А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 11
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C, fXt t 2 T g { ), .. E(Xt ; Xs)(Xu ; Xv ) = 0 v < u 6 s < t, v u s t 2 T .E. 5.8. + fn gn2N { PnitS . + Sn = k=1 k , Xn = e n =EeitSn , n 2 N, t 2 R. C, (Xn Fn)n2N { () , Fn = f1 : : : n g,Pn 2 N.= (. O?, . 547]), ) 1k=1 k":1) ..,2) ,3) ( ) ). ', 0 1 - .. 0.92E. 5.9. + fYt t 2 T Rg { () ) , EjYtj2 < 1, t 2 T . C, ) h : T ! R , Xt = jYtj2 ; h(t) { Ft = fYs s 6 t s 2 T g, t 2 T , , EjYt ; Ys j2 = h(t) ; h(s) s 6 t s t 2 T:(5.32) 95.10.
fYt t 2 T Rg { % ). EevYt < 1 Eev(Yt;Ys) < 1 v 2 R s 6 t (s t 2 T ):(5.33) Zt = fv (t)evYt , t 2 T , fv : T ! R. % fZt t 2 T g fv (t) 0, t 2 T , , s 6 t (s t 2 T )E(evYt =evYs ) = EevYt =EevYs (5.34)fv (t) = c=EevYt t 2 T(5.35) c { .2 D Zt t 2 T { , EZt = EZt0 t t0 2 T . +"fv (t)EevYt = fv (t0)EevYt0 t 2 T:D fv (t0) = 0, fv (t) 0, t 2 T . + fv (t0) 6= 0, fv (t) 6= 0 t 2 T ,, , (5.35). ', Ft = fZs s 6 t s 2 T g fYs s 6 t s 2 T g t 2 T:9 ) fYt t 2 T g (5.33), s 6 t (s t 2 T )E(Zt jFs) = ffv((st)) E(ev(Yt;Ys)evYs fv (s)jFs) = Zs ffv((st)) Eev(Yt;Ys ):vv@ , (Zt Ft)t2T { , Eev(Yt ;Ys) = ffv ((st)) s 6 t (s t 2 T ):(5.36)v0 (5.35) (5.34).B&%. 9 (5.34) (5.35), , (5.36). 2E. 5.11. (5.34), ) fYt t 2 R+g { , ) fYt t 2 R+g { ?E.
5.12. O. O?, . 166]] + fWt t > 0g { m- f (x) { Rm, .. f (x) x 2 Rm " x. C, f (Wt) { .934 % 5.11 %% %E. 5.13. O -] C (Xn Fn)n2Z supn EjXn j < 1 , ) Yn Zn ( (Fn)n2Z ) , Xn = Yn ; Zn, n 2 Z+.E. 5.14. O M] + (Xn Fn)n2Z { . @ ) Xn = Mn +Rn, (Mn Fn)n2Z{ (Rn Fn)n2Z { %, .. , Rn ! 0 ..
n ! 1.B&= % % % , & % % (&%).E. 5.15. + (Xn Fn)16n6N { h : R ! R+ { ) . @ u 2 R t > 0P (1maxX > u) 6 Eh(tXN )=h(tu)(5.37)6n6N n+++++( , Eh(tXN ) < 1).E. 5.16. + (Xn Fn)16n6N { . @e (1 + EjX j log+ jX j)E 1maxjXj6(5.38)nNN6n6Ne;1 log+ x = log x x > 1 log+ x = 0 x < 1.C (5.38) , ) .E. 5.17. OC] + 1 : : : N { Pn Ek = 0, k = 1 : : : N .
+ Xn = k=1 k , k = 1 : : : N . @E 1maxjX j 6 8EjXN j6n6N n(5.39)+ 1, C" :, )) ] 4 { ', . 2 ) O?].A 95.18 (F,). X = (Xn Fn)n2Z+ { X0 = 0. p > 1 Ap Bp , ) X , ,ppApk OX ]nkp 6 kXnkp 6 Bpk OX ]nkp n 2 Z+(5.40)P OX ]n = nk=1 (^Xk )2 { % X (OX ]0 = 0). : (5.40) Ap = O18p3=2 =(p ; 1)];1 , Bp = 18p3=2 (p ; 1)1=2. > , 5.9, Xn = max jXk j p > 106k6nppApk OX ]nkp 6 kXnkp 6 Bpk OX ]nkp Ap = Ap, Bp = (p=(p ; 1))Bp .94(5.41)E. 5.19. + (. O?, . 532]), ), p = 1 (5.40) .0 9-, (5.41) p = 1 A1, B1, ) X .9 % & & , %, A 95.20 (F, { 9- { "). Y : O0 1] ! O0 1]{ #) $%, O0 1), , Y(0) = 0,Y(;1) = Y(1) Y(2t) 6 cY(t) t > 0 c > 0.
0 < A < B < 1, ) c, , # X = (Xn Fn )n2ZAEY(S1 ) 6 EY(X ) 6 B EY(S1)P (^X )2)1=2, ^X = X ; X , (X = 0), k 2 N. X = supn jXn j, S1 = ( 1kkkk ;10k=1E. 5.21. + fWt t > 0g { m- . C, (kWtk Ft)t>0 { ( .. ), k k { Rm. * 5.15, h(x) = etx (p t > 0), , s > 0 x > ms+ sd ;m=2P ( sup kWtk > x) 6 ex2e;x2=(2s):(5.42)t20s]E. 5.22. O. O?, . ]] + fWt t > 0g { m- , m > 3.
C, jWtj ! 1 .. t ! 1. C, 1 Wt 0 . =7, m = 1 .( % %, ,%%, % % . + (E F P ) { N { , ) .C - A F A = fA Ng. + (Ft)t2R+ { ) - (E F P ). - Ft+ = \s>t Fs, t 2 R+. 4 5.5, % &%, & %%, -&Ft ( , N F0). @, ,N Ft+ t 2 R+. C (.. f < tg 2 Ft t 2 R+) -F + = fA 2 F : A \ f 6 tg 2 Ft+ t 2 R+g:(5.43)', F + .E.
5.23. + n, n 2 N { = inf n n . C, { , F + = \n Fn+ . D n { , < n f < 1g, F + = \n Fn . 95.24. (Xt Ft)t2R+ { , #) .. . { % ( $% (Ft)t2R+ ) , 6 6 c .., c { . E(X jF+ ) = X ..,(5.44) - F + (5.43) ( f = 1g f = 1g,#) # , X = 0 X = 0).952 Tn = 2;n Z+ = fk2;n k 2 Z+g, n 2 N. =, (Xu Fu)u2Tn{ . = (n) = 2;n O2n +1], (n) = 2;n O2n +1], n 2 N, O] { . =, (n) (n) 2 Tn (n) # , (n) # n ! 1 ! 2 E. - , (n) 6 (n) .., n 2 N. ', (n) (n) { (Fu)u2Tn , k 2 Z+ f (n) 6 k2;n g = f < k2;n g 2 Fk2;n( (n), n 2 N).
+", 5.5 (), " Z+, Tn)E(X (n)jF(n)) = X(n) .., n 2 N:(5.45) m 2 Z+ E(Xc(m)jF(n)) = X(n) .. n > m c(m) = 2;m O2mc + 1]. M 5.14, , E(X jA) - A F , 2 S (X { ). 9 Xt t 2 R+ , X(n) ! X .. L1(E F P ) n ! 1. 0, X { . + E(Xc(m) jF (n)) = X (n) .. n > m, , X (n) ! X .. L1(E F P ) n ! 1. 5.23 F+ = \n F(n), " X 2 F+ jB(R). 0, (5.44) , EX 1 A = EX 1 A A 2 F+ .+ F+ F(n) n 2 N, (5.45) , EX (n)1 A = EX(n) 1 A n 2 N. = n " L1 X (n) X X(n) X . 2@ 95.25. C ( C5.24) ) (5.44) EX = EX .
+ ", C5.24, 5.23, , (5.44) F+ = \n F(n).B&% % 0 { & (1.18). *,Yt = y0 + ct ; St t > 0 St =XtXj =1j P(5.46) fXt t > 0g (1.17) ( 0j=1 j = 0). %, Xt = Nt, t 2 R+, fNt t > 0g { %% . @ 8.5 ( ) , fNt t > 0g (1.17), fj gj2N { , . @ Nt t > 0 .. , , " fYt t 2 R+g. (, ...
fj gj2N ) fj gj2N, fj gj2N fNt t > 0g ( - (E F P )). 95.26. : (5.46) % fYt t 2 R+g ).962 ', fZt t 2 R+g { ) h(t), t 2 R+, { , fZt+h(t) t 2 R+g{ ). +" ) fSt t > 0g. 2.11 " ( ), (2.17). +(v) = Eeiv1 , v 2 R. 9 ) fj gj2N, 0 6 s < t v 2 REeiv(St;Ss ) ==1Xkm=0E expfiv=Xk+mj =k+11Xk=01Xkm=0Eeiv(St;Ss )1 fNs = kg1 fNt ; Ns = mg =j gP (Ns = k)P (Nt ; Ns = m) =P (Ns = k)1X; s))m e;(t;s) = e(t;s)((v);1): 2 (5.47)((v))m ((t m!m=0' , (v) = Eev1 < 1 v > 0:(5.48)2 j , j ( 1 > 0 ..,, (5.48) v 6 0).
, (5.47), , 0 6 s < t v 2 REe;v(Yt;Ys ) = e(t;s)g(v) g(v) = ((v) ; 1) ; vc:(5.49)* (5.49), , Y0 = y0, , Ee;vYt = etg(v);vy0 , t 2 R+.0, (5.33) (5.34) , 5.10 v 2 R Zt = e;vYt;tg(v) Ft = fZs 0 6 s 6 tg fYs 0 6 s 6 tg t 2 R+:4 % = inf ft > 0 : Yt < 0g inf ft > 0 : Yt 2 (;1 0)g:(5.50)+ fYt t > 0g .. , (;1 0) { , 4.4 ( ), , .. f < tg 2 Ft t 2 R+. C5.25 ( = 0 6 ^ t 6 t, , 0 t ^ { ) t v 2 R+e;vy = EZ0 = EZt^ > E expf;vYt^ ; (t ^ )g(v)g1 f 6tg >0> E expf;vY ; g(v)g1 f 6tg > Ee;g(v)1 f 6tg > 06infs6t e;sg(v)P ( < t):+ (5.51) , Y 6 0 ..
*, v > 0 t > 0P ( 6 t) 6 e;vy0 sup esg(v):06s6t(5.51)(5.52)97%& %,, %&c > a a = E1 > 0:(5.53)@ g0(v) = 0(v) ; c g0(0) = a ; c < 0. - , g00(v) = 00(v) > E12 v > 0, 00(v) = E12ev1 ( E1 > 0 , E12 > 0). 0, ) v0 > 0, , g(v0) = 0. v = v0, (5.52) P ( 6 t) 6 e;v0 y0 t 2 R+. = , P ( < 1) 6 e;v0 y0 :(5.54)3%, A 95.27. > { , (1.18) % (1.17) > 0, j , j 2 N, -# # (5.48) (5.53).
% )# $ (5.54), y0 { , v0 > 0 () g (v ) = 0, v 2 R+ ($% g (5.49)).', & %% % %, ' ,%%. + (Gt)t2T R { #) - (E F P ), .. Gt Gs F s < t, s t 2 T . + (Xt )t2T{ , (Gt)t2T . @ (Xt Gt)t2T ( , ), (Xt Gt)t2U U = ;T = f;t t 2 T g (, ). , (Xn Gn)n2Z+ { , E(Xn jGn+1) = Xn+1 , n 2 Z+. ' "=" ">", ( "6" { ). + Rm (m > 1).E. 5.28. + 1 : : : N { ... Rm (m > 1) Ek1k < 1, k kP{ . + Xn = (1=n) nk=1 k , GnN = fXn : : : XN g. C, (Xn GnN )16n6N { .E. 5.29.
+ fn n 2 Ng { (.. N 2 N (i1 : : : iN ) f1 : : : N g (i1 : : : iN ) =D (1 : : : N )). + m 2 N g : Rm ! R. M U - Unm = (Cnm);1X16i1 <:::<im 6ng(i1 : : : im ) n > m:+ Gnm = fUkm k > ng, n > m. C, (Unm Gnm)n>m { . ' Cnm = n!=(m!(n ; m)!).A 95.30. (Xn Gn)n2Z+ { . .. ) X1 = limn!1 Xn 2 O;1 1). ", , )lim EXn = c > ;1(5.55)n!1 X1 2 (;1 1) Xn ! X1 L1(E F P ) n ! 1. "Xn > 0 .. n 2 Z+ EX0p < 1 p 2 (1 1), Xn ! X1 Lp(E F P ) n ! 1.982 C (a b) N 2 N N (a b) (. .
??) (XN GN ) : : : , (X0 G0). @EN (a b) 6 E(X0 ; a)+=(b ; a) 5.10. 0,1(a b) := Nlim (a b) < 1 ..!1 N( ) ). @ , 5.11 , .. ) X1 = limn!1 Xn . @ , (Xn+ Gn)n2Z+ . +",E(nlimX + ) 6 liminf EXn+ 6 EX0+ < 1:!1 nn!1= P (X1 = 1) = 0. + .%, (5.55). + EXn > EXn+1 , EXn+ > EXn++1 n 2 Z+,EjXn j = 2EXn+ ; EXn 6 2EXn+ ; c 6 2EX0+ ; c n 2 Z+:*, supn EjXnj < 1. 9' < %%, fXn n 2 Z+g.C " > 0 m = m(") 2 N , EXn < c + " n > m.@ > 0 n > m ( (5.1) ),EjXn j1 fjXn j > g = EXn 1 fXn > g ; EXn 1 fXn 6 ;g == EXn 1 fXn > g + EXn 1 fXn > ;g ; EXn 66 EXm 1 fXn > g + EXm 1 fXn > ;g ; c == EjXm j1 fjXn j > g + EXm ; c 6 EjXmj1 fjXnj > g + " < 2" ( n > m), supn P (jXn j > ) 6 ;1EjXn j ! 0 ! 1.
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