А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 2
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4 & %%, %% %% , . . F P - 0 0. @ -% : T = E = O0 1], F = B(O0 1]), P { 3 O0 1],Xt(!) = 1 ftg(!) (. (1.15)), Yt(!) 0 t 2 T ! 2 E. =, X , Y .+ X Y (, , t 2 T ) (Xt Bt)) . , PX = PY BT . D X Y () , (Xt Bt) t 2 T { , " " .14+ X = fXt t 2 T g Y = fYt t 2 T g, (E F P ) ) (Xt Bt) t 2 T , , P (! : Xt(!) 6= Yt(!) t 2 T ) = 0.. 1 " . 2 " X Y. C, s 2 T f! : Xs (!) 6= Ys (!)g f! : Xt(!) 6= Yt(!) t 2 T g(- F ). .E. 1.1. D (Xn Bn) n 2 N { , %,%, %% ( ) Qn { Bn (n 2 N), (E F P ) .".
Xn 2 FjBn, L(Xn ) = Qn n 2 N.2 ) ) .A 91.2 (* @, . O?, . 266]). (X1 B1) Q1 n 2 N xk 2 Xk , k = 1 : : : n (Xn+1 Bn+1 ) Qn (x1 : : : xn Q ), Qn ( : : : Q B ) 2 Fn jB (R) # B 2 Bn+1 , Fn = nk=1 Bk (. -, ) B1 : : : Bn). Bk 2 Bk ,Wk = 1 : : : n (n > 2)Pn(B1 : : : Bn ) =ZB1Q1(dx1)ZB2Q2(x1Q dx2) : : :ZBnQn(x1 : : : xn;1Q dxn ) (1.20)( (1.20) 1). (E F P ) )# .. Xn , L(X1 : : : Xn ) = Pn n 2 N.E. 1.3.
+ - A 2 S, ) E, , 2A - ( 1.5).E. 1.4. + G g : Rd ! R, . +, G 2 BT (Xt = R t 2 T = Rd). (PX(G), X = fXt = (1 + f (t)) t 2 Rdg, f : Rd ! R )( ) , { , O-1,2].E. 1.5. C, (O0 1] B(O0 1]) P ), P { 3, fXt t 2 Rg , . . , P (Xt = 0) = P (Xt = 1) = 1=2 t( 1.9).E. 1.6.
C, (...) .. X (t !) t 2 R ! 2 E, , 1 X ( !) R.15E. 1.7. + C (R) { , R. ' " ", ) , . . DJ = fx 2 C (R) : x(t) 2 Bt t 2 J g, Bt 2 B(R) t 2 J J 2 F (R), YQJ (DJ ) :=t2JQ0(Bt) Q0 { B(R). C, QJ J 2 F (R) C (R) , &<% % %%,< ( 1.9).E. 1.8. - (1.17) (1.18)?E. 1.9. + k = 1 (1.16), .
. fj g { ... , d = 1. +Xn(t !) = p1n Sn (O0 t] !) t 2 O0 1]:+ " , ) { 3 RQ ) {"#) ", ..X(B ) = 1 B (j ) B R:j 2Z= , ) ) ) . 1.9 ( C O0 1] 0 DO0 1]) 2,3 O?].E. 1.10. + k = 1 (1.14). +Fn(x !) = Pn ((;1 x] !) x 2 R:+ Fn(x !), $% . , j O0 1], j { , .
. P (j = 1) = p, P (j = 0) = 1 ; p,0 < p < 1?- M B(Rk) - (.-.) Q, B(Rk), " > 0 S1(") : : : SN(") 2 B(Rk), N = N ("), , B 2 M Si(") Sj(") Si(") B Sj(") Q(Sj(") n Si(")) < ".A 91.11 (. O?, . 421]). M { , .-. Q. Pn (B ! ) { %, $ (1.14), P1 = Q. sup jPn (B !) ; Q(B )j ! 0 n ! 1eB 2M(1.21)e ! 2 E, E n E E0 P (E0 ) = 0. 2 , (E F P ) , (1.21) #1.E. 1.12. C, M = f(;1 x] x 2 Rkg, .
. (;1 x1] (;1 xk ], (x1 : : : xk ) 2 Rk, .-. - () Q. @ % % " { 0% .. n ! 1 " Fn(x !) := Pn ((;1 x] !) F1 (x).16E. 1.13. C, Q = P 3 Rk M { , (1.21) 1 ( , ) . 1.12, k > 2).E. 1.14. +, , ) . 1.13, , Q 3 ( k > 2).* " , " , , ) , ., ., O?], O?].+ A B(O0 1]d), d > 1. C F : A ! R kF kA = sup jF (A)j.A2A+ F(A) = jA()j, A 2 A, > 0 A() = fx 2 Rd : (x A) < g, (x A) = inf y2A (x y), { j j { 3 Rd.+, A B(O0 1]d) , kF kA ! 0 ! 0 + :(1.22)E. 1.15. C, (1.22) , A { Oa1 b1] Oad bd] O0 1]d.+ ) .A 91.16 ( &,= , . O?]). fXj j 2 Nd g { , ) ...
c. A B (O0 1]d) # (1.22). kn;dSn () ; j jckA ! 0 .. n ! 1P Sn (A) = j 2nA Xj , nA = fx = ny y 2 Ag, n 2 N, . . n;d(1.16), { #) .= , , . O?], O?]. % (1.17) . + , 1 {, . . k, k 2 Z, > 0 { , " " ( 1 { , k = 0 1 2 : : : ). .= $%# u(t) t > 0, (0 t], .
. u(t) = EXt .E. 1.17. C, u(t) < 1 t > 0.* ) % % %.A 91.18 (F-, ., ., O?, . 46]). 4 % - % u(t + s) ; u(t) ! s=c$ s > 0, c = E1t ! 1 #1=1 = 0). 4 % . 1 .= . O?], O?].( $ s,17E. 1.19. =7, Z (B ), (1.19), B 2 B(X ) (Z+ A), Z+ = f0 1 : : : g = f0g N, A { - Z+ ( , fZ (B ) B 2 B(X )g { , B(X )).%, { - (X B), .. X = 1m=1 Xm, Xm 2 B 0 < (Xm) < 1, m 2 N.
" %< (1.19) %, < &. ( (E F P ) Y (1) X1(1) X2(1) : : : Y (m) X1(m) X2(m) : : :, Y (m) + m = (Xm) L(Xj(m)) = ()=(Xm ) Bm = B \ Xm m j 2 N:+Zm (D !) =Z (B !) =XY (m) (!)j =11Xm=11 D (Xj(m)(!)) D 2 Bm ! 2 E:Zm (B \ Xm !) B 2 B ! 2 E:(1.23)=, Z (B !) . D - * @ (. . 1.1), , 1.15 ' %%, , %% (X B).E. 1.20.
C, (B ) = 1 B 2 B, Z (B ) = 1.., (B ) < 1, Z (B ) < 1 ..A 91.21 (. O?, . 24]). , .. % (1.23), :1. 4 # n 2 N # #) Z (B1 ) : : : Z (Bn ) .2. 4 B (B ).B1 : : : Bn 2 B -2 B Z (B ) -Z ( !) # % - B.9, < % = 1 , , < &% %%,< . Z (B ) , " ", B 2 B. " ) .E. 1.22. O. O?, .
370]] + B 2 B , 0 < (B ) < 1, {- (X B). + B = nq=1Bq , Bq 2PB, Bj \ Bq = q 6= j(q j = 1 : : : n). @ k1 : : : kn 2 Z+ k = nq=1 kq 3. 18P (Z (B1) = k1 : : : Z (Bn) = knjZ (B ) = k) =k! (B1) k1 (Bn ) kn (B ) : (1.24)@ , B = B(Rd), d > 1, % (1.24) , " %", = '% B ( 7) %' ' %%, %, <% k , B .= k ! k ! (B )1n0 , , , ) (., ., O?, ?] ).@ , .E.
1.23. + X = fXt t 2 T g Y = fYt t 2 T g, (E F P ), (X B) t 2 T , , PX = PY BT , X Y ".E. 1.24. + () X = fXt t 2 T g D L 2= BT , f! : X ( !) 2 Dg 2 F f! : X ( !) 2 Lg 2= F .E. 1.25. + " X = fXt t 2 T g Y = fYt t 2 T g, D 2= BT , A = f! : X ( !) 2 Dg 2 F B = f! : Y ( !) 2 Dg 2 F , P (A) 6= P (B ).9 1.24 1.25 , ' %, %-% '%, <-&, -% ', % %%, %'% & %, & fX (tj ) tj 2 T j = 1 : : : ng n 2 N.E. 1.26.
+ " , - .+ & = - %% %, % ( ') &% , % &% "=" %.* , , -, " .= ) . = %, %% ' t 2 T <% Xt t 2 T , ', "%" (E F P ) ( )!). @ , ' %%, X = fXt t 2 T g %% %% (E^ F P ). T , ( ) - ) (E^ F P ) = (E F P ) (E0 F 0 P 0). " X E E0 Y (!e) = X (!) !e = (! !0) 2 E E0:(1.25)@, , PY = PX BT .19M O?] . 120 { 130. 2 , ) " O?].+, %% &% . + (Xt Bt)t2T (Yt Et)t2T { ht : Xt ! Yt, ht 2 BtjEt t 2 T . 0) 2.E. 1.27. C V T (V T ) hV : XV ! YV ,hV (x) = y x = (xt t 2 V ) y = (ht(xt) t 2 V )(1.26)+, hV 2 BV jEV V T (EV { - QYV = t2V Yt).E.
1.28. + QJ { (XJ BJ ) J 2 F (T ).C, J = QJ h;J 1 { (YJ EJ ) J 2 F (T ), hJ (1.26).E. 1.29. + (Xt Bt)t2T (Yt Et)t2T { $ ( (Xt Bt) (Yt Et) t 2 T ), .. t 2 T ht : Xt ! Yt, , ht 2 BtjEt h;t 1 2 Et jBt.+, (YT ET ) ) J = QJ h;J 1 J 2 F (T ).@ Q = hT QJ J 2 F (T ) ( hT = (h;T 1 );1).E. 1.30. + (Yt Et) (Zt At) t 2 T , .. Yt Zt Et At t 2 T( EV AV V T ). + (YJ EJ ) J , J 2 F (T ). C, PJ (A) = J (A \ YJ ) A 2 AJ(1.27) (ZJ AJ ) J 2 F (T ). + ", ,PJ = J EJ J 2 F (T ).E. 1.31.
+ (ZT AT ) ) P , ) PJ ,;1;1J 2 F (T ), ) (1.27). C, PhT (.. P (hT ) ) (XT BT ) QJ (XJ BJ ), J 2 F (T ).', % ,.. Ys = Xg;1(s) s 2 S , g : T ! S { .+ , , , , fXt t 2 U g fXt t 2 V g U V T U V . (T ) (T ) ( , (x y ) = 0, x = y ).(, Xt = R EXt2 < 1 t 2 T , " " ()(s t) = (E(Xt ; Xs )2)1=2:(1.28) " , M.3. C, =.D. 3 C. M", 20. B , %% %, %& %%, - , %, % ' & .
+ , + 1, . @ ) O?], O?].= , ) . + %,.. (., ., O?], O?]). (. O?], O?] O?], O?]).21 2. . . - # . -# .. ! ! . CT . /# %. % . 0 Qt1 :::tn t1 : : : tn 2 T . 1 (Rn B(Rn)). 0 . % 2 2. .3 ! ( ).C - .
2.1. Xt, t 2 T , | t - Bt = B (Xt). XJ ,J 2 F (T ), J (y() z()) = max (y(t) z(t))t2J t . BJ .(2.1)= B(XJ ), . . % - -2 =, J . 0 " yn() 2 XJ , n ! 1 yn(t) (Xt t ) t 2 J . 3 , (XJ J ) | .%+& Jt : XJ ! Xt, Jt y = y(t), ;1Gt 2 B (XJ ) (XJ J ) (Xt t ) t 2 J . +" Jt Gt 2 B(Xt). + 1.2 Jt;1Bt 2 B(XJ ) Bt 2 B(Xt). * (1.2) , BJ B(XJ ). +, , Xt, t 2 T .+ G | (XJ J ). (XJ J ) G 7 - ( . O?, .
104]). ( B" (y) = fz 2 XJ : J (z y) < "g %& XJ . 0, G 7 BJ . +" B(XJ ) BJ . 2@ 2.2. 4 , &, % %% & -& % -&, '%% ( %, %) =. - . 2.3 (% % ). (X ) | Q | B (X ). # " > 0 B 2 B (X ))# F" G" , F" B G" Q(G" n F") < ".2 D F , F" = F .
= G = fz 2 X :0 (z F ) < g, (z F ) = inf f(z y): y 2 F g. 4 G , G G < 0 22T G = F . 0, Q(G ) # Q(F ) ! 0 >0. +" = (") , G" = G(") Q(G" n F" < ".C , Y , ) 2.3, -. @ , B(X ) Y , Y , B(X ) -, ) . + Bn 2 Y , n 2 N. C " > 0 Fn" Gn" , Fn" Bn Gn" Q(Gn" n Fn") < "2;n;1 .S F , n , Q S1 F n F < "=2 (F F" =n"0n" ""n6n0n=1 7 ).
S1S1 G" = Gn" . @ F" Bn G" Q(G" nF" ) < ". =, Y Xn=1 n=1 . 2 2.4 (E). (X ) | Q | B(X ). B 2 B(X ) # " > 0 ) K" B , Q(B n K" ) < ".S12 C n 2 N X = B1=n(ynm ), B"(y) = fx 2 X :m=1(x y) < "g, ynm , m = 1 2 : : : , 1=n- (. . y 2 X ynm, (y ynm) < 1=n). C Sjn " > 0 jn , QB1=n(ynm) > 1 ; "2;n;1 . +m=1T1 Sjn B (y ). @ OR ] R R" =""1=n nmn=1 m=1 Xjn11Q(X n OR"]) 6 Q(X n R") = Qn=1 m=1B1=n(ynm)6n=1"2;n;1 = "=2:4 OR"] | , . . (. . " > 0 ) "-), , 2=n- ynm, m = 1 : : : jn .
+ 2.3 F" B , Q(B n F") < "=2. @ K" = F" \ OR"] Q(B n K" ) < ". 2', X = Rq OR" ] 2.4 , .@ ' % 1.7 ( " ).2 D Q QJ ), ;1 BJ 2 CT , BJ 2 BJ (J 2 F (T )), C = TJ(. (1.5));1 BJ ) = Q ;1 (BJ ) = QJ (BJ ):Q(C ) = Q(TJTJ+" %% , %, Q & CT , -';1 B J 2 F (T ) B 2 BJ :Q(C ) := QJ (B ) = TJ(2.2);1 Bi , Ji 2 F (T ), Bi 2 BJ ,2 %. C, C = TJiii = 1 2. +, QJ1 (B1) = QJ2 (B2).
J = J1 J2 (2 F (T )). +23;1 Bi = ;1 ( ;1 Bi ), i = 1 2, . . h NC = TJTJ JJii M M1 M2 M h;1 (M1) = h;1(M2) () M1 = M2,;1 B1 = ;1 B2 . , JJJJ21 F (T ) ( (1.9) V U T , U 2 F (T )), ;1 (B1 ) = QJ ( ;1 B1 ) = QJ ( ;1 B2 ) = QJ ;1 (B2 ) = QJ (B2 ):QJ1 (B1) = QJ JJ2JJ1JJ2JJ21=, Q | CT , Q(XT ) = QJ (XJ ) = 1 J 2 F (T ). < %%, Q CT . Ci 2 CT ,i = 1 2, C1 \ C2 = ?. M , , ;1 Bi , J 2 F (T ), Bi 2 BJ , i = 1 2, B1 \ B2 = ?. @Ci = TJ;1 (B1 B2 )) = QJ (B1 B2 ) = QJ (B1 ) + QJ (B2 ) = Q(C1) + Q(C2):Q(C1 C2) = Q(TJD Q CT , -Q fCT g, .
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