А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 9
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4.16. C y > 0, 0 6 a < b < 1 6 b ; aPsupst2ab]js;tj6p c0(b ; a) ;y =16jw(s) ; w(t)j > y 6 y e :2* (4.68) 4.16 , c 1, = (c) > 1 k 2 Npi 6 c5 expf; Log Lognk g i = 3 4:0 1 { -, , gn 2 K 2" .. ! n > N (" !). = , " > 0 . 3%, (4.60)%.9' (4.61).E. 4.17. =7, (4.61) , x 2 K , Z 1 dx 2 dt = a2 < 10 dt P (gn 2 fxg" ..) = 1.76(4.69)', fgn 2 fxg"g = fkgn ; xk < "g f1maxjg (i=m) ; x(i=m)j+6i6m n+jx(t) ; x(s)j + 1max6i6msupst201]js;tj61=msups2(i;1)=mi=m]jgn (s) ; gn (i=m)j < "g:+ x (. (4.69))supjx(t) ; x(s)j 6 a=pmst201]js;tj61=m , 1 Bn = f1maxjg (i=m) ; gn ((i ; 1)=m) ; (x(i=m) ; x((i ; 1)=m))j < "=(8m)g:6i6m n4 , gn 2 K "0 "0 > 0 ..
! ( ! ) n > N ("0 !), "pjgn (t) ; gn (s)j 6 jt ; sj + 2"0 s t 2 O0 1]:(4.70)1 , , .. Bn n ml (l 2 N). G%, %, % Bn i > 2, &<%%,&% Bml , l 2 N. 9 (4.70), = "=(8mpq), ,P (Bn ) > Pjmax max g(j)(i=m)26i6m 16j 6q nm q>;gn(j)((i; 1)=m) ;(x(j) (i=m);x(j)((i; 1)=m))j < >YY pi=1 j =1P ( 2mLog Lognjx(j)(i=m) ; x(j)((i ; 1)=m)j <p< w(1) < 2mLog Logn(jx(j)(i=m) ; x(j)((i ; 1)=m)j + "= ): ) .E. 4.18.
C 0 6 u < v < 1 Zv 21pe;s =2ds > p1 e;u2 =2(1 ; e;(u2;v2 )=2):2 uv 2+ , Z 1 dx 2q XmX(j)(j)2lim m(x (i=m) ; x ((i ; 1)=m)) = dt dtm!10j =1 i=1(, x K ), , P (Bn ) > expf; Log Logng(4.71) 2 (a 1), = (x() m)P l.= , l P (Bml ) = 1 (4.71) %1X F { 0%: A1 A2 : : : P (An) = 1,n=1 P (An ..) = 1. 2D) ) %, % 0 { % (. O?]), - { + { 2 { J.77A 94.19.
(t) t > 0 { #) $% , (t) ! 1 t ! 1. 0 I () < 1pP (w(t) > t(t) ..) = 1 I () = 1Z 1 (t)I () =expf;(t)2=2gdt:t1', " , 4.16, p I () , (t) = (1 + ") 2tLog Logt, " > 0 " < 0, ( { $%, P (w(t) > (t) ..) = 0, { P (w(t) > (t) ..) = 1, tn !p1 w(tn)).@ C4.19 " = 0, .. (t) = 2tLog Logt,, ( ) . C4.19 : '%, % & & , ' (4.59), %fn(t) = pWn(nt(n)) t 2 O0 1] n 2 N { , C4.19.A 94.20 (O?]).
C (ffn g) = KR .., KR R = R(), # $% x , Z1jdx=dtj2ds 6 R2 0( R = 0, KR = 0 , .. -$%, #, R = , KR = (C O0 1])d). R2() = inf r > 0 : I ( r) < 1fgfZ 1 (t)1gexpf;r(t)2=2gdt:I ( r) =t1E. 4.21. O. O?]] C, (t)R();1 = liminft!1 2Log Log t( 0;1 = 1 1;1 = 0).* , (J'+3). (, O?], J'+3, . O?] . = J'+3 , , . O?].E. 4.22.
+ a = inf ft > 0 : w(t) = ag, a 2 R. C, Da = a21.E. 4.23. + U { " w (0 t). C, rx2P (U 6 x) = arcsin t x > 0:78 5. ($4, , . . -! / .%. / >. #. % . % . >/ . 4# # / . ( . > . ?2 @# { ?. # L1(* F P ). > (. ./, { %, , % & , &= & % %, < & '.+ (E F P ) fFt t 2 T Rg, .. ) - Ft F , t 2 T (Fs Ft s 6 t, s t 2 T ). :, (Xt Ft)t2T , Xt : E ! R, , s t 2 T , s 6 t ) 1) Xt 2 FtjB(R), , fXt t 2 T g fFt t 2 T g,2) EjXtj < 1,3) E(XtjFs) = Xs ..D 3) , E(XtjFs) > Xs .., (Xt Ft)t2T .
' 3) E(Xt jFs ) 6 Xs .. . =, (Xt Ft)t2T % % % %, %, (;Xt Ft)t2T { &% (" ). 4, , .', 3) , s t 2 T , s 6 t A 2 Fs ZAXs dP =ZAXtdP:(5.1)0 (5.1) "6".D (Xt Ft)t2T { () (Gt)t2T , Gt Ft, Xt 2 GtjB(R) t 2 T , (Xt Gt) { (). 2 "" : - A1 A2 F E(E(jA1)jA2) = E(E(jA2)jA1) = E(jA1) ..(5.2) , (Gt)t2T # $%#FtX = fXs s 6 t s 2 T g, t 2 T .
"4" , .. FtX Gt t 2 T 1). , , , , (Xt t 2 T ) { (). @ , . T , .. (E F Ft P ) $ . - ,79 P P . @ $ (E F Ft P ). D , , (Xt Ft P ) { , , 3) P -..= Rm, , 1),2),3) .C T = Z+ 3) () t = s + 1.* , ( ), E(^XnjFn;1 ) = 0 .., ^Xn = Xn ; Xn;1 n 2 N:(5.3)+ (n Fn)n2Z+, n { - Fn (n 2 Z+), -#, E(n jFn;1) = 0 ..,n 2 N. @ , (5.3) , (Xn Fn)n2Z+ { , (^Xn Fn)n2Z+ { - (^X0 := 0). .1.
+ fXt t 2 T Rg { ), EXt = c t 2 T (c = const). @ (Xt)t2T { . C, s 6 t,s t 2 T (Ft)t2T E(XtjFs) = E(Xt ; Xs + Xs jFs) = E(Xt ; Xs) + Xs ..(5.4)4 , Xt ; Xs { Fs (. 4.2) E(jA) = E, A E ). 0, %%. fNt ; ENt t 2 R+g, fNt t > 0g { , %' % %. C (5.4) , . , t Xt Ft Xt ; Xs Fs. , Sn = 1 + : : : + n, n 2 N, , 1 2 : : : { En = 0, n 2 N. ', Fn = fS1 : : : Sn g = f1 : : : n g,n 2 N.2. Q+ n , n 2 N { En = 1 n.
+Xn = nk=1 k , Fn = f1 : : : ng, n 2 N. @ (Xn Fn)n2N { (7).loc3. + (E F ) Q P , (Ft)t2T Q P , ..Qt := QjFt Pt := P jFt , t 2 T . C , Q P Ft t 2 T . + (E A). (, ( # ), (A) = 0 (A) = 0. 2 , ) AjB(R)- g (g > 0 -..), R , d=d, , (B ) = B gd B 2 A.
*, gt = dQt=dPt . @(gt Ft)t2T { . C s 6 t, s t 2 T , B 2 Fs ( B 2 Ft), , ZBgs dPs = Qs(B ) = Q(B ) = Qt(B ) =ZBgtdPt:loc= (5.1). ', Q P , Q P .4. M ) , % . + ("n)n2N { , ) 1 -1 ( (E F P )). 0, "n = 1 n-80, "n = ;1 . + Vn { " n- , n > 2 ( ) "1 : : : "n;1 V1 : : : Vn;1 . @ , V1 f"n gn2N. + F0 = fV1g, Fn = fV1 "1 : : : "ng,n 2 N. *, V = (Vn )n2N , ..Vn 2 Fn;1jB(R), n 2 N ( (Vn Fn;1)n2N). , V0 const, F0 = f Eg Fn = f"1 : : : "ng, n 2 N.
0 (, ) n Xn = Xn;1 + Vn "n =nXk=1Vk ^Yk n 2 N (X0 = 0)(5.5) ^Yk = Yk ; Yk;1 , Y0 = 0, Yk = "1 + : : : + "k , k 2 N. +(Xn )n2Z+ V Y (V Y ). 0, EVk < 1, k 2 N ( Vk > 0 ), (5.5)E(^Xn jFn;1) = Vn E"n .. n 2 N Vn { "n Fn;1 n 2 N. @, (Xn Fn )n2N { , E"n = 0, .. P ("n = 1) = P ("n = ;1) = 1=2 n 2 N, (Xn Fn)n2N { , P ("n = 1) > 1=2 n 2 N ( ).
0 , () .M , V1 = 1 Vn = 2n;1 1 f"1 = ;1 : : : "n;1 = ;1g n > 2:C , ( ) ). ) , : E ! f0 1 : : : 1g. +, { - (Fn)n2Z+. C,f = 0g = 2 F0 f = ng = f"1 = ;1 : : : "n;1 = ;1 "n = 1g 2 Fn n 2 N:D , P ( = n) = 2;n , n 2 N, , , P ( < 1) = 1.+ " X = 1, .. f = ng, n 2 N X = Xn = ;n;1Xk=12k;1 + 2n;1 = 1:*, P (X = 1) = 1 EX = 1, , EXn = 0 n 2 N .M, ( %, ) . ( , . = , (5.5) , ).5. + (Ft)t2T { { . + Xt = E(jFt), t 2 T . @ (5.2) , (Xt Ft)t2T {.
2 ,.I< '%, %%, &% %%81 5.1. (Xt Ft)t2T { . h : R ! R { $%, Yt = h(Xt ) t 2 T . (Yt Ft)t2T {. " $% h , , (Xt Ft)t2T { .2 9 q (. O?, . 250]) , s 6 t (s t 2 T )h(Xs ) = h(E(XtjFs)) 6 E(h(Xt )jFs):(5.6)D (Xt Ft)t2T { h , (5.6) h(Xs ) 6 h(E(Xt jFs)) .. 20 ), ' 9&.A 5.2 (9&). (E F P ) ( n) % (Xn )n2Z+ , $% (Fn )n2Z+ . ) (..) X = M + A, .. Xn = Mn + An n 2 Z+, (Mn Fn )n2Z+ { , (An Fn;1 )n2Z+ { % A0 0 F;1 = f Eg. : , (Xn Fn )n2Z+ , % A , .. ^An > 0 ..
n 2 N.2 D X , ^An = E(^Xn jFn;1) n 2 N (5.3), (A0 = 0)An =nXk=1E(^Xk jFk;1) n 2 N(5.7) .C X , ) A (5.7) A0 = 0. @ M = X ; A , 1),2), , E(^Mn jFn;1) = E(^Xn jFn;1) ; ^An = 0 .., n 2 N:(5.8)9 , ) , (5.7) (5.8). 2+ , < %, % 5.2 . + S0 = 0, Sn = "1 + : : : + "n, "1 "2 : : : { , P ("n = 1) = P ("n = ;1) = 1=2, n 2 N. M C ( 5.1) Xn = jSn j, (F0 = f Eg,Fn = f"1 : : : "ng, n 2 N), n 2 Z+.
* ^Xn = jSnj ; jSn;1j, n 2 N. @^Mn = ^Xn ; ^An = ^Xn ; E(^XnjFn;1 ) = jSn j ; E(jSn jjFn;1) n 2 N:(5.9)@ , jSnj = jSn;1 + "nj = (Sn;1 + "n)1 fSn;1 > 0g + 1 fSn;1 = 0g ; (Sn;1 + "n)1 fSn;1 < 0g:(5.10)+"82E(jSn;1 + "njjFn;1) = E((Sn;1 + "n )1 fSn;1 > 0gjFn;1 )++E(1 fSn;1 = 0gjFn;1 ) ; E((Sn;1 + "n )1 fSn;1 < 0gjFn;1 ) == Sn;1 1 fSn;1 > 0g + 1 fSn;1 = 0g ; Sn;11 fSn;1 < 0g n 2 N:(5.11)4 , E("njFn;1) = E"n = 0. * (5.9) { (5.11) , Mn =0 (5.7), An =nXk=1nXk=1(sgnSk;1 )^Sk 8 1 x > 0<sgnx = : 0 x = 0;1 x < 0:E(^Xk jFk;1) =nXk=1(E(jSk;1 + "k jjFk;1) ; jSk;1j):* (5.11) , E(^Xk jFk;1) = 1 fSk;1 =0g k 2 N, " An = Ln (0), Ln (0) = #fk 1 6 k 6 n : Sk;1 = 0g..
Ln (0) { " fSk g06k6n;1 . @ ,jSnj =nXk=1(sgnSk;1)^Sk + Ln (0)(5.12) A . * (5.12) , ELn (0) = EjSnj:(5.13)+ , r(5.14)ELn (0) 2 n n ! 1:@ ( ) ) ) , " (5.14) 6.7.@ 5.3. 5.2 , A0 0 A0 = 0 ..( ). C , , A1 = E(^X1jF0) + A0 F0jB(R) { . T , <%, % %% %% (E F P ) ,% (Ft)t2T R %' , .. - Ft P - F ( (E F )). = N = fA 2 F : P (A) = 0g. C -A A = fA Ng. + Ae = fA N A 2 A N 2 Ng.
=, Ae -, Ae A. 0 , A N Ae, " A Ae. 0, A = Ae. + F t = fFt Ng, () ,) (Ft F t t 2 T ).83 5.4. (Xt Ft)t2T { (). (Xt F t)t2T { ().2 =, (Xt F t)t2T 1) 2) (, Ft F t,t 2 T ). s 6 t, s t 2 T , E(XtjF s ) = Xs .. C,Xs 2 F sjB(R) A 2 F s B 2 Fs C 2 N , A = B C( " F BC = ). + (5.1), ZAXs dP =ZBXs dP =ZBXtdP =ZAXtdP:(5.15)4 , ) E P (C ) = 0, E1 C = 0. C (5.15) "6". 2D (Xn Fn)n2Z+ { , ) (An Fn;1)n2Z+, ) C (A0 = 0, F;1 = f Eg), ("" ) ).
+ M = (Mn Fn)n2Z+ { , ..EMn2 < 1, n 2 Z+. @ 5.1 , M 2 = (Mn2 Fn)n2Z+ {. + C Mn2 = mn + hM in , mn { , hM in {, M . * (5.7) n 2 NhM in =nXk=1E(^Mk2jFk;1) =nXk=1E(Mk2;Mk2;1jFk;1) =nXk=1E((^Mk )2jFk;1) (5.16) E(Mk Mk;1 jFk;1) = Mk;1E(Mk jFk;1) = Mk2;1 , k 2 N. 0%%% '% %%, . , M0 = 0 Mn = 1 + : : :+ n , n 2 N, 1 2 : : : { Pn2 Ek < 1 (k 2 N), (5.16) , hM in = k=1 Dk = DMn ,k 2 N. % X = (Xn )n2Z+, OX ]n =nXk=1(^Xk )2 n 2 N (OX ]0 = 0):(5.17)@ &% % % %,% () %, ,< %. +(Xn Fn)n2Z+ { , .. (Xn )n2Z+ (Fn)n2Z+. 4 " .
4 , 6 k.. k 2 N. + X (!)(!) = 0, (!) = 1 (P ( = 1) = 0 ).A 5.5 ( & &, 9&). (Xn Fn)n2Z+ { , ) . #) :841)(Xn Fn)n2Z+ { ()82)EX = (>)EX # > ..8 ,3)E(X jF ) = (>)X ^ , { , { - F = fA 2 F : A \ f 6 ng 2 Fn ng.: $ ">" ( ) # 1).2 1) ) 2). + 6 k ..
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