А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 12
Текст из файла (страница 12)
= , Xn ! X1 L1(E F P ) n ! 1. K X1 2 L1,, , jX1 j < 1 .. D Xn > 0 .. n 2 Z+ EX0p < 1 p 2 (1 1), 5.9, XN : : : X0 E(0maxX p ) 6 (p=(p ; 1))pE(X0p ):6n6N n9 N , E( sup Xnp) < 1. - n2Z+ 5.12, Xn ! X1 Lp(E F P ) n ! 1. 2@ 95.31. 9 (5.55) .% 95.32. EjX j < 1 (Gn)n2N { #) - (E F P ). G1 = \1n=1 Gn . n ! 1Xn := E(X jGn) ! E(X jG1).. L1(E F P ):(5.56)2 =, (Xn Gn)n2N { . C5.31 Xn ! X1 .. L1 n ! 1. 3 (), X1 2 G1jB(R). @ A 2 G1 EX 1 = lim E(E(X 1 AjGn)) = EX 1 A: 2EX1 1 A = nlim!1 n A n!1+, % ( 5.17) % % 0 1 0 ( 4.11). + fk gk2N { 99 Rm (m > 1), Fn = fk k 6 ng, Gn = fk k > ng, n 2 N.C A 2 G1 = \1n=1 Gn P (A) = E1 A = E(1 A jFn) ! E(1 AjF1) = 1 A .. n ! 1(5.57) F1 = _1n=1 Fn = fk k 2 Ng. 0, P (A) 0 1.
4 , A 2 F1, G1 F1 , A Fn n 2 N. : N ! N () 2 R(N)), (n) 6= n () ) n 2 N. + fk gk2N { (E F P ). @ = (1 2 : : : ) FjB(R1) { " (). + = (1 2 : : : ) = (1 2 : : : ) N. = F - G = f;1(B ) B 2 B(R1) : P (;1 (B )^();1(B )) = 0 2 R(N)g:A 95.33 ( 0 1 K,<%% { -'). fk gk2N { ... . - G , .. 0 1.E. 5.34.
( 5.29) + C5.30 , Unm ! U1m .. n ! 1:0 ) C5.33 , fk gk2N { ..., U1m = Eg(1 : : : m).E. 5.35. + (Xn Gn)n2Z+ { , (5.55). C, X1 6 E(Xm jG1), m 2 Z+, G1 = \1n=1 Gn (X1 = limn!1 Xn) .. L1(E F P ) C5.30). * , (Xn Gn)n2Z+f1g { . +, (Xn Gn)n2Z+ { , (Xn Gn)n2Z+f1g { .E. 5.36. 4 C5.30 (5.55)?% 95.37 ( &,= , 0). { ...
Rm (m > 1), E1 = a 2 Rm.1 2 : : : nX1Xn := n k ! a .. L1(E F P ) n ! 1:k=11) a = 0 2 Rm. 5.282(Xn GnN )16n6N { () , GnN = fn : : : N g,n = 1 : : : N . , Xn = E(X1jGnN ) .. n = 1 : : : N . + 3( ), N , Xn = E(X1jGn).., Gn = fk k > ng. C5.32, E(X1jGn) ! E(X1 jG1) .. L1(E F P ) n ! 1, - G1 = \1n=1 Gn 0 1 - (5.57). 0, E(X1jG1) = 0 .. 2A 95.38 (F-- 9&&). fXn n 2 Ng { (E F P ) , Xn ! X1 ..
n ! 1 E(supn jXn j) < 1. (Fn )n2N { - F , #, # ( F1 = _1n=1 Fn F1 = \1n=1 Fn). limE(XjF)=E(XjF).. L1 (E F P ):(5.58)nk11nk!11002 =U = mlimsup E(Xn jFk ) V = mliminf E(Xn jFk )!1!1 kn>mkn>m( U V ) .. f sup E(Xn jFk )gkn>m fkninf>m E(Xn jFk )g). + Ym = sup Xn , m 2 N. ', EjYn j 6 E(supn jXnj) < 1,n>mm 2 N.
0, m 2 N 5.16 C5.32E(Ym jFk ) ! E(Ym jF1) .. L1(E F P ) k ! 1:(5.59)+ Xn 6 Ym n > m, E(Xn jFk ) 6 E(Ym jFk ) .. n > m, k 2 N.@ (5.59) U 6 mlimsup E(YmjFk ) 6 lim sup E(YmjF1) ..!1m!1k >m* , Ym # X1 , E(YmjF1) # E(X1 jF1) ..
m ! 1 ().K , V > E(X1 jF1). +" U = V .. C L1- (5.58). 23< '% % &% x5, . 7O?]. + , .A 95.39 (O?, . ??]). (Xn Fn)n2Z+ { fAn g { . A1 = limn!1 An . fXn g .. fA1 < 1g Xn = o(f (An )) n ! 1 # $% f : R+ ! R+, #) #Z10( , $%(1 + f (t));2dt < 1f (t) = t1=2(log+ t), > 1=2).+, % C5.38 '% %, , " .E. 5.40. + (Ft)t2R+ { ) - ( ) (E F P ). @ F t+ = Ft+, .. - ,% % %.2 =, Ft F t, " Ft+ F t+ Ft+ F t+ = F t+ (.. F t N { , F t+ N ).B&%. Ft+ Ft+h h > 0.
0, Ft+ F t+h, Ft+ F t+. 2:, () (Xt Ft)t2R , (Ft)t2R .E. 5.41. + (Xt Ft)t2R { () ) .. . + t 2 R+ ) = (t) > 0, E(sups2tt+] jXs j) < 1 (, sups2tt+] jXs j ). @ (Xt F t+ ) { ().+++1012 5.4 5.40 F t = Ft, t 2 R+.M 0 6 s < t sm # s m ! 1 tn # t n ! 1 , sm < t m 2 N tn < t + (t), n 2 N. @, C5.38E(Xt jFs+) = mnlimE(Xtn jFsm ) > (=) mlimX = Xs ..,!1!1 sm(5.60) (5.60) (=) . 2E.
5.42. 9, ) (, .. , .. ). ) %, ,%%.A 95.43 (. O?, . 16]). X = (Xt Ft)t2R+ { $% (Ft)t2R+ . % X -$%# , , , $% t 7! EXt R+ R . " $% Yt , t 2 R+, ), , t 2 (0 1) (, % cadlag RCLL) $% (Ft)t2R+ . (Yt Ft)t2R+ {. " , ) ( , , , ., . O?], O?, ?], O?]). = . O?].102 6. % $* +$ # . # ..
A. % . !. # C (T X ). "# # # . $ . /{. 4 > ( ( ), . # %. 3 . " ( ) % , & , .G % & % . + (X ) | - B(X ) Qn Q | (X B(X )). :, Qn, n 2 N, Q ( Qn ) Q), f 2 Cb(X R), . . f : X ! R,ZXf (x)Qn(dx) !ZXf (x)Q(dx) n ! 1(6.1)( , f 2 Cb(X C ), . . f : X ! C Q , ..
, , , Q"," > 0). 3% f Q ( ) & &%,%' hf Qi.D ), , 6.1. hf Qi = hf Qei f 2 Cb(X R). Q = Qe.2 + , (4.15), , Q(F ) = Qe(F ) F . 2.3 , Q = Q~ B(X ). 2A 6.2. Q Qn, n 2 N, | (X ). . Qn ) Q (n ! 1) #):1. lim sup Qn(F ) 6 Q(F ) # F 2 B(X )8n!12. liminf Q (G) > Q(G) # G 2 B(X )8n!1 n3. nlimQ (B ) = Q(B ) # B 2 B(X ) , Q(@B ) = 0,!1 n @B | % B (@B B (X )).
; B , $#) 3 , # Q- .2 + , (6.1) , %,% <& f 2 Cb(X R)lim suphf Qni 6 hf Qi:n!1(6.2)C, (6.2) f ;f , liminfhf Qni > hf Qi.n!1+ Qn ) Q. 9' %' 1. C F " > 0 f"F (x) = '((x F )=") 2 Cb(X R), '(t) = 1 t 6 0, '(t) = 1 ; t103 0 < t < 1 '(t) = 0 t > 1. @ Qn(F ) = h1 F Qni 6 hf"F Qni, 1 F 6 f"F ( (1.15)). (6.2)lim sup Qn(F ) 6 lim suphf"F Qni 6 hf"F Qi:n!1n!1= , 3 hf"F Qi ! h1 F Qi = Q(F ) " ! 0.%, 1.
9', % Qn ) Q. ( (6.2) 0 < f (x) < 1, af (x)+ b, a > 0, b 2 R. k 2 N Fi = fx : f (x) > i=kg, i = 0 : : : k. i ; 1 i= Ci = k 6 f (x) < k , . . Ci = Fi;1 n Fi, i = 1 : : : k. @kXi;1i=1k Q(Ci) 6ZXf (x)Q(dx) 6kXii=1k Q(Ci):(6.3)PkPkPk@ , i Q(Ci) = i (Q(Fi;1) ; Q(Fi)) = 1 + 1 Q(Fi). Kk k i=1i=1 ki=1 k (6.3) ZkkX1X11k i=1 Q(Fi) 6 f (x)Q(dx) 6 k + k i=1 Q(Fi):X(6.4)' (6.4) Qn, kkXX1111lim suphf Qni 6 k + lim sup k Qn (Fi) 6 k + k Q(Fi) 6 k1 + hf Qi:n!1n!1i=1i=19 k, (6.2).
0, Qn ) Q.2%%, 2 1 ( ).', % 1 % 3 . = B B , OB ] | . @, 1 2, Q(OB ]) > lim sup Qn(OB ]) > lim sup Qn(B ) >n!1n!1> liminf Q (B ) > liminf Q (B ) > Q(B ): (6.5)n!1 nn!1 n( Q(OB ]) = Q(B ) = Q(B ), Q(@B ) = 0 (OB ] n B 2 @B B n B 2 @B )," (6.5) 3.%, 3, ', % 1. F F " = fx 2 X : (x F ) < "g, " > 0. ', @F " fx : (x F ) = "g, " @F " \ @F = ? " 6= . *, Q(@F ") > 0 ".
+" "k # 0, Q(@F "k ) = 0, k 2 N. @ lim sup Qn(F ) 6 limn!1 Qn (F "k ) = Q(F "k ) n!1 k. = , Q(F "k ) ! Q(F ) k ! 1. 2+ (E F P ), (En Fn Pn), n 2 N, .".X : E ! X Xn : En ! X , . . F jB(X ) Fn jB(X )-. 0.". Xn ) # X (104DXn !X ), PXn ) PX n ! 1 (. (1.5)). + (2.10), , DXn ! X , Enf (Xn ) ! Ef (X ) f 2 Cb(X R) ( f 2 Cb(X C )). ' En Pn .A 6.3. (X B(X )), (Y B(Y )) | h |D X Y . " Xn ! X (Xn X # X ),D h(X ) n ! 1:h(Xn ) !(6.6)E $: Qn ) Q (X B (X )), Qn h;1 ) Qh;1 Y B(Y )) n ! 1.2 g : Y ! R.
@ gh 22 Cb(X R) .0, Eg(h(Xn )) ! Eg(h(X )), n ! 1, (6.6).+ 1.6 .". Xn , X , Qn = PXn , n 2 N, Q = PX .@ , Qnh;1 = Ph(Xn) Qh;1 = Ph(X ): 2 6.4. Qn ) Q n ! 1 , fnk g N fn0k g #, Qn0k ) Q k ! 1.2 ( . C . C, , Qn 6 )Q n ! 1. @ )f 2 Cb(X R) fmk g N , " > 0jhf Qmk i ; hf Qij > " k 2 N:, fm0k g fmk g , Qm0k ) Q k ! 1.+ . 20 fQ 2 Sg (X B(X )) , Qn (n 2 N) ) Q0n () " "). =, ) . 3 ( , ) , ) , ) .
+ " A 6.5. 4 , fQngn2N (X B (X )) , , . $% H Cb (X R), 1) ) limn!1 hh Qn i h 2 H,2) # Q Q (X B (X )) .eQ = Qe B(X ).hh Qi = hh Qei h 2 H(6.7)1052 ( ( H = Cb(X R) 6.1).9%%%,. * fnk g N fn0k g , Qn0k ) Q k ! 1. @ hf Qn0k i ! hf Qi f 2 Cb(X R), 1) , lim hh Qn i = hh Qi h 2 H:(6.8)n!1C, Qn 6 )Q n ! 1. @ 6.4 fmk g N , Qmk ) Qe k ! 1 Qe 6= Q. * 6.8 , hh Qi = hh Qei h 2 H. 0 2) Q = Qe.
+ . 2M% %, X = fXt t 2 T g X (n) = fXt(n) t 2 T g,n 2 N, (E F P ) (En Fn Pn) ) t 2 T Xt. 0 1.4 X 2 FjBT X (n) 2 FnjBT n 2 N. D X (n) X BT , & %, & -& % ( ) %%. 4 & , %'%,, %XT % %<, % BT = B(XT ). " , , ( 1) XT . , T { C (T X ){ T X ( t 2 T ), , .. C (3.20). = B(C (T X )) - (C (T X ) C ). + BT (C (T X )) { - C (T X ), ..-, " " t;11:::tk (B ), B 2 B(X k),t1 : : : tk 2 T , k 2 N ( X k (2.1)), t1:::tk : C (T X ) ! X k t1:::tk x = (x(t1) : : : x(tk)) x 2 C (T X ):(6.9)A 6.6.
C (T X ) { . . #) :a) B (C (T X )) = BT (C (T X )),b) Q Q, , ..eB(C (T X )) # -e t;11:::tkQt;11:::tk = QQ = Qe B(C (T X )), t1 : : : tk 2 T k 2 N(6.10)c) fQn gn2N B (C (T X )) n ! 1 , )# , .. t1 : : : tk 2 T k 2 N Qn t;11:::tk gn2N .2 a) B(C (T X )) BT (C (T X )) 2.2 3.9, , BT (C (T X )). 0 , " " B(C (T X )) 1.2 t1:::tk t1 : : : tk 2 T ,k 2 N.
+" BT (C (T X )) B(C (T X )).106b) D Q Qe " " C (T X ), - BT (C (T X )), , a) BT (C (T X )).c) ( 6.3 (6.9).C H = fh = gk t1:::tk gk 2 Cb(X k R) t1 : : : tk 2 T k 2 Ng:0 (2.10) h = gk t1:::tk 2 H hh Qni = hgk Qnt;11:::tk i:(6.11)0 fQnt1:::tk gn2N (6.11) 1) 6.5.
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