А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 10
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6 .., k 2 N. @X ; X ====kXm=1kXm=1kXm=1k mXX;1m=1 j =01 f =mg1 f=jg (Xm ; Xj ) =1 f =mg1 f<mgXm ;1 f<mg 1 f =mgXm ;1 f<mg1 f >mgXm ;kXm=1k;1XkXj =0m=11 f=jg1 f>jgXj =1 f=m;1g 1 f >mgXm;1 =1 f<mg 1 f >mgXm;1 =kXm=1+ (5.18) kXm=11 f<m;1g 1 f >mgXm;1 0,E(X ; X ) =k;1Xm=0nXm=11 f<mg1 f >m+1gXm 1 f<mg 1 f >mg^Xm:kXm=1(5.18)1 f<mg1 f >m+1gXm:EO1 f<mg 1 f >mgE(^XmjFm;1)] = (>)0:2) ) 3).
+ = ^ A 2 F. 3 , A = 1 A + 11 A A = 1 A + 11 A = A ^ k = A ^ k , 6 6 k ! 2 E. +" EX = (>)EX .0,@ ,EX 1 A + EXk 1 A = (>)EX1 A + EXk 1 A :E(E(X jF)1 A) = EX 1 A = (>)EX1 A:m=0 EjXm j < 1 6 k .. , E(X jF ) 2 F jB (R)( ).3) ) 1). n m, 0 6 m 6 n (m n 2 Z+). =,F = Fm, " 3) 1). 2% 5.6. (Xn Fn)n2Z+ { () , # #) n , (.. n 6 n+1 .. n 2 Z+ n 6 kn .. kn 2 N) (Xn Fn )n2Z+ P4 , EjX j 6 k().85% 5.7 (, , %, 9&).
(Xn Fn)n2Z { . # N 2 N u > 0uP (0maxX > u) 6 EXN 1 f max Xn > ug 6 EXN+ (5.19)6n6N n+06n6NuP (06minX 6 ;u) 6 ;EX0 + EXN 1 f min Xn 6 ;ug 6 ;EX0 + EXN+ :n6N n06n6N(5.20)2 = = minfn : Xn > ug ^ N . @EXN > EX = EX 1 A + EX 1 A > uP (A) + EXN 1 A : A = f0maxX > ug. 0,6n6N nuP (A) 6 EXN ; EXN 1 A = EXN 1 A 6 EXN+ :C (5.20) , = minfn : Xn 6 ;ug^N . 2+ ) - .% 5.8. (Xn Fn )n2Z+ { EjXn jp < 1 p > 1 n 2 Z+. u > 0 N 2 Z+;p EjXN jp :P (0maxjXj>u)6un6n6N2 C , (jXn jp Fn)n2Z+ { p > upg: 2f0maxjXj>ug=fmaxjXjnn6n6N06n6N% 5.9.
(Xn Fn)n2Z { % . EjXn jp < 1 n 2 f0 : : : N g, N 2 N p 2 (1 1).+ k 0maxjX jk 6 (p=(p ; 1))kXN kp6n6N n p, ,kkp = (Ejjp)1=p.(5.21)2 1 ) ( 5.1) Xn > 0 .. n 2 Z+. + p (. O?, . 223]), EO(0maxX6n6N n)p] = pZ1Z10up;1P (0maxX > u)du 66n6N nup;2E(XN 1 f max Xn > ug)du =006n6Npp;1 ) 6 p (EX p )1=pOE(max X )p ](p;1)=p:= p ; 1 E(XN (0maxX)nn6n6Np;1 N' (5.19)uP (0maxX > u) 6 EXN 1 fmax06n6N Xn>ug6n6N n6p86 , B = inf fn : Xn 2 B g B 2 B(R). 4 , > 0 .., ) E, E =Z10E(1 f>ug)du:(5.22)+ O?] .
223 1. = Pm = j=1 cj 1 Aj , . 2( k max06n6N jXn jk1 .9 % &% &% N (a b) " " &% % a b % 0 N . . + (Xn Fn)n2N { - (a b) { . = (") k , k 2 N, 0 = 0 2m;1 = minfn : n > 2m;2 Xn 6 ag 2m = minfn : n > 2m;1 Xn > bg m 2 N( k j , j > k, , ) ). (. . 5.1) N (a b) = 0 2 > Nmaxfm : 2m 6 N g 2 6 N: 5.10 ( , 9&). 4 . - ++EN (a b) 6 E(XN ; a) 6 EXN + jaj :(5.23)b;ab;a2 + N (a b) (Xn Fn)n2N N (0 b ; a) ((XN ; a)+ Fn)n2N, , a = 0 Xn > 0,n 2 N. + X0 = 0, F0 = f Eg.
+ i 2 Ni = 1 fm < i 6 m+1 mg:@bN (0 b) 6NX(Xi ; Xi;1 )i:i=1S', fi = 1g = m { ffm < ig n fm+1 < igg 2 Fi;1, i 2 N.+"bEN (0 b) 66NXi=1NXi=1EO(Xi ; Xi;1 )i] =E(E(Xi jFi;1) ; Xi;1 ) =NXi=1NXi=1EOi(E(Xi jFi;1) ; Xi;1)] 6(EXi ; EXi;1 ) = EXN : 2A 5.11. (Xn Fn)n2N { , supn EjXn j < 1. # % ) X1 = lim Xn , EjX1 j < 1.n!187inf X , X = lim sup Xn . C P (X < X ) > 0. +2 + X = limn!1 nn!1fX < X g = ab2Qa<bfX < a < b < X g Q { , P fX < a < b < X g > 0 a < b.
5.10 N 2 NEN (a b) 6 (EXN+ + jaj)=(b ; a):= 1(a b) = Nlim (a b). @!1 NE1(a b) 6 (sup EXN+ + jaj)=(b ; a):N@ , (Xn Fn)n2Nsup EXn+ < 1 , sup EjXnj < 1nnEXn+ 6 EjXn j = 2EXn+ ; EXn 6 2EXn+ ; EX1.(5.24) 0, E1(a b) < 1.., P fX < a < b < X g > 0.
@ ,P fX < X g = 0. + J EjX1 j 6 supn EjXnj < 1. 2@ 5.11 ( , (;Xn Fn)n2Z+ { , (Xn Fn)n2Z+ { ). + " sup EjXn j < 1 n ( { ).% 5.12. (Xn Fn)n2N { % , supn EjXn jp < 1 p 2 (1 1). )X1 = nlimX .., Lp.!1 n2 + 5.9E(sup jXn jp) 6 (p=(p ; 1))p sup EjXn jp:n2Nn2N+ 5.11 Xn ! X1 .. n ! 1.
@ , jXn ; X1 jp 6 2p;1(jXn jp + jX1jp) 6 2p sup jXn jp:n2N(5.25)(5.26)9 (5.25), (5.26) . 24 % % % % % ) . 5.13 (% ",% { 4%). + fk(n) Q k n 2 Ng, ) -, ) P P(1)(1)Sn;1 (n)E1 = > 0. + S0 = 1, S1 = 1 Sn = k=1 k n > 2 ( 0k=1 k(n) := 0,n 2 N). 4 , Sn n- ( n- ). K , ' <& % % % %(1)%% 1 , % n = 0 %%, . 9 , .88', (S0 : : : Sn;1) n fk(n) k 2 Ng.
4 Xn = Sn =n n 2 N ', % &<% % Fn = fX0 : : : Xn g, n 2 Z+. * n 2 NXSn;1E(Xn jFn;1) = E(SXn;= ;n E1k=11X;n=1 fSj =1k(n) S0 : : :n;1 =j g@ , jXk=1k=1k(n)jX0 : : : Xn;1 )=n =j1 XX(n);n Sn;1 = E 1 fSn;1=jg k S0 : : : Sn;1 =j =0jE(k(n) S0 : : : Sn;1 ) = ;n+1k=11Xj =1j 1 fSn;1=jg = Sn;1 =n;1 :sup EjXn j = sup EXn = 1:n2Z+n2Z+0 ( 5.11), Xn ! X1 .. n ! 1 EX1 < 1. =, n ! 1Sn ! 0 .., < 1 Sn ! 1 .., > 1: = 1 ) (., ., O?, x36]), Sn ! 0 .. n ! 1.
2B= %, % %% L1(E F P ).A 5.14. (Xn Fn )n2Z+ { . #) :Xn = E(X jFn), n 2 Z+, X { 82) fXn g 83) ) X1 , EjX1 ; Xn j ! 0 n ! 184) supn EjXn j < 1 X1 = lim Xn ( ) .. n!1 5.11) Xk = E(WX1 jFk ), k 2 Z+. , (Xk Fk )k2Z+f1g, F1 = n2Z+ Fn, .2 1) ) 2). C n 2 Z+ a > 0, b > 0 EjXnj1 fjXn j > ag 6 EjX j1 fjXnj > ag =1)+",= EjX j1 fjXn j > a jX j 6 bg + EjX j1 fjXnj > a jX j > bg 66 bP (jXn j > a) + EjX j1 fjX j > bg 66 ab EjXn j + EjX j1 fjX j > bg 6 ab EjX j + EjX j1 fjX j > bg:lim sup EjXnj1 fjXn j > ag 6 EjX j1 fjX j > bg:a!1 n89= , b > 0 .2) ) 3). M fXn g , supn EjXn j < 1. 0, .. ) X1 = nlimX . ( ) .. !1 n L1(E F P ).3) ) 4). * 3) , supn EjXn j < 1.
+" .. )Y = nlimX , EjY j < 1. ( Y = X1 .. C n k 2 Z+!1 nEjE(Xn jFk ) ; E(X1 jFk )j 6 EjXn ; X1 j:( E(Xn jFk ) = Xk .. n > k. 0, Xk = E(X1 jFk ) .., k 2 Z+. C 4) , X1 2 F1jB(R) 4.7 ( Fk , k 2 N , " - F1 ).4) ) 1) { . 2+ % %Q . + 1 2 : : :{ ... , P (1 = 0) = P (1 = 2) = 1=2. + Xn = nk=1 k . @fXn = 0g fXn+1 = 0g P (Xn 6= 0) = 2;n n 2 N , , Xn ! 0 .. n ! 1. - , EjXn ; 0j = EXn = 1, n 2 N.
= 5.14.A 5.15 (). EjX j < 1 (Fn)n2N { #) -1 (E F P ). F1 = _n=1 Fn . n ! 1Xn := E(X jFn ) ! E(X jF1 ) .. L1(E F P ):(5.27)2 (Xn Fn )n2N { % % 5.14.+" .. (5.27). 0 Xn ! X1 .. ( L1(E F P )) n ! 1. + Xn 2 FnjB(R), Xn 2 F1jB(R), n 2 N, , , 4.7, X1 2 F 1jB(R), F 1 { F1. C , X1 = E(X jF1) .. , EX 1 A = EX1 1 A A 2 F1, E(X jF1) = E(X jF 1).. 3 , A = 1n=1 Fn F1 = (A). +" (. 4.2) " > 0 A 2 F1 A" 2 A, P (A^A" < ". @ , A" 2 Fn n > N , N = N (" A).C,jEX 1 A ; EX 1 A" j 6 EjX j1 AA" jEX1 1 A ; EX11 A" j 6 EjX1 j1 AA" :+ , Ejj1 B ! 0 , P (B ) ! 0,, ) . + n > NjEX 1 A" ; EX1 1 A" j = E(E(X jFn )1 A" ) ; EX1 1 A" j 66 EjE(X jFn) ; X1 j = EjXn ; X1 j ! 0 (n ! 1): 2* , , , .
B ,%% % ( ) . @, ) .% 5.16. (Xs Fs)06s6t { , #) .. . # c > 0P ( sup Xs > c) 6 EXt+ =c:06s6t90(5.28)2 C u > 0, Xs , 0 6 s 6 t, f sup Xs > ug = 1n=1 06k2;n6t fXk2;n > ug fXt > ug:06s6tM, (5.19) ( X0 : : : XN Xs1 : : : Xsm , s1 < : : : < sm) P (Nn=1 06k2;n 6t fXk2;n > ug fXt > ug) 6 EXt+ =u:=, P (Nn=1 Bn) ! P (1n=1 Bn ), N ! 1. = u # c. 2% 5.17.
(Xs Fs)06s6t { (.. EXs2 < 1 s 2 O0 t]), #) .. . c > 0P ( sup jXs j > c) 6 c;2EXt2:06s6t(5.29)2 - 5.8, , (Xs2 Fs)06s6t { . + " (5.28). 2 .E. 5.1. + fk gk2N { P , Fn = f1 : : : n g, Xn = nk=1 k , n 2 N. C, (Xn Fn)n2N { , En+1 fn(1 : : : n ) = 0(5.30) fn : Rn ! R n 2 N.* (5.30) ) .C k , k 2 N, ( ) , covg(n+1 )fn(1 : : : n ) = 0 n g : R ! R fn : Rn ! R.E.
5.2. C, (n Fn)n2N { -, (Snm Fn )n>m { m 2 N, Snm =X16i1 <:::<im 6ni1 im n > m: , n Fn = f1 : : : n g, n 2 N.E. 5.3. + a b . * , ) c . = Xn (n 2 N) { ) n- ) c (X0 := a=(a + b)). C, (Xn )n2Z+ {.91E. 5.4. + (E F P ) = (O0 1] BO0 1] ), { 3. + fTn n 2 Ng{ ) O0 1], , Tn 0 = tn0 < : : : < tnmn = 1 Tn Tn+1, n 2 N, mn > 2. -, ^n1 = O0 tn0], ^nk = (tnk;1 tnk ], 2 6 k 6 mn, n 2 N.C f : O0 1] ! R Xn (!) =mnXf (tnk ) ; f (tnk;1)tnk ; tnk;1k=11 nk (!) n 2 N:C, (Xn Fn)n2N { .E.
5.5. + ) f 3, .. jf (x) ; f (y)j 6 Ljx ; yj x y 2 O0 1] L > 0.+ Tn , 06max(t ; tnk;1 ) ! 0, n ! 1. C, k6mn nk_1n=1Fn = BO0 1] fXn gn2N . @ 5.16 , Xn ! X1 .. L1 n ! 1, X1 2 BO0 1]jB(R).0, Oa b] O0 1]ZbaXn d = EXn 1 ab] ! EX1 1 ab] =ZbZbaX1 d n ! 1:(5.31)=7, Xn d ! f (b) ; f (a) n ! 1. = (5.31) ,a f { , ) X1 .E.
5.6. + fXn n 2 Ng { , (E F P ), fn0(x1 : : : xn) fn1(x1 : : : xn) { 3 (Rn B(Rn)), n 2 N. M 0 (X (! ) : : : X (! ))fnLn (!) = f 1(X1 (!) : : : Xn (!)) n 2 Nn1n, fn1 { (X1 : : : Xn ), n 2 N , , fn0(z) = 0, fn1(z) = 0 ( Ln(z) := 0). C, (Ln Fn)n2N { , Fn = fX1 : : : Xn g, n 2 N.E. 5.7. + (Xt Ft)t2T { , .. ((ReXt ImXt),Ft)t2T { R2. @ E(XtjFs) = Xs .. s 6 t, s t 2 T(E(Y + iZ jA) := E(Y jA) + iE(Z jA) Y Z - A F ).+ EjXtj2 < 1 t 2 T .
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