А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 8
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2% 4.14. t y > 0P (M (t) > y) = 2P (W (t) > y):(4.25)2 x = 0 (4.22). @P (W (t) < y M (t) > y) = P (W (t) > y)P (M (t) > y) = P (M (t) > y W (t) < y) + P (M (t) > y W (t) > y) == P (W (t) > y) + P (W (t) > y) = 2P (W (t) > y)(, P (W (t) = y) = 0 y 2 R t > 0). 2', M (t) ) 6.% 4.15. 4 y > 0 0 6 a < b < 1P ( sup jW (t) ; W (a)j > y) 6 4P (W (b ; a) > y) 2P (jW (b ; a)j > y):(4.26)a6t6b2 4.3P ( sup jW (t) ; W (a)j > y) = P ( sup jW (s)j > y) 6a6t6b06s6b;a6 P ( sup W (s) > y) + P (06infW (s) 6 ;y):s6b;a06s6b;a+ , sup (;W (s)) = ; inf W (s) t > 0 ;W s20t]s20t] , (4.25).
2C t > 0 Log t = ln(t _ e). = (4.26) ) .A 4.16 ( % ). ( # %(t)lim sup (2t LogWLogt)1=2 = 1(4.27)W (t)liminf= ;1:t!1 (2t Log Log t)1=2(4.28)t!1676(1;")p2t Log Log t(1+")p2t Log Log t~0W (t)tt0(" !). 5.2@ 4.16 , p ) %& (1 + ") 2t Log Log t ( " > 0, t0(" !)). 1p %& %& (1 ; ") 2t Log Log t(. . 5.2).
C " ( ) 3, , " ) j.( ) , . A W0(t) = W (t) ; tW (1) t 2 O0 1]:(4.29)( , W0(0) = W0(1) = 0, (4.29). .E. 4.1. + , 4.1, , fW (t) 0 6 t 6 1g .. , ) : > 1=2 (.. 3.8 ??).E. 4.2. + 3.8 C2.24 , () q 2 N, 0 6 s < t < 1E(W (t) ; W (s))2q = (2q ; 1)!!(t ; s)q :(4.30)68= H (!) ! 2 E ( , E = C O0 1)) t 2 O0 1), : . 9 3.8, 4.2 , P (H = O0 1)) = 1 < 1=2. 9 4.1 , P (H = ) = 1 > 1=2. * ?? , P (t 2 H1=2) = 0 t > 0. = C" O?] 1983 . , P (H1=2 6= ) = 1.@ , %%,- ( ) % < & ' % . 2 K. .
0 (. O?]).A 94.3. X { .., #) EjX j < 1. )# ( ) W (t) t > 0, .. , X =D EX + W ( ):(4.31)" EX 2 < 1, ) E E = DX:(4.32)2 + (E F P ) X fW (t) t > 0g ( ). C , EX = 0 X { ( 0). + X a b (a < 0 < b, EX = 0). DP (X = a) = p P (X = b) = 1 ; p(4.33) EX = 0 p = b ;b a 1 ; p = b;;aa :(4.34) 4.4 ab = inf ft > 0 : W (t) 2 fa bgg .C, ab , .. ab < 1 .. (" k < 1 k 2 N. C 4.12). 1 , , Eabm 2 N fab > mg fjW (n) ; W (n ; 1)j 6 b ; a n = 1 : : : mg:P (jW (n) ; W (n ; 1)j 6 b ; a n = 1 : : : m) = P (jj 6 b ; a)m N (0 1). +" ()k < 1 (k 2 N):(4.35)P (ab = 1) = 0 Eab9 , : W (ab) = a W (ab) = b( 1). * (4.35), P (W (ab) = a) = pab P (W (ab) = b) = 1 ; pab:(4.36)69C, EW (ab) = 0(4.37) (4.36) , pab = b=(b ; a) , (4.33), (4.34), X (4.31) .
%'% 4,: 1 2 : : : ... { ( N) - Fn = f1 : : : n g,n 2 N, Ej1j < 1 E < 1. @E(1 + : : : + ) = E E1:C,Ej1 + : : : + j ==1Xn=11Xn=1E(j1 + : : : + j1( = n)) =E(E(j1 + : : : + n j1( = n)jFn )) ==1Xn=1P ( = n)Ej1 + : : : + n j 61Xn=11Xn=1(4.38)E(j1 + : : : + n j1( = n)) =E(1( = n)E(j1 + : : : + njjFn )) =1Xn=1nP ( = n)Ej1j = E Ej1j:(4.39)4 , ) (., ., O?]).
) .K (4.39) ( ) (4.38).E. 4.4. C % %'% 4,. K , ,) (4.38), D1 < 1, E(1 + : : : + ; E1)2 = E D1:(4.40)@ n 2 N nm = W (m=n) ; W ((m ; 1)=n),m 2 N (n) = inf fm : + : : : + ab= (a b)g:n1nm 2(4.41)E. 4.5. C, n 2 N ab(n) { (n) < 1. - Fk(n) = fn1 : : : nk g, k 2 N Eab(n)* " , (4.35) (4.38), , n1 + : : : + nab(n) = W (ab =n) En1 = 0, (n)EW (ab=n) = 0:(4.42)E. 4.6. 9 , , (n)=n ! ab ..
n ! 1:(4.43)ab70* 4.6 , (n)EW (ab=n) ! E(Wab) n ! 1:(4.44)C " , .. f 2 Sg , Zlim supj jdPc!1 2 fjj>cg = 0:(4.45)D* (., ., O?, . ??]), n ;! ( ) fn g , En ! E n ! 1(4.46)( , n > 0, (4.45) (4.46)). = .. .E. 4.7. C, sup Ejj < 1 > 12 f 2 Sg.E. 4.8. 0 ) (4.35), ) ,jW (ab(n)=n)j 6 b + a + jnab(n) j(n) =n) n 2 Ng , fW (ab.9 (4.43), 4.8, (4.44). (4.42) (4.37). @ , (4.33) , X =D W (ab):(4.47)M & . = F (x) = P (X 6 x).
9, EX = 0 , .. X , c=Z(;10](;y)dF (y) =Z(01)zdF (z) 6= 0:(4.48)+ f : R ! R { . @cEf (X ) = c+Z(;10]f (y)dF (y)=Ef (X ) = c;1ZZ1Z;1f (x)dF (x) =(01)zdF (z) =ZZZ(01)(01)f (z)dF (z)dF (z)Z(;10]Z(;10](;y)dF (y)+dF (y)(zf (y) ; yf (z)): (4.49)dF (z)dF (y)(z ; y) f (y) z ;z y + f (z) z;;yy :(01)(;10](4.50)71( (E0 F 0 P 0) (Y Z ) R2 , P 0((Y Z ) 2 B ) = c;1ZZ(z ; y)dF (z)dF (y):B \f(;10](01)g(4.51)3 , (4.51) . @, " , (4.50), f 1.C y < 0 < z, (4.36), pyz = z=(z ; y) (4.47) f (y) z ;z y + f (z) z(;;yy) = Ef (W (yz )):(4.52)2 y 6 0 < z, 0b = 0 b > 0.M (E^ F P ) = (E F P ) (E0 F 0 P 0):@, (4.51), (4.52), (4.37) (2.10), (4.50) Ef (X ) = Eef (X ) = E0Ef (X ) = E0Ef (W (YZ )) = eEf (W (YZ ))(4.53) E0 P 0, eE { Pe = P P 0.
+" 2.9 PeX = PeW (YZ ):(4.54)f(t !e) := W (t !),C (4.31) (), W0 t > 0 !e = (! ! ) 2 Ee , , (E^ F P ). C , (E F P ) E P .E. 4.9. * ( 4.4), , Eab = ;ab:(4.55)%, %, EX 2 < 1. (4.55) J, EeYZ = E0E(YZ ) = E0(;Y Z ):(4.56)+ (4.56), (4.51) (4.48), EeYZ = E(;Y Z ) ===72ZZ(;10](;10]Z(;10]dF (y)(;y)dF (y)(;y) ;y +y2dF (y) +Z(01)ZZ(01)(01)dF (z)z(z ; y)c;1 =dF (z)c;1z2=z2dF (z) = EX 2 = EeX 2 : 2A 94.10 (, N?]). X1 X2 : : : { %- , .
. , .. Tk , k 2 N fW (t) t > 0g, fXk k 2 Ng =D fW (Tk ) ; W (Tk;1) k 2 Ng % Tk ; Tk;1 , k 2 N (T0 0)EXk2 < 1, E(Tk ; Tk;1) = EXk2.(4.57), 2 C C4.3. + Fk .. Xk (k 2 N). + (E F P ) (Yk Zk ), k 2 N, , (Yk Zk ) (4.51), F Fk . + f(Yk Zk ) k 2 Ng fW (t) t > 0g.
+Tk = inf ft > Tk;1 : W (t + Tk;1) ; W (Tk;1) 2= (Yk Zk )g k 2 N:(4.58) (4.58) (Yk Zk ) , Yk Zk .@ fTk k 2 Ng { . 2+ W (t) = (W (1)(t) : : : W (q)(t)) q- & ' (.. W (j)(t) t > 0, j = 1 : : : q, , , q .". C O0 1)). (C O0 1])q , kx()k = sup jx(t)jt201] j j { Rq . = gn(t) = p W (nt) t 2 O0 1] n 2 N2nLog Logn(4.59) Logz = (log z) _ 1 z > 0 Log Logz = Log(Logz) ( , (4.59) , n > 3).A 94.11 (, % , I%-).; fgn g (C O0 1])q , .. K = fx : x(t) =RRZt0h(s)ds s 2 O0 1] RZ10jh(s)j2ds 6 1gttt 0 h(s)ds = ( 0 h(1)(s)ds : : : 0 h(q) (s)ds). 4 , K # $% x , x(j )(0) = 0 j = 1 : : : q Z10jdx=dtj2ds 6 1dx=dt = (dx(1)=dt : : : dx(q)=dt).73@ 94.12.
@ C4.11 ( , - ) gT (t) = p W (Tt) T > 0 t 2 O0 1]2T Log LogT T , gTn (), fTng { , Tn > 0 Tn ! 1 n ! 1.E. 4.13. C, K (" j") { (C O0 1])q.E. 4.14. + C (fxng) { fxng (X ). + h { X (Y ). C, C (fhxng) = h(C (fxng)). h(x()) = x(1)(1) x 2 (C O0 1])q:@ C4.11 4.14 (7 ) ) ) 4.16.% 94.15. W (t) { . # ftn g, p tn ! 1 n ! 1, fW (tn )= 2tn Log Log tn n 2 Ng # 1 O;1 1], .. "." .2 C 4.11.
3 , %' "C (fgn g) = K ..", < < %=:1) " > 0P (gn 2= K " ..) = 0(4.60) B " { " "- B , ".." (fAn ..g = \n j>n Aj ),2) x 2 K " > 0 1gnk ( !) 2 fxg" k > N(4.61) fnk g N ", x " !.*, , " > 0 & %, P (gn 2= K ") &,= n. M O0 1] i=m i = 0 1 : : : m, m . J g 2 (C O0 1])q () gb (C O0 1])q, ) (i=m g(i=m)) i = 0 1 : : : m.
C r > 1 P (gn 2= K ") 6 P (r;1bgn 2= K ) + P (r;1 bgn 2 K kgn ; r;1bgn k > ") =: p1 + p2:C,74Z 1 1 dbg 2p1 = P (r;1gbn 2= K ) = P ( r dt dt > 1) = P (2md > 2r2Log Logn)0(4.62)Z 1 1 dbg 2q Xm 2X;1 2 (j)(j)dt=q=(2LogLogn)g(i=m);g((i;1)=m)mdnn0 r dt j =1 i=1 gn(t) = (gn(1)(t) : : : gn(q)(t)), 2d { , ) - d (d 2 N) . (, 2d 8 zd=2;1e;z=2<p2d (z) = : 2d=2 ;(d=2) z > 00z < 0:R ;() = 01 x;1e;xdx > 0.
+" ( )d=2;1 ;x=2P (2d > x) 2xd=2;1;(e d=2) x ! 1:0, c1 ( c , ) n) np1 6 c1 expf;rLog Logng:(4.63)C,p2 6 P (r;1bgn 2 K (1 ; r;1)kr;1 bgn k > "=2) + P (kgn ; gbn k > "=2):(4.64) r = r(") 1, , nP (r;1 bgn 2 K (1 ; r;1)kr;1gbn k > "=2) = 0(4.65) r;1bgn 2 K kbgn k 6 r. C, x() 2 K , 06s6t61 Z t Z t !1=2dxdu 6 (t ; s)1=2jx(t) ; x(s)j = du du 6 (t ; s)1=2 dxss du RRR t dx du = ( t dx du : : : t dx q du). ' (1)s du( )s dus du 4.15, P (kgn ; bgn k > "=2) 66qmXi=1P(supt2(i;1)=mi=m]mXi=1P(supt2(i;1)=mi=m]jgn (t) ; gn ((i ; 1)=m)j > "=4) 6ppjw(t) ; w((i ; 1)=m)j > ("=4) m=q 2Log Logng) 66 qmc2 expf;Log Logng"2m=(16q)g 6 c3 expf;rLog Logng(4.66) w { m > 16qr";2, ,pqg:fy 2 Rq : jyj > "g fy 2 Rq : 1maxjyj>"=i6i6q(4.67)75@ , c > 1, & nk = Ock ], O] { , (4.62) { (4.66)XkP (gnk 2= K ") 6 c4Xkexpf;rLog Lognk g < 1:%,, F { 0% gnk 2 K " ..
! k > N (" c !).M% %, gn n 2 Onk nk+1]. 9 (4.67), P (n 6maxkgn ; gnk k > ") 6k n6 nk+1!w(n)w(n)p6 qP nk 6max; p2n Logk Logn > "= q 6pn6nk 2nLog Lognkk!+16 qPkwp(n) ; w(nk )k > "=(2pq) +maxnk 6n6nk+12nLog Lognp;1=2;1=2qP n 6max(2nLog Logn) ; (2nk Log Lognk ) kw(nk )k > "=(2 q) 6k n6nk" p+16 qPsupst20nk+1 ]js;tj6nk+1 ;nkjw(s) ; w(t)j > 2pq 2nk Log Lognk ++qP sup jw(t)j > 2p" q ((2nk Log Lognk );1=2 ; (2nk+1 Log Lognk+1 );1=2) =: q(p3 + p4 ):t20nk ](4.68)@ , 4.15.E.
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