А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 25
Текст из файла (страница 25)
() 11.3). + 0 1 : : : { ... , ) fNt t > 0g, ) 1 1 < 2 < : : :(. 8.5). + X (t !) = k (!) k (!) 6 t < k+1(!), k = 0 1 : : :(0 = 0). D fk (!)g1k=0 ) ( 0), X (t !) = 0 t > 0. 0 E02 < 1, fX (t) t > 0g.E. 11.5. O ] * 11.4, : R ! R % .X (t) =1Xk=0k (t ; k ) t 2 R(11.42) .. (). ( r " , lims!1 r(s s + t), , "" , " s.
( ) (11.42)?+ (11.42) t > 1 k > 0, k 2 Z+ ) O0 1) () > 0, (0) = 1, ) . k , k 2 N ". + " k k , . 0 " . @ X (t) { t. =) (11.42), ., (., ., O?],O?]).E. 11.6. ( = { 9(. . 3.19).216E. 11.7. + ( 11.8) R(t), t 2 R, , t = 0.E. 11.8. + h1(t) : : : hn (t) { , P T . C, r(s t) = nj=1 hj (s)hj (t), s t 2 T , - . 9 , ) " . * " . 10.20.E.
11.9. C, Pn (z1 : : : zm) { n- ( m ) " rk (s t), k = 1 : : : m, s t 2 T { , Pn (r1(s t) : : : rm(s t)) . C fX (t) t 2 T g " .E. 11.10. + fXn n 2 Zg { a R(n), n 2 Z. C, N ! 1;1;11 NX1 NXL2 ()X;!a,(11.43)N k=0 kN k=0 R(k) ! 0:E. 11.11. + ) N XN1 X1 X R(k)(1 ; jkj=N ) 6 cN ;R(k;j)(11.44)N 2 k=1 m=1N jkj6N ;1 c > 0 N 2 N. C, ;11 NX(11.45)N k=0 Xk ! 0 .. N ! 1:+, (11.44) , R(n) = O(n; ) n ! 1.A 911.12 ("=). fX (n) n 2 Zg { % % a = 0 % $% R(n), n 2 N ( R(0) = 1). (11.45) , G(f0g) = 0 ZlimZ (d) = 0 ..,n!1 0<jj62;n Z { , $#) (11.12), G { .C " O?] k;1 kj=1 Xj 2n 6 k < 2n+1 , n = 0 1 : : : ) jj62;n Z (d) + k , limk!1 k = 0 ..
L2(E).RPE. 11.13. + fXn n 2 Ng { R(n), n 2 Z. C, ;11 NX2(11.46)N k=0 R (k) ! 0 N ! 1 , RbN (m), (11.24) , .. m 2 ZEjRbN (m) ; R(m)j2 ! 0 N ! 1:217E. 11.14. + fXn n 2 ZgP{1 ) (11.15).C , k=;1 jck j < 1 "k , k 2 Z, ... , ) E"40 < 1. + fbN () { (11.25). (lim cov(fbN () fbN ( )) 2 O; ].N !1E. 11.15.
+ Z { , ) = { 9 > 0 (. 3.19). C, Z 1 eit ; 1 i + W (t) = Z (d) t > 0(11.47);1 i( (eit ; 1)=i = t = 0) ( fW (t) t > 0g ) - C2.22).* , (11.47) , = { 9, 3.19 , = { 9 . 13 3, ) ( ) " .+ 10.6, (11.47) ( C10.7)Z 1 eit ; 1W ( t) =V (d) t > 0(11.48);1 i V K, ) 10.6, (10.28) h() = (i + )=, 2 R.@ 911.16. 0 4.1 ..
t 2 R+. 3 (), t 2 R+ ) W (t) . @ , , t (11.48), " ."W_ (t) =Z1;1eitV (d):(11.49)2 , eitQ (11.49) , V 3. @ " "" ) , . O?], " O?].M,%%, ' 11, %<% % , %' .4 C (s t) = (Cjk (s t))mjk=1, s t 2 T " % ( ) n 2 N z1 : : : zn 2 C m , t1 : : : tn 2 TnXjk=1218zjC (tj tk)zk > 0(11.50)$ ". X = fX (t) t 2 T g C n L2-, EkX (t)k2 < 1 t 2 T . ' ( ,) kzk = (z z)1=2 (z w) =mXk=1zk wk z w 2 C m :(11.51)= # $%# # %# () $%# L2- X , a(t) = EX (t) 2 C m r(s t) = E(X (s) ; a(s))(X (t) ; a(t)):(11.52)E. 11.17.
C, R(s t), s t 2 T ( ") L2-X = fX (t) t 2 T g , .+ T { , L2- X = fX (t) t 2 T g C m % . , a(t) = a 2 C m r(s t) = R(s ; t) s t 2 T:(11.53)4 C (t), t 2 T , % , r(s t) := C (s ; t), s t 2 T .4< & %%, % ( ).:, "(n), n 2 Z, C m . ( " , , ), (11.53)a = 0 R(0) = I R(n) = 0 n 6= 0(11.54) I { m- .() % ) X (n) =1Xk=;1Ak "(n ; k) n 2 Z(11.55) f"(n) n 2 Zg { , Ak , k 2 Z { (), )C m C m . M (11.55) , ) .E.
11.18. C, L2(E F P ) (11.55) 1=2PP1m21=22, k=;1 jAk j < 1, jAj := (TrAA ) =jq=1 jajq jA = (ajq )mjq=1. ( ).D T { , L2- X = fX (t) t 2 T g C m t 2 T , EkX (s) ; X (t)k2 ! 0 (s t) ! 0:(11.56)( T (11.56) t 2 T .C , 219A 911.19. 1% R(t), t 2 Rd, % $% -, X = fX (t) t 2 Rdg , R(t) =Zi(t)G(d) t 2 RdedR G { % .(11.57)B(Rd). GK 11.1.0 X , ) C11.19, Z G (9'1T) . 2 ) .J. :.A 911.20 ("=). X = fX (t) t 2 Rng { % % $%R(t), t 2 Rn ( , R(0) = 1). "ZYn(0<jj61 k=1log log(3 + jk j;1))2G(d) < 1G { , T1 ! 1 : : : Tn ! 1(T1 : : : Tn);1ZT10Z Tn0X (t1 : : : tn)dt1 dtn ! (!)..(11.58)G(f0g) = 0.9 C11.20 , O?]. =9'1T .
O?], O?].- X = fX (t) t 2 Rdg , r(s t) t ks ; tk. D ) , r(s t) = R(ks ; tk) s t 2 Rd. @ (11.57) ) , A 911.21 (. O?], . 1, . 262). 4 , $% R(u), u 2 R+,E # .. , % $% , X = fX (t) t 2 Rdg, , m Z 1 I(m;2)=2(u)(11.59)(u)(m;2)=2 Q(d) u 2 R+ I (x) { $% , Q {% B(R+), Q(R+) = G(Rm) = R(0).9 X , Rd E ) Rd.
+ (11.53) ), ..R(u) = 2(m;2)2;20(S ;1X (St1) : : : S ;1X (Stn )) =D (X (t1) : : : X (tn))220 ) S Rd t1 : : : tn 2 Rd (n 2 N). @ ,) %# ), ..(X (t1 +h);X (t0 +h) : : : X (tn +h);X (tn;1 +h)) =D (X (t1);X (t0) : : : X (tn );X (tn;1)) n 2 N t0 : : : tn h 2 Rd, (., ., O?]).)%< %-%< < '% G(B ) = (Gjk (B ))mjk=1,B 2 K (K { S) % , G(B ) B 2 K.C X = fX (t) t 2 Rdg C m 2 C m Y = fY (t) = (X (t) ) t 2 Rdg.= ().
+ ", , A 911.22. 4 , $% R(t), t 2 Rd % $% X = fX (t) t 2 Rdg C m , (11.57), R , m m $%, G { m m - $% B(Rd).=) 1 { ] 4.: ( C m ) Z K S L2- fZ (B ) B 2 Kg, , EZ (B )Z (C ) = G(B \ C )(11.60) G { - K, () . 0% % %, && % % .* 10.13 C11.22 A 911.23. X , $#) X (t) =Z 411.22, -ei(t)Z (d) t 2 RdRd(11.61) Z { B (Rd) G, #) # % $% . L2 OX ], .. L2 (E F P ) Xt , t 2 Rd, L2 ( ) = L2 (R B (R) ), = TrG, , 1) X (t) $ ei(t),2) Yk $ gk (), Yk 2 L2 OX ], gk 2 L2 ( ), k = 1 2, Yk =EY1Y =2Zgk ()Z (d)(11.62)Zg1 ()g2()G(d):(11.63)RdRd221+ , A 911.24 (0%, { I). , X = fX (t) t 2 Rdg ,..
O;u1 u1] : : : O;ud ud] . X (t) =XYd sin(uk tk ; nk ) n1 nd uk tk ; nk X u1 : : : ud n=(n1 :::nd )2Zd k=1(11.64) t = (t1 : : : td ) 2 Rd . 2, ( ) t 2 Rd ..* ) O?], O?], O?].B, ' ' % &. + (""), X = fX (t) t 2 T g Y = fY (t) t 2 S g.1, , Y = AX , ). D , A { , , ( ) X A, A " ( ). 1, % A X % ' % t 2 S < .. Y (t). 9 , ( , ). * < %. + X 0(t) () L2- X , t 2 R, , X (t) () B = L2(E F P ) (. . ??), ..EjX 0(t) ; (X (t + h) ; X (t))=hj2 ! 0 h ! 0:(11.65)@ ( ( { 3 (8.53)). D (X (t + h) ; X (t))=h h ! 0, , ) .E.
11.25. C, fNt t > 0g > 0 t 2 R+ ..*, & %%, %, " . - , k- t ( )) X (k;1)(t) " .222E. 11.26. + X = fX (t) t 2 Oa b]g { " L2- r(s t). C, X 0(t) ) t 2 (a b) e2, ) ) @ r(s t) (t t) (@s@t ) " ), @e2r(s t) := lim (r(s + h t + u) ; r(s + h t) ; r(s t + u) + r(s t))=hu:(11.66)hu!0@s@tK , .C " << , % < -% . 911.27.
J $% f (t), t 2 Oa b], #) H ( ) ) t0 2 Oa b] , ) limst!t0 (f (s) f (t)).E. 11.28. + ) X 0(t). @ ) dEX (t)=dt EX 0(t) = dEX (t)=dt:- ,(s t) EX 0(s)X 0(t) = @e2r(s t) EX 0(s)X (t) = @r@s@s@t(11.67) ) ) , ), .2r(s t)E. 11.29. C, ) (s t) @ @s@te2r(s t) ) " @ @s@t , 2 . + L - X = fX (t) t 2 Rg, ) u 2 R, ) v 2 R.
4 X ?E. 11.30. C, X = fX (t) t 2 Rg X (k)(t) ) t , Z1;12k G(d) < 1(11.68) G { X . 1 X (k) ? D , ?( (..A(X + V ) = AX + AV 2 C X V 2 DomA) % % . C L2- X = fX (t) t 2 Oa b]g Rb a X (t)dt ( )), M Oa b] X () 223L2(E) = L2(E F P ), . . ??. C , a = x(0n) < : : : < x(knn) = b si 2 Ox(i;n)1 x(in)], i = 1 : : : kn (n kn 2 N)Z b2knX(n)(n)E X (t)dt ; X (si )^xi ! 0ai=1 ^n ! 0(11.69) ^x(in) = x(in) ; x(i;n)1, i = 1 : : : kn ^n = max16i6kn ^x(in).E. 11.31.
C Oa b] x(in) i = 0 : : : kn t(in), i = 1 : : : kn ^n ! 0 n ! 1. C, L2- X = fX (t) t 2 Oa b]g, ) r, Oa b], .. (11.69), , 9 n!1limm!1kn XkmXi=1 j =1r(s(in) s(jm))^x(in)^x(jm):(11.70)=, L2- X = fX (t) t 2 Oa b]g, ) r, (11.70) , ) M,..9Z bZ baar(s t)dsdt:(11.71)', O?] X = fX (t) t 2 Oa b]g, Oa b], (11.71), . O?, . 200].
, B ( ) ) ). , X = fX (t) t 2 Oa b]g Lp- p > 1, .. EjXtjp < 1 Rbt 2 Oa b], (Lp) a X (t)dt, M Lp(E).RD (Lp) ab X (t)dt % M &, & % %% ( " ). C O?], . 2, x2.1, , , . 47 , . 348 349 E. 11.32. + fX (t) t 2 Oa b]g Lp(E) p > 1. @ (Lp)ZbaX (t)dt)(!) =ZbaX (t !)dt(11.72) 3, . 9 X (t) (11.72) ! ), . ' , C2.15 Oa b] Lp(E) .224+ E. 11.33.
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