn) < 1, n 2 N. f g 2 L2(S). @11PPf fn, g gn , fn = f jn , gn = gjn , L2(S).n=1n=1@(Jf Jg) == Nlim!1XNXnn=1Jn f n XmJmgm = Nlim!1(Jnfn Jngn ) = Nlim!1NXn=1XNn=1Jn fnNXm=1NJm gm =hfn gn i = Nlim!1X XNn=1fn m=1gm = hf gi: (10.19)4 , Jn fn ? Jmgm n 6= m (. 10.9) , hfn gniL2 (n) = hfn gniL2 (). *, J | " . =, L2(S) J L2(E), L2Z . C B 2 K 1P (B ) = (Bn ) < 1, Bn = B \ Sn, n 2 N.
C, 1 B 2 L2(S A ) n=11P1 B 1 B , L2(S). 0, (10.15)n=1nJ1B =1Xn=1Jn 1 Bn 1P (10.8) Jn1 Bn = Z (Bn ), 10.1 Z (Bn) = Z (B ) (n=1 L2(E)). 0, B 2 K J 1 B = Z (B ), . . (10.18) J K M. - , B C 2 M J (Z (B ) Z (C )) = (J 1 B J 1 C ) = h1 B 1 C i = (B \ C ): 2@ 10.11. = Z %, EZ (B ) = 0 B 2 M. + Z K M , EZ (B ) = 0 B 2 K, " B 2 M.A 10.12.
| - % K S( A = fKg). ) (E F P ) % Z , E -% M = fB 2 A : (B ) < 1g, Z .1882 Z . + E = S, F = A. JB ) . 0 < (S) < 1 ( (S) = 0 ), P (B ) = ((S)p' Z (B !) = (S)1 B (!), B 2 A. =, Z (B ) 2 L2(E). C, B1 B2 2 A p 2Z(Z (B1) Z (B2)) = ( (S)) 1 B1 1 B2 dP = (S)P (B1 \ B2) = (B1 \ B2):%, %, (S) = 1 S1 S2 : : : | S, (Sn ) < 1, n 2 N.C A 2 A '1XP (A) = (A \ Sn ) :n=1(Sn )2n@ P | A (P (S) =Z (B !) =1 pXnn=11P2;n = 1). B B 2 Mn=12 (Sn )1 Bn (!) Bn = B \ Sn, n 2 N, L2(E) ( ) , "1 p11XXXn)2nn= (B ) < 1):E( 2 (Sn )1 Bn ) = 2 (Sn )E1 Bn = 2n (Sn) (S(B)2n nn=1n=1n=1@ , (Z (B ) Z (C )) =1Xn=12n (Sn)(1 Bn 1 Cn )L2() ==1Xn=12n (Sn)P (Bn \ Cn ) =1Xn=1(Bn \ Cn) = (B \ C ) Bn = B \ Sn, Cn = C \ Sn, n 2 N.
( %< , ) . +, Z (B ) ; EZ (B ), B 2 M, (). +" (E0 F 0 P 0) , E0 = 0, E0(2) = 1, E0 P 0. + Z { (E F P ) . (E^ F P ) = (E F P ) (E0 F 0 P 0) " Ze(B !e) := Z (B !)(!0) B 2 M, !e = (! !0) 2 E E0. 3 , Ze ( ) . 2@ .. X (t !), t 2 T , ! 2 E,. .
X (t !) = X1 (t !) + iX2(t !), X1(t ) X2(t ) F j B(R)- t 2 T . (, X (t), ) EjX (t)j2 << 1, t 2 T , % $% r(s t) = cov(X (s) X (t)) = E(X (s) ; EX (s))(X (t) ; EX (t)) s t 2 T: (10.20)1 & %%, % , . .EX (t) = 0 t 2 T( X~ (t) = X (t) ; EX (t)).189A 10.13 (0). % $% %% X (t ! ), t 2 T , (E F P ), , . . r(s t) =Zf (s )f (t )(d) s t 2 T(10.21) f (t ) 2 L2 (S) = L2 (S A ) t 2 T , | - . ) % Z , -% M = fA 2 A : (A) < 1g , ) , . (E^ F P ) = (E F P ) (E0 F 0 P 0), #) # , , X (t) =Zf (t )Z (d) "@ U() 2L2(SA ):Zt 2 T:(10.22)f (t )U()(d) = 0 8t 2 T(10.23)((10.23) , @ U: U ? f (t ) L2(S) t 2 T ), . (E F P ) Z .@ 10.14. D X (t), t 2 T , (10.22), 10.10 (10.21).2 %, (10.23). C t 2 T G : f (t )7!X (t )(10.24) :nXG(k=1ck f (tk )) =nXk=1ck X (tk )(10.25) ck 2 C n, tk 2 T , k = m1 : : : n.
+ %%, -% ,nPPP. . ck f (tk ) = dl f (sl ), dl 2 C , sl 2 T , l = 1 : : : m, ck X (tk ) =k=1l=1mP= d X (s ). C " , l=1lXnk=1lck f (tk )=XXnmk=1 l=1mXl=1 Xn Xmdl f (sl ) =Zck dZl f (tk )f (sl )(d) =k=1 l=1mck dZlEX (tk )X (sl ) =k=1 l=1XXck dZlr(tk sl) =n+ (10.26), , nXk=1190k=1ck f (tk ) ;mXl=1Xnk=1ck X (tk )mXl=1nm XXdlf (sl ) = ck X (tk ) ; dlX (sl ):k=1l=1dlX (sl ) :(10.26)*, . - G L2Of ], . .Pn L2(S) ck f (tk ).
+ " (10.23) k=1L2Of ] = L2(S A ) G(L2Of ]) = L2OX ], L2OX ] | X (t), . . L2(E F P ) c1X (t1) + : : : + cnX (tn ) (ci 2 C , ti 2 T , i = 1 : : : n).C B 2 M, , 1 B 2 L2(S), Z (B ) = G1 B :(10.27)@(Z (B ) Z (C )) = (G1 B G1 C ) = h1 B 1 C i = (B \ C ):0, Z | M .4 Z , Z (B ) 2 L2OX ] B 2 M, E = 0 2 L2OX ], .. X . @ h 2 L2(S) Jh =Zh()Z (d):+ 10.10 J | L2(S) L2Z .*, G : L2(S)7!L2OX ] L2(E) J : L2(S)7!L2Z L2(E):+ " (10.27) (10.18) G1 B = J 1 B B 2 M.( L2(S) ( 10.6).
0, G = J L2(S) , , L2OX ] = L2Z . * (10.24) , Jf (t ) = X (t), . . (10.22) .%, %, (10.23) . @ L2Of ] 6! L2(S). M L2(S) " L2Of ], . . L2Of ], - g(u ) 2 L2(S), u 2 T 0, T 0 \ T = ? (. O?]). (s t) =Zg(s )g(t )(d) = (g(s ) g(t ))L2() s t 2 T 0:=, ck 2 C , tk 2 T 0, k = 1 : : : n, n 2 N,2Z Xnck cZl(tk tl) = ck g(tk ) (d) > 0:k=1kl=1nX 3.3 ) fY (t) t 2 T 0g, (E0 F 0 P 0), cov(Y (s) Y (t)) = (s t) s t 2 T 0: (E^ F P ) = (E F P ) (E0 F 0 P 0). @ t 2 T , s 2 T 0 X (t) Y (s) " ( !~ = (! !0) 2 E~ X (t) = X (t !~ ) = X (t !), Y (s) = Y (s !~ ) = Y (s !0)).191 (T T 0) E~ . .@((t !~ ) = X (t !0) t 2 T Q0Y (t ! ) t 2 T :8><r(s t) s t 2 T 0Qcov((s) (t)) = >(s t) s t 2 T Q: 0 s 2 T t 2 T 0* ,Zcov((s) (t)) = s 2 T 0 t 2 T:h(s )h(t )(d)(h(t ) = f (t ) t 2 T Q0g(t ) t 2 T :4 , f (t ) ? g(s ) L2(S) t 2 T s 2 T 0.
- , L2Oh] = L2(S),. . @ U 2 L2(S): U ? h(t ) t 2 T T 0. 0, ) Z M (E^ F P ), L2O] = L2ZZ(t) = h(t )Z (d) t 2 T T 0:( t 2 T (t !~ ) X (t !~ ). 0, t 2 TX (t !~ ) = X (t !) =Zh(t )Z (d) =Zf (t )Z (d):z Z , , (10.23). 2 .E. 10.1. C Z , K S, .. ! 2 E, .. K? 0) Z ?E. 10.2. +, " , (10.2) ). + B K S Z (B ) 2 L2(E F P ) , (Z (B ) Z (C )) = 0, B C 2 K B \ C = ? (-, ( ) = E 2 L2(E F P )). + Z (10.3). @ (B ) := EjZ (B )j2 K " (10.2).E. 10.3. + Z 2 R { L2- (.
. 28) ,1) EjZ ; Z j2 ! 0 2 R # ,2) ) , .. 1 < 2 < 3E(Z1 ; Z2 )(Z3 ; Z2 ) = 0:( K = f(a b] ;1 < a 6 b < 1g, (a a] = ?, Z ((a b]) := Z (b);Z (a).C, Z { .192D Z, 2 R, (10.17)R Rf ()dZ .E. 10.4. + Z { B(R), EjZ (R)j2 < 1.+ Z = Z ((;1 ]), 2 R. C, Z, 2 R { 1), 2), ) .E. 10.5. + Z, 2 R { ), (, ).+ t 2 T g(t ) : R ! C ZRjg(t )j2(d) < 1 { , ) Z , Z, 2 R (. 10.3).
C, Y = fY (t) =ZRg(t )dZ t 2 T g .E. 10.6. + Z { , EB 2 M ) - (S A) (, - B 2 M= fA 2 A : (A) < 1g). + h : S ! C , h 2 L2(S A ). -ZN = fB 2 A : jh()j2(d) < 1gB ZV (B ) :=1 B ()h()Z (d) B 2 N :(10.28)=7, V (, Z ) - (B ) =ZBjh()j2(d) < 1 B 2 N :C, g : S ! C , g 2 L2(S fNg ), Zg()V (d) =Zg()h()Z (d):(10.29) - 11 , &% ,' % .M L2Oa b], ) , 3 Oa b]. 0 " (f g) =Zbaf (t)g(t)dt f g 2 L2Oa b]:(10.30)0 , ) 10.13.193 910.7. C, O0 2] - 1X1 ; eikt z 2 k=;1 ik kW (t) = p1(10.31) zk { , t 2 O0 2] ( k = 0 (1;eikt)=ik = ;t).2 C s u 2 O0 2] 1X1 0s](u) = p1ck (s)eiku:2 k=;1(10.32)J p12 eiku , k 2 Z, L2O0 2] s 2 O0 2] (10.32) " , " JpZ 21ck (s) = p1 0s](u)e;ikudu = p1 (1 ; e;iks)=ik k 2 Z2 02(c0(s) = s= 2).
M + s t 2 O0 2]1X1(1 ; e;iks )(1 ; eikt) :minfs tg = (1 0s] 1 0t]) = 2k2k=;1= 10.13, S = Z, f (t k) = p12 1;eik;ikt , t 2 O0 2], k 2 S ) S, .. (fkg) = 1, k 2 Z. 2E. 10.8. C, (10.31) .. zk N (0 1), k 2 Z. @ , , zk , k 2 Z, (10.31) , ) minfs tg, s t 2 O0 2].C &% % %', % %% ,&% %%. + "E. 10.9. + X = fX (t) t 2 Oa b]g { L2-, Oa b]. C, " , Oa b] m(t) = EX (t) Oa b] Oa b] r(s t) = cov(X (s) X (t)).0 Oa b] Oa b] r 1, L2Oa b] ):Af (s) =Zbar(s t)f (t)dt f 2 L2Oa b](10.33)r " A.E.
10.10. C, (10.33) A ( ).194- A { : { j (. O?]), Z bZ baajr(s t)j2dsdt < 1:(10.34)', A , .. (. O?, . 531]) A " r(t s), r(s t) = r(t s) s t.( (. O?], . 4, x6) .A 910.11 (",&% { I%). A { () H . ) fn gn2J A, #) , , # h 2 H h=Xn2Jcnn + u(10.35) cn 2 C , n 2 J , u 2 KerA, .. Au = 0. ; J ( (10.35) k k = ( )1=2). , A n . # % 0 , j1 j > j2 j > : : : .
" J , limn!1 n = 0.=, A H , , ) ,, (n u) = 0, n 2 J .C A (10.33) H = L2Oa b] KerA fk gk2M , M { ( M J ), KerA = 0, M = ?).C fngn2J fgn2J M H , (10.35) A:Xh=ck k ck = (h k ) k 2 J M:(10.36)k2J MD J M , " J L2Oa b]. 10.9 10.10 r X = fX (t) t 2 Oa b]g (10.36) t 2 Oa b]Xr(t ) =ck (t)k()(10.37)k2J M'ck (t) ="Zbar(t s)k (s)ds = k k (t) = k k (t)r(t ) =Xk2Jk k ()k(t):(10.38)(10.39)D J { , (10.36), (10.39) L2Oa b] t 2 Oa b]. 4, k , k = 0, k 2 M .195E. 10.12. C, k > 0 k 2 J - r. %= (10.39), A 910.13 ()).
r { Oa b] Oa b] % $%. s t 2 Oa b]r(s t) =Xk 2Jk k (s)k (t)(10.40), J { , (10.40) # Oa b] Oa b]. H, - k k # #) $% (10.33). A 910.14 (0 { -). X = fX (t) t 2 Oa b]g { Oa b] % L2 -%, #)%# $%# r. t 2 Oa b]X (t) =Xpk 2Jk k (t)zk..,(10.41) k , k { , (10.40), fzk k 2 J g { % . 4 J (10.41) .p2 + T = Oa b], S = J f (t k) = k k (t) t 2 T , k 2 S. @ (10.40)) (10.21), { ) S, .. (fkg) = 1 k 2 S.= (10.13). 2E. 10.15. ( , EjX (t) ;Xpk6nk k (t)zk j2 C10.14, (10.41) Zb1zk = pX (t)k (t)dt(10.42)k a ( )) M.
C " 7, Oa b] Oa b] r k 2 C Oa b], k 2 J .- , , (10.41) zk , k 2 J , t 2 Oa b]. 910.16. ( - { 3" W (t), t 2 O0 1].2 = , ) O0 1]O0 1] r(s t) = minfs tg,196s t 2 O0 1]. @ k > 0 k 2 J . 10.12, k 2 C O0 1] .10.15, > 0 2 C O0 1] Z10..r(s t)(t)dt = (s) s 2 O0 1]Zs0t(t)dt + sZ1s(t)dt = (s):(10.43)(10.44)+ (10.44) s 2 O0 1] ( ), , Z1s(t)dt = 0(s) s 2 O0 1]:0 s " , 00(s) = ;(1=)(s):=) (10.45) pp(s) = A cos(s= ) + B sin(s= ) A, B { .
9, (0) = 0, 0(1) = 0, ppA = 0 (B= ) cos(1= ) = 0k = ((k + 1=2));2 k 2 Z+ = f0 1 : : : g:9 R 1 2 (s)ds = 1 0 k(10.45)(10.46)pk (t) = 2 sin((k + 1=2)s) s 2 O0 1] k 2 Z+:(10.47)3 , k k , k 2 Z+, (10.44). @ , 1pXk + 1=2)t) z W (t) = 2 sin(((10.48)kk=0 (k + 1=2)t fzk k 2 Z+g { 10.15 , (10.48) O0 1].
2E. 10.17. C, X = fX (t) t 2 T g { , L2OX ] , .. n Y1 : : : Yn 2 L2OX ] (Y1 : : : Yn) .@ 910.18. +) , L2OW (t) t 2 O0 1]]{ . 0 10.15 fzk k 2 Z+g { . + , 5.8, , (10.48) t 2 O0 1] , zk N (0 1), k 2 Z+, (E F P ), .197E. 10.19. 4 , (E F P ) zk N (0 1), k 2 Z+, (10.48) O0 1]?E. 10.20. =7, J, C10.16, KerA = 0. + X = fX (t) t 2 Oa b]g, (10.33) KerA 6= 0.E. 10.21. ( - { 3" W0(t) = W (t) ; tW (1) t 2 O0 1]: W (t), t 2 O0 1], (10.48).E. 10.22. ( O0 c] - { 3" = { 9, ) r(s t) = e;js;tj, > 0, s t 2 R.E. 10.23. + X = fX (t) t 2 Rg { r(s t) = R(s ; t). + X { , ..
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