А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 14
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K (z), z.4 , , ) (6.29) (., , O?, . 96]), " ! 0+, 6.14.0. C6.14 . 2 .E. 6.1. C, Qn ) Q n ! 1 ( B(X ), (X ){ ) , (6.1) .% , .. f : X ! R, kf k1 := sup jf (x)j < 1 L(f ) := supfjf (x) ; f (y)j=(x y)g < 1:(6.30)x2R114x6=yE. 6.2. + f : X ! Y , X , Y { (f ). +, Df = fx : f xg 2 B(X ). C, Qn ) Q , (6.1) f 2 B(X )jB(R) , Q(Df ) = 0.E. 6.3.
+ X = Rm . 1 , Qn ) Q, (6.1) f 2 C01(Rm), .. ?B & % (6.1) %% <& ( ) . " E. 6.4. + Q Qn n 2 N { (X B(X )). C, 6.2 , Qn(X ) ! Q(X ) n ! 1. (X A) N A #) , N A. +X { . - M B(X ) #) () , Q Q1 Q2 : : :, Qn(B ) ! Q(B ) B 2 M Q(@B ) = 0 Qn ) Q.E.
6.5. +, , ) , , ) . =7, .E. 6.6. ( 6.7). C, 1R ( 1X((x1 x2 : : : ) (y1 y2 : : : )) = 2;k 1 +jxjkx; ;ykyj j kkk=1 - ) Qn ) Q n ! 1 , . *, R1 % , < < %,.E.p 6.7. ( (5.14)). + pDD j j Sn= n ;! N (0 1) n ! 1 , ,jSj=n;!n6.3. - ,fjSnj=pngn2N . +pp" EjSn j= n ! Ejj = 2=, n ! 1.C Rm F ' (. . ??). D A 96.8 (., ., O?, . 344]).
(. n ;!Rm # #) :1. Fn (x) ! F (x) n ! 1 x 2 Rm, $% F .2. 'n () ! ' () n ! 1 2 Rm.> , n Rm 'n () ! '() 2 Rm, $% ' 0 2 Rm, '() = ' () D n ;! n ! 1.115DE. 6.9. O - { 9] C, n ;! (Rm B(Rm)) D , (a n) ;! (a ) n ! 1 a 2 Rm ( ) { Rm.2 , ' %, % % Rm < & % .: ) .A 96.10 (9&). % X = fXt t > 0g t Rm ). " % (..), % Y = fYt = Xt ; X0 t > 0g . 0 6 s 6 t < 1E(Xt ; Xs) = Mt ; Ms D(Xt ; Xs ) = Gt ; Gs (6.31) $% M : R+ ! Rm, #) $%G : R+ ! Rm2 (R+ = O0 1)).J " X0 % ' , % fXt t > 0g &% (7). D X C6.10 Oa b], fYt = Xt ; Xa t 2 Oa b]g, Xet = Xt+a t 2 O0 b ; a] Xet = Xb t > b.
@ C6.10 , ) , 2.+, %% ( 6.11) % supp t201] W (t). + Sn () Xni = Xi= n, i = 1 : : : n, X1 X2 : : : , P (X1 = ;1) = P (X1 = 1) = 1=2.E. 6.11. ( 4.14 C, jP (0maxS > j ) = 2P (Sn > j ) + P (Sn = j )(6.32)6k6n k S0 = 0, Sk = X1 + : : : + Xk , k > 1 fXn g .@ , z > 0Sk > z) = P ( max S > j ) = 2P (S > j ) + P (S = j )pP ( sup Sn(t) > z) = P (0maxnnnnn6k6n n06k6n kt201]pp jn { , z n ( jn = ;O;z n], O] { ). + z+@, , z > 0ppP (Sn > jn ) = P (Sn= n > jn = n) ! P ( > z) N (0 1).
@ , . 2 ) " .DE. 6.12. + n ;! , n , n 2 N , F (x) R. @sup jFn (x) ; F(x)j ! 0 n ! 1:x2R116C (6.32) (4.25) E. 6.13. C, 6.11 Xi , i 2 N, Sk = X1 + : : : + Xk , k > 1, maxP (Sn = j ) ! 0 n ! 1:j+, - X1 X2 : : : ..., P (X1 = ;1) = P (X1 = 1) = 1=2 S0 = 0,Sk = X1 + : : : + Xk , k > 1. = mn = 0minS Mn = 0maxS 6k6n k6k6n k W (t) t > 0, m = t2infW (t), M = sup W (t). 01]t201] h : C O0 1] ! R3,h(x()) = (t2infx(t) sup x(t) x(1)):01]t201] 6.3 6.11, pppDh(Sn ()) = (mn= n Mn = n Sn= n) ;!(m M W (1))(6.33)p Sn (t) 0 6 t 6 1 { (k=n Sk = n), k = 0 : : : n. ( , M , (., ., O?, . 18-21]) ) (6.33) ) .A 96.14 (). 4 a < 0 < b, a < r < s < b N (0 1) P (a < m 6 M < b r < W (1) < s) =;1Xk=;11Xk=;1P (r + 2k(b ; a) < < s + 2k(b ; a)) ;P (2b ; s + 2k(b ; a) < < 2b ; r + 2k(b ; a)):* , ' %%, - % %% %% % %.
@,P z+@ ... , Xk k 2 N,D X n ! 1, X N (0 1). 0 1, n;1=2 nk=1 Xk ;!PP ) (7) .. Y , n;1=2 nk=1 Xk ;!Y n ! 1 ( , , r).B& % % % %% %% (X A). = , hf Qni hf Qi f .= B(X R) { , ) AjB(R) { f : X ! R, k k1 , . (6.30). + P = P (X ){ (X A). C P Q 2 P P Q kP ; Qkvar = supfjhf P i ; hf Qij : f 2 B(X R) kf k1 6 1g:E. 6.15. +, (P Q) = kP ; Qkvar P , # P , U (Q ) = fP 2 P : (P Q) < g, Q 2 P , > 0 ( .
O?, . 2, x5]). %117E. 6.16. C, (P Q) = 2 sup jP (A) ; Q(A)j:A2AE. 6.17. + P Q 2 P P , Q 2 P (.. P Q - Q ), = (P + Q)=2). + g = dP=d, h = dQ=d. C, (P Q) = kg ; hkL1 (), L1() = L1(X A ). +, (P Q) 6 2, , P ?Q ( , .. A 2 A , P (A) = 1 Q(A) = 0).E.
6.18. C, (P (X ) ) { % %%.C, " &,, X { %. T (P (X ) ), X { ?* % % &, %%.A 96.19. Sn = Pnk=1 Xk , X1 : : : Xn { ,P (Xk = 1) = pk P (Xk = 0) = 1 ; pk , k = 1 : : : n. Y { P .. = p1 + : : : + pn . jP (Sn 2 B ) ; P (Y 2 B )j 6 nk=1 p2k <&B R.C " O?, . ], . C , A 96.20 (I).
Qn n 2 N 1 { (X A), Qn - (X A). dQn =d ! dQ1 =d .. ( ). Qn Q1 %.0 > { 9d(P Q). * , . 6.17, pd2(P Q) = 12 h(pg ; h)2 i:E. 6.21. +, d { P (X ), ) .', d 9 1/2. 2 2 (0 1) H (Q P Q) = h(g) h1; Qi:@ d2(P Q) = 1 ; H (1=2Q P Q).pE. 6.22. C, 2d2 (P Q) 6 kP ; Qkvar 6 8d(P Q) P Q 2 P (X ).C, P = P1 : : : Pn , Q = Q1 : : : Qn, H (Q P Q) =Ynk=1H (Q Pk Qk ):+ d H O?, .
3, x9,10].@ % -% %% %%. "118E. 6.23. + Xn ! X1 .. n ! 1, Xn : E ! X , Xn 2 FjB(X ) n 2 N f1g. @ L(Xn ) ) L(X1 ), n ! 1.2 3 Ef (Xn ) ! Ef (X1 ) f 2 Cb(X R). 29, ) " , .., (). = ' % &%%, ,%%, A 96.24 (). (X ) { , Qn ) Q1 n ! 1. ) (E F P ) Xn : E ! X , Xn 2 FjB(X ), L(Xn ) = Qn n = 1 2 : : : 1 Xn ! X1.. n ! 1.2 % < % ,< &X . C k 2 N X Gkm ,m 2 N 2;(k+1) , m Gkm = X . 4 2;(k+1) 2;k , X Bkm, Qn(@Bkm) = 0 k m 2 N n 2 N f1g (7 ).+ Dk1 = Bk1 m > 2 Dkm = Bkm n mr=1;1 Bkr . + k 2 N Dkm m 2 N X , diamDkm 6 2;k , m 2 N(diamD = supf(x y) : x y 2 Dg, D 2 X ) Qn (@Dkm) = 0 n k m. 4 '% Si1 :::ik = \kj=1 Djij ( { N). = ) k X , 1) Si1:::ik \ Sj1:::jk = (i1 : : : ik ) 6= (j1 : : : jk )Q2) j Sj = X , j Si1 :::ik j = Si1 :::ik Q3) diamSi1:::ik 6 2;k k i1 : : : ik 2 NQ4) Qn(@Si1:::ik ) = 0 k i1 : : : ik 2 N, n 2 N f1g.@ & %%, < %,%, (E F P ), E = O0 1), F = B(O0 1)) P = j j, j j 3 (, , % -& F ).
C n 2 N 1 k 2 N O0 1) )(n)(n)^(i1n:::ik = Oai1 :::ik bi1 :::ik ) (n) i1 : : : ik 2 N ^i :::ik = Qn(Si :::ik ), ,11^(1n) = O0 Qn(S1)) ^(2n) = OQn(S1) Qn(S1) + Qn(S2)) : : :K ^(in) ^(ijn) ( , ^(in)) ..)* ^(i1n:::ik .C k 2 N Si1:::ik xi1:::ik n 2 N f1g E = O0 1) )Xnk (!) = xi1:::ik ! 2 ^(i1n:::ik119)( Qn(Si1:::ik ) = 0, ^(i1n:::ik = { Oa a), ).
=, Xnk 2 FjB(X ) k n.', (Xnk (!) Xnk+m (!)) 6 2;k ! 2 O0 1) k n m(6.34) 2) 3) . X )Xn (!) = klimX k (!) ! 2 O0 1) n 2 N f1g(6.35)!1 n 4.6 Xn 2 FjB(X ) n 2 N f1g. + 4) 3 6.2, , k i1 : : : ik 2 N (n) (1) Qn(Si :::ik ) = ^i :::ik ! ^i :::ik = Q1(Si :::ik ) n ! 1:1111) 6= ), 0,Q1(Si1:::ik ) > 0 (, ^(i11:::ik (1) ! 2 ^i1:::ik (B { " B ), nk (!) , ! 2 ^(i1n:::) ik n > nk (!) (7 ). ( ! Xnk (!) = X1k (!) , (6.35), (Xn (!) X1 (!)) 6 (Xn (!) Xnk (!)) + (Xnk (!) X1k (!)) + (X1k (!) X1 (!)) 6 2;k+1(1) n > nk (!). + E0 = \1k=1 i1 :::ik ^i1 :::ik . =, P (E0 ) = jE0 j = 1 Xn (!) ! X1 (!) ! 2 E0 n ! 1. @ X1 2 FjB(X ) 4.6.B%, %,, % L(Xn ) = Qn n 2 N f1g.
+ n m i1 : : : im k > mP (Xnk (n) 2 Si :::im ) = ^i :::im = Qn (Si :::im ):111(6.36)<& %% '% G X ' %%,, &; % '% Si :::im ( m i1 : : : im). + " ( 3,4O?, . 2, x5]). GN ! G, GN N .0, P (Xnk 2 G) > P (Xnk 2 GN ) N , (6.37), liminf P (Xnk 2 G) > liminf P (Xnk 2 GN ) = Qn(GN ):k!1k!11@ , lim infk!1 P (Xnk 2 G) > Qn(G).
0 2 6.2, L(Xnk ) ) Qn k ! 1 n 2 Nf1g. ( Xnk ! Xn.. k ! 1 n, " . 6.22 , L(Xnk ) ) L(Xn ).= . 2@ C6.24 . 6.2. * " ( ..), D X , X , EX ! EX . C X = R, Xn ;!nn 96.25. (X ) { , Qn ) Q P . supfjhf Qni ; hf Qij : f 2 GC g ! 0 n ! 1(6.37) GC { $% f : X ! R, #) kf k1 6 C .1202 + (E F P ) ( , !) -, ) C6.24. @ ( (2.10)), , supfjEf (Xn ) ; Ef (X1 )j : f 2 GC g ! 0 n ! 1: " > 0 = (") > 0,, jf (x) ; f (y)j 6 " f 2 GC , (x y) 6 . @ jEf (Xn ) ; Ef (X1 )j 6 Ejf (Xn ) ; f (X1 )j1 f(Xn X1) 6 g ++ Ejf (Xn ) ; f (X1 )j1 f(Xn X1 ) 6 g 6 " + 2CP ((Xn X1 ) > ):P= , Xn ! X1 .., Xn ;!X1 , ..P (! : (Xn (!) X1 (!)) > ) ! 0 > 0 n ! 1(Xn X1 ) ( X ), 0 .., . 2@ % & %.
%, ( ) (X ) { , %%. CB X " > 0 B " = fx 2 X : (x B ) < "g, (x B ) = inf f(x y) : y 2 B g. P (X ) , { (P Q) = inf f" > 0 : P (B ) 6 Q(B ") + " Q(B ) 6 P (B ") + " B 2 B(X )g: (6.38)E. 6.26. C, ( ) P (X ). C, ) ( ) (P Q) = inf f" > 0 : P (F ) 6 Q(F ") + " F Xg:(6.39)E. 6.27. C, (P (X ) ) { .A 96.28. Qn ) Q , (Qn Q) ! 0 (n ! 1).2 + (Qn Q) ! 0. @ (6.41) " > 0 F X Qn(F ) 6 Q(F ") + " n > n("). 0,lim supn!1 Qn(F ) 6 Q"(F ) + " " > 0. = " 1 6.2.B&%.
+ Qn ) Q. 0 f"F (), 6.2, " > 0 F { X , , 0 6 f"F () 6 1 jf"F (x) ; f"F (y)j 6 ";1(x y) x y 2 X (7). 0, " > 0 M" = ff"F () F Xg G1, GC C6.25. * (6.37) , " > 0^n(") = supfjhf Qni ; hf Qij : f 2 M"g ! 0 n ! 1:(6.40)C " > 0 F , , 1 F () 6 f"F () 6 1 F " (), ^(n") Q(F ") > hf"F Qi > hf"F Qni ; ^(n") > h1 F Qni ; ^n(") = Qn(F ) ; ^(n"):(6.41)= , ^(n") 6 " n > n0(") (6.39). 2121E. 6.29. M BL X , kf kBL = kf k1 + L(f ), . (6.30).C, kP ; QkBL = supfjhf P i ; hf Qij : f 2 BL kf kBL 6 1g % P (X ), %, % %' -% &%. 1 , P Q 2 P (X ) kP ; QkBL 6 2(P Q) '((P Q)) 6 kP ; QkBL '(t) = 2t2=(t + 2), t > 0.A 96.30 (I%, N?]).
; , { > 1 { (#) ). F ,(P Q) = inf f{(X Y )g(6.42) (X Y ) ( ) , L(X ) = P , L(Y ) = Q, { (X Y ) = inf f" > 0 : P ((X Y ) > ") < ".0 (6.42) ) - (.. ), ) , ) " (X Y ), (.. L(X ) L(Y )) , " X Y . * ) O?], O?]. =, % % , % . (, , = ' %, %,< % % %% 9 { . + Xni i = 1 : : : mnQ n 2 N{ , , EXni = 0, EjXPnijs < 1 s > 2. 12 = 1, 2 = DX . +PXni , mi=1n nininimnLns := i=1 EjXni js. 2 { ,( s=2PP 2, Lns = mi=1n EjXni ; EXni js= mi=1n ni ). 3 ( ), ,Lns ! 0 n ! 1 3 (6.18). ' , 3 , z+@ (. 2), "# ": " > 0max P (jXni j > ") ! 0 n ! 1:16i6mn(6.43)( z+@ (6.43) O?], O?].122A 96.31 (F, O?]).
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