А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 7
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.F Gn . K, n (!) 6 n+1 (!), . . Gn+1 Gn , n 2 N. 0,)1(!) = nlim (!) 2 O0 1] 1 (!) 6 F (!):(4.8)!1 nD 1 (!) = 1, F (!) = 1. M E0 = f! 2 E~ : 1 (!) < 1g. Xt(!) Xn (!)(!) 2 OGn ] ( Gn ) n 2 N. +" Xk (!)(!) 2 OGn ] k > n (Gk Gn , " OGk ] OGn]), , XtX1(!)(!) 21\OGn ] = F:n=1@ , F (!) 6 1(!). 9 (4.8), , F (!) = 1 (!) = nlim (!) ! 2 E~ :!1 n(4.9)3. @ , fF = 0g = fX0 2 F g ( F ). *, fF 6 0g 2 fX0g = F60.
+, 0 < t < 1f! : F (!) 6 tg =1\f! : n (!) < tg:n=1(4.10)D n (!) < t, n 2 N, (4.9) F (!) 6 t. D F (!) 6 t, n (!) 6 F (!),n 2 N. D F (!) = 0, n (!) = 0, " ! fn (!) < tg. + ! 2 E0 = f0 < F (!) 6 tg. C, n(!) < F (!) ! n 2 N. C ! 2 E0 n (!) ! F (!) > 0.
+" nXn(!)(!) 2 @Gn ( Gn ), . . " , ) Gn , ) Gn . C, Gn | , , , Xt(!) 2= Gn t < n (!).9, @Gn fx : (s F ) = 1=ng, , n(!) < n+1 (!) ! 2 E0 n ((Xn(!)(!) F ) = 1=n, (Xn+1 (!)(!) F ) = 1=(n + 1)," n (!) 6= n+1(!)), , n (!) < F (!) n > 1 ! 2 E0.T1= , fn (!) < tg 2 F6t (4.10). 2n=1+ | - Ft, t 2 T . B F %, &% A, % % A \ f 6 tg 2 Ft <&t 2 T . 3 , F -.
( , , % &.60A 4.5 (% %). | F6t = fW (s): 0 6 s 6 tg, t > 0. Y (t) = W (t + ) ;; W ( ), t > 0, %, - F( F6t, t > 0). ,W (t + (!) !) ; W ( (!) !) ! 2 E = f < 1gY (!) =(4.11)0 ! 2= E :1P2 n (!) = k2;n 1 Akn , A1n = f 6 2;n g,k=1Akn = f(k ; 1)2;n < (!) 6 k2;n g k > 2. =, n (!) # (!) n ! 1 ! 2 E . - , n 2 N n F6t, .
. t > 0fn 6 tg = f 6 k2;n g 2 Fk2;n Ft k = maxfl : l2;n 6 tg: ( !) W (t + n (!) !) ! W (t + (!) !) n ! 1 t > 0 ! 2 E . + n 2 N t > 0, z 2 R, ! 2 E1f! : W (t + n (!) !) 6 zg = f! : W (t + n (!) !) 6 z n (!) = k2;n g =k=1=1k=1f! : W (t + k2;n !) 6 z n(!) = k2;n g 2 F : (4.12)4 , | , f = yg = f 6 yg n f < yg 2 F6y1 S y 2 R, f < yg = 6 y ; k1 2 F6y (F6y; k1 F6y k=1111y > 0 k, y ; k > 0Q y ; k < 0, 6 y ; k = ?).( ) . 4.6.
(M A) | , (N ) | - B (N ). Fn : M ! N #A j B(N )- n = 1 2 : : : . Fn(x) ;!F (x) 2 N x 2 M (n ! 1). F A j B (N )- .2 3 , B 2 B(N )fx : F (x) 2 B g =1 1 \\k=1 m=1 n>mfx : Fn(x) 2 B (1=k)g B (") = fx 2 N : (x B ) < "g, (x B ) = inf f(x y): y 2 B g. = 1.2. 2@4.7. + 4.6 (M A) P Fn(x) ;! F (x) n ! 1 P -.. x 2 M. @ F A jB(N )-, A | A P .2 + Fn(x) ! F (x), n ! 1, x 2 M0, P (M0) = 1.
z0 2 N Fen(x) = Fn(x) x 2 M0 Fen(x) = z0 x 2 M n M0, n 2 N.61@ Fen(x) ! Fe(x) x 2 M, Fe(x) = F (x) x 2 M0 Fe(x) = z0 x 2 M n M0, Fe A j B(N )- . @ , B 2 B(N ) F ;1(B ) = fM0 \ Fe;1(B )g f(M n M0) \ F ;1(B )g 2 A M0 2 A, Fe;1(B ) 2 A ( 4.6) (M n M0) \ F ;1(B ) M n M0,P (M n M0) = 0. 2+ 4.7 (4.12) Y (t) t. D F jB(R) - , ~ = .., .. ~ F j B(R) - (F ). = , Y () .9', % Y % % F % &, % Y | .- 4.3, , A 2 F , n 2 N, 0 6 t1 < : : : < tm, B 2 B(Rm)P (A \ f 2 B g) = P (A)P ( 2 B )(4.13) = (Y (t1) : : : Y (tm)).
(4.13) B( 2.3 B " > 0 F" B , P (B n F") < "). + (4.13) E1 A 1 f2Bg = E1 A E1 f2Bg:(4.14)C , f : Rm ! R E1 A f () = E1 A Ef ():(4.15)* (4.15) (4.14). fk (x) = '(k(x B )), '(t) = 1 t 6 0,'(t) = 1 ; t t 2 O0 1] '(t) = 0 t > 1, (x B ) = inf f(x y): y 2 B g, | . 0 (( B ) B ), " 3 ,, fk (x) # 1 B (x), k ! 1.0 3 AE1 A f (n )E1 A f () = limn n = (W (t1 + n) ; W (n ) : : : W (tm + n ) ; W (n)) ! n ! 1 ! 2 E . @ , 3E1 A f (n ) =1Xk=1E1 A f (n )1 fn=k2;n g=1Xk=1E1 A\fn=k2;ngf (nk )(4.16) nk = (W (t1 + k2;n ) ; W (k2;n ) : : : W (tm + k2;n ) ; W (k2;n )). D | fFt t 2 T Rg A 2 F , A \ f < tg = 1S1=A \ 6 t ; q 2 F6t t 2 T A \f = tg = (A \f 6 tg) n (A \q=1\ f < tg) 2 F6t.
+" A \ fn = k2;n g 2 F6k2;n A 2 F . + 4.3- F6k2;n nk nk , 62(W (t1) : : : W (tm)). @ , (4.16) ) :Ef (W (t1) : : : W (tm))1Xk=1E1 A\fn=k2;ng = Ef (W (t1) : : : W (tm))E1 A : A = E, f , m 2 N 0 6 t1 < : : : < tm, Ef (Y (t1) : : : Y (tm)) = Ef (W (t1) : : : W (tm)):(4.17)@ (4.15) . * (4.17), , , Y W . 0, Y | . 2% % , ' % % & '.
+ " ( 14).+ | - F6t = fW (s):0 6 s 6 tg. C ! 2 E = f < 1g (P (E ) = 1) % &Z (t !) =(W (t !)0 6 t 6 (!)2W ( (!) !) ; W (t !) t > (!)r6(4.18)W (t !) (!)X (t !)Z (t ! )-t. 5.1 Z (t !) = W (t !) ! 2 E n E . (, %<% % -& F .A 4.8 ( %'). % fZ (t) t > 0g .2 C t > 0 Z (t !) = W (t !)1 f >tg ++(2W ( (!) !) ; W (t !))1 f<tg. +", (4.12), , Z (t ) t . - , Z ( !) ! 2 E. 9 ?? , Z (0 !) 0, , Z ( !) .". C0O0 1) = ff 2 C O0 1): f (0) = 0g, (3.20), Kn = O0 n], n 2 N.63+ Y (t !) | , (4.11). ) % & X (t !) = W (t ^ (!) !), t > 0, t ^ s = minft sg (X (t !) == W (t !) ! 2 E n E ). K , X ( !) |.".
C0O0 1).= ( b f g)h : Y ! C0O0 1), Y = O0 1) C0O0 1) C0O0 1), h(b f () g())(t) = f (t)1 0b](t) + (f (b) + g(t ; b))1 (b1)(t):', ! 2 Eh( (!) X ( !) Y ( !)) = W ( !)h( (!) X ( !) ;Y ( !)) = Z ( !):@ 4.9. D {.". (E F P ) (X B) = .., P = P B.+" & & % % E = E.@ , ': ) h 2 B(Y ) jB(C0O0 1)) ( , C0O0 1) (3.20)Q . 2.1)Q ) .". ( X Y ) ( X ;Y ) (Y B(Y )).C, PW = P(XY )h;1 PZ = P(X;Y )h;1 . ). H 2 C0O0 1) r > 0. 2.2 , A = f(b f g) 2 Y : (h(b f g) H ) 66 rg 2 B(Y ). C , t 2 M , M | O0 1), At = f(b f g) 2 Y : 6 h(b f g)(t) 6 g 2 B(Y ) ;1 < < < 1. C , t th : Y ! R , t = 01)t, .
. 4.9' %' &). - 4.10. X (t ) F jB(R)- .. t > 0.2 n (!) =1Xk;12n 1 O k2;n1 2kn ) ( (!)):@ n(!) " (!) ! 2 E (n ! 1), , W (t ^ n(!) !) ! W (t ^ (!) !) t > 0 ! 2 E (W , X , ! 2 E). 4.7 F j B(R)- W (t ^ n() ) (-& F %% ).C B 2 B(R), C = f! : W (t ^ n() ) 2 B g s 2 O(m ; 1)=2n m=2n ), n m > 1,1k=1C \ f 6 sg = f 6 s n = k 2;n 1 W t ^ k 2;n 1 2 B g =k=1k ; 1 k ;1k=n 6 < 2n Wn ^ t 2 B 2216mk6;m;11m ; 1 2n 6 6 s W 2n ^ t 2 B64(4.19) k ; 1 ( m == ?). = , W 2n ^ t 2 B 2? k k ; 1kk;12 F6 k;n F6s k 6 m, 2n 6 < 2n = < 2n n < 2n 2 F6 kn F6sm ; 1S1 1S1 212 k 6 m ; 1 2n 6 6 s 2 F6s (f < ug = 6 u ; q 2 F6u u > 0Qq=1f < 0g = ?). 2+ 4.6, F A, , X ( !) F jB(C0O0 1))- .".', ..
2 F jB(R), s t > 0 f 6 sg\f 6 tg == f 6 s ^ tg 2 F6s^t F6t.+ 2.1, , ( X ( !)) F j B(R) B(C0O0 1))- .". + 4.5 .". ( X ) Y . 0, P(XY ) = P(X ) PY = P(X ) W. =,;Y F , . . ;Y , P(X;Y ) = P(X ) P;Y = P(X ) W = P(XY ), . 2B&M (t !) = sup W (s !) t > 0:s20t]3 , M (t ) t > 0 .., ..Gtf = sup f (s) f 2 C0O0 1)s20t](4.20) C0O0 1) R.C ) .A 4.11 ( 0). (E F P ) ..
fXt t 2 T Rg , .. Xt : E ! Xt , Xt 2 FjBt, t 2 T . *F>t = fXs s 2 T \ Ot 1)g: - F 1 := \t2T F>t , .. P (A) # % # A 2 F 1 ( T \ Ot 1) = , F>t := ).2 4, , A 2 F 1 ', % A % % A. * A 2 F>t t 2 T . C " > 0 4.2 ) A" &, ) F>t, .. A" fXt : : : Xtn 2 B g, B 2 B(t1 : : : tn),t 6 t1 < : : : < tn ( T ), n 2 N, , P (A4A") < " ( 1 4.1 2.5). 0,jC (A A) ; C (A A")j 6 2P (A4A") < 2" C (A D) = P (AD) ; P (A)P (D) A D. 0 4.2 , - B(t1 : : : tn) F>t t > tn.
+" A A", .. C (A A") = 0. @ , C (A A) = P (A) ; P (A)2 = 0( "). 2 4.12. a > 0. a(!) = inf ft > 0 : W (t !) = ag { ( $% Ft = fW (s) 0 6 s 6 tg).652 * 4.2 , a { . 9', % a(!) < 1.. C a > 0 pP (a < 1) > P ( sup W (t) > a) > P (W (n) > a n ..) >t201)> P (lim sup n;1=2W (n) > a) > lim sup P (n;1=2W (n) > a) = P ( > a) > 0n!1n!1(4.21) N (0 1) ".." ( n). 4 , .. Yn n 2 N c 2 RP (lim sup Yn > c) = P (\1n=1 m>n fYm > cg) = nlimP (m>n fYm > cg)!1n!1 P (m>n fYm > cg) > P (Yn > c) n 2 N, P (lim sup Yn > c) > lim sup P (Yn > c):n!1n!1@ , c 2 R nXpfW (n)= n > c ..g f X =pn > c ..g 2 F 1k=1k - F 1 Xk = W (k) ; W (k ; 1) k 2 N:C,flimsupn!1nXk=1nXppX = n > cg flim sup X = n > cg 2 Fkn!1 k=mk>m m 2 N.
+ pP flim sup W (n)= n > cg 2 f0 1g:n!1* (4.21) , " ( P (a < 1)) 1. 2A 4.13. t x y > 0 P (W (t) < y ; x M (t) > y) = P (W (t) > y + x):(4.22)2 D y = 0, (4.22) ) P (W (t) < ;x) == P (W (t) > x). + y > 0. + 4.12 y = inf fs > 0: W (s !) = yg F6t = fW (s): 0 6 s 6 tg.
+ Z (t !) (4.18) = y . =, y (!) == inf fs > 0: Z (s !) = yg F6(Zt ) = fZ (s): 0 6 s 6 tg, " y (!) y (!) y > 0.', fy 6 tg = fM (t) > yg t y > 0. +" B 2 B(C O0 1)), t > 0P (y 6 t W () 2 B ) = P ( sup W (s) > y W () 2 B ) = P (W () 2 B~ \ B )s20t]66 B~ = G;t 1 (Oy 1)) 2 B(C O0 1)), . (4.20).
+ 4.8, , P (y 6 t Z () 2 B ) = P (Z () 2 B~ \ B ) = P (W () 2 B~ \ B ):*, .". (y W ) (y Z ) .0, x 2 R, t y > 0P (y 6 t Z (t) < y ; x) = P (y 6 t W (t) < y ; x):(4.23) W W (y (!) !) = y y > 0, ! 2 E . +" t > y (!) Z (t !) = 2W (y (!) !) ; W (t !) = 2y ; W (t !). @ , y > 0 x 2 R (4.23) P (M (t) > y W (t) < y ; x) = P (y 6 t Z (t) < y ; x) == P (y 6 t W (t) > y + x) = P (y 6 t W (t) > y + x) == P (M (t) > y W (t) > y + x):(4.24)D x > 0, P (M (t) > y W (t) > y + x) = P (W (t) > y + x) (4.24) (4.22).
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