А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 18
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@P (t = 0) = P (1 > t) =Z1te;xdx = e;x:= Sk = 1 + : : : + k , k > 1. + k 2 N 8 (x)k;1<;xpSk (x) = : (k ; 1)! e x > 00x < 0:(8.13)M (8.13) " . + (8.13), k > 1P (t = k) = P (Sk 6 t Sk+1 > t) = P (Sk 6 t Sk + k+1 > t) =ZZZt (u)k;1Z1=pSk (u)pk+1 (v) du dv = (k ; 1)! e;u du e;v dv =u6tu+v>t= e;tZt (u)k;10t;u0k ;t(t) e :du=(k ; 1)!k!', % ft t > 0g % t ; s t;s 0 6 s < t.
2 (P (t1 = k1 t2 ; t1 = k2 : : : tn ; tn;1 = kn) =Ynj =1qkj ((tj ; tj;1)))(8.14)1476vtt-u. 8.3 n > 2, 0 = t0 6 t1 < : : : < tn k1 k2 : : : kn > 0, 8 k e;> > 0 k = 0 1 : : : <qk () = > k0! < 0 k = 0 1 : : : : k = 0 k = 0 1 : : : :C, (8.14), P (t2 ; t1 = k2) =1Xk1 =0P (t1 = k1 t2 ; t1 = k2) == qk2 ((t2 ; t1))1Xk1 =0qk1 (t1) = qk2 ((t2 ; t1)):*, (8.14). = A , ) (8.14). @ A = ft1 = k1 t2 = k1 + k2 : : : tn = k1 + : : : + kn g. Dk1 = : : : kn = 0, P (A) = P (1 > tn) = e;tn = e;t1 e;(t2;t1 ) : : :e;(tn;tn;1) (8.14) .
9,= %,% % n.+ 2 6 m 6 n k1 = 0, : : : , km;1 = 0, km > 1, kj > 0, m < j 6 n.@A = ftm;1 < 1 6 tm Skm 6 tm Skm +1 > tm : : : Skm +:::+kn 6 tn Skm+:::+kn +1 > tng P (A) = E(E(1 A j 1)). + (7.13), E(1 A j 1 = x) == E1 ftm;1 < x 6 tm x + 2 + : : : + km 6 tm x + 2 + : : : + km +1 > tm : : : x + 2 + : : : + km +:::+kn 6 tn x + 2 + : : : + km +:::+kn +1 > tng == 1 ftm;1 < x 6 tmg P (Skm ;1 6 tm ; x Skm > tm ; x : : : Skm+:::+kn ;1 6 tn ; x Skm +:::+kn > tn ; x):(8.15)4 , 2 3 : : : , 1 2 : : : ( , fS~k g, S~k = 2 + : : : + k+1 , , fSk g).
C, , E(1 A j 1 = x) = 1 ftm;1<x6tm g148 P (tm ;x = km ; 1 tm+1;x ; tm ;x = km+1 : : : tn ;x ; tm;1 ;x = kn ) == 1 ftm;1<x6tm gqkm;1((tm ; x))Ynj =m+1qkj (O(tj ; x) ; (tj;1 ; x)]): (8.16) (8.16) tm;x x > tm. = x (8.15) . +" , 1 ftm;1 <x6tm g tm;1 < x 6 tm. @ ,P (A) = E1 ftm;1 <16tm gqkm ;1((tm ; 1))=Ztmtm;1e;x ((t(mk ;;x))1)!km ;1;tm= (ke ; 1)! mZtmtm;1m((tmYnj =m+1ne;(tm;x)dx; x))km;1dxqkj ((tj ; tj;1)) =Yj =m+1Ynj =m+1qkj ((tj ; tj;1)) =qkj ((tj ; tj;1)) =km= e;t1 e;(t2;t1 ) : : : e;(tm;tm;1 ) ((tm ; tm;1))km !Ynj =m+1qkj ((tj ; tj;1)):*, (8.14) k1 = : : : = km;1 = 0, km > 1, kj > 0, m < j 6 n(2 6 m 6 n).
+ k1 > 1. @ 0 6 x 6 t1 P (1 A j Sk1 = x) == E1 f0 6 x 6 t1 x + k1 +1 > t1 x + k1 +1 + : : : + k1 +k2 6 t2x + k1 +1 + : : : + k1 +k2 +1 > t2 : : : x + k1 +1 + : : : + k1 +:::+kn 6 tnx + k1 +1 + : : : + k1 +:::+kn +1 > tng == E1 f0 6 x 6 t1 1 > t1 ; x Sk2 6 t2 ; xSk2 +1 > t2 ; x : : : Sk2+:::+kn 6 tn ; x Sk2+:::+k1 +1 > tn ; xg == 1 f0 6 x 6 t1gP (t1;x = 0 t2;x = k2 : : : tn;x = k2 + : : : + kn ) == 1 f0 6 x 6 t1gP (t1;x = 0 t2;x ; t1;x = k2 : : : tn;x ; tn;1;x = kn ) =Y= 1 f0 6 x 6 t1ge;(t1;x) qknj ((tjj =2; tj;1)):(8.17)+ (8.17) , k1 = 0.@ , (8.13)E1 f06Sk1 6t1 ge;(t1;Sk1 ) =Zt (x)k ;1;x e;(t ;x) dx =e(k1 ; 1)!0Zt (x)k ;11111= e;t10dx= (t1) e;t1 :(k1 ; 1)!k1!1k1* (8.17) (8.16) (8.14) k1 > 1. 2C % T = O0 1) ( T = f^k k = 0 1 : : : g, ^ > 0), .
. , 149s t s + h t + h 2 T (0 6 s 6 t), x 2 X , B 2 BP (s x t B ) = P (s + h x t + h B ):(8.18)+ " t ; s, P (s x s + t B ), P (x t B ), t 2 T . C pij (t) = pij (s s + t), s t 2 T , P (t) | , ) pij (t).3 , " (8.4) P (s + t) = P (s)P (t) s t 2 T:(8.19)* , , '% # P (t), t 2 T , . . , i j 2 X t 2 Tpij (t) > 0Xjpij (t) = 1 pij (0) = ij(8.20)( (8.20) , P (0) = I | ). ', '% &%, - < % = %% (., ., . 7.25).
T = O0 1), T " .A 8.6 (- %). j0 2 X -h > 0pij0 (h) > 8i 2 X :(8.21) ( ) # lim p (t) = pj t!1 ijt>0i j )(8.22)jpij (t) ; pj j 6 (1 ; )t=h](8.23)O] | % .( " , &,= -%, ,<, & 7&%8, % %%.2 = t 2 Tmj (t) = infi pij (t) Mj (t) = sup pij (t):i=, mj (t) 6 pij (t) 6 Mj (t) i j 2 X t 2 T . +, mj (t) % Mj (t) & t ! 1 Mj (t) ; mj (t) ! 0 t ! 1. @ (8.22) .C s t 2 T , (8.19), (8.20),mj (s + t) = infiXMj (s + t) = supi150pik (s)pkj (t) > mj (t) infikXkXiXkpik (s)pkj (t) 6 Mj (t) suppik (s) = mj (t)kpik (s) = Mj (t):C, t h t ; h 2 T , (8.19), Mj (t) ; mj (t) = sup pij (t) + sup(;prj (t)) == sup(pij (t) ; prj (t)) = supir= supirX+kXriirk(pik (h) ; prk (h))pkj (t ; h) =(pik (h) ; prk (h))pkj (t ; h) +6 sup Mj (t ; h)irX+kX;k(pik (h) ; prk (h))pkj (t ; h) 6(pik (h) ; prk (h)) + mj (t ; h)X;k(pik (h) ; prk (h)) PP + k, pik (h) ; prk (h) > 0, ; | k,kkPP p (h) ; p (h) < 0.
+ p (h) = p (h) = 1, ikrkX+kk(pik (h) ; prk (h)) +X;kikk(pik (h) ; prk (h)) = 0:+"Mj (t) ; mj (t) 6 (Mj (t ; h) ; mj (t ; h)) supir@ , j0 X+k(pik (h) ; prk (h)) 6rkX+(pik (h) ; prk (h)):P+, (8.22)X+kpik (h) 6 1 ; pij0 (h) 6 1 ; P j0 + , , (8.22), kX+X+0,k(pik (h) ; prk (h)) 6kpik (h) ; prj0 (h) 6 1 ; :Mj (t) ; mj (t) 6 (1 ; )(Mj (t ; h) ; mj (t ; h)):(8.24)C Ot=h] , Mj (u) ; mj (u) 6 1, u = t ; Ot=h]h, (8.24) (8.23). 2% 8.7. # i j 2 X (8.22).
j 2 X )lim p (t) = pj (8.25)t!1 j pj (t) = P (Xt= j ), (8.21), jpj (t) ; pj j 6 (1 ; )t=h]:P2 + pj (t) = pi(0)pij (t). +" i3 j 2 Xpj (t) ; pj =Xipi (0)(pij (t) ; pj ) ! 0 t ! 1:151D (8.21), j XXjpj (t) ; pj = pi(0)(pij (t) ; p) 6 (1 ; )t=h] pi(0) = (1 ; )t=h]: 2jii% 8.8. # i j 2 X t 2 TXpj =iipi (s)pij (t) > slim!1P*, pj > pi pij (t). C, ipj > -Xi(8.26)P (t) ( P (t)).2 (8.25) j 2 X N 2 NX(8.22).pi pij (t). . p %pj = slimp (s + t) = slim!1 j!1Xi6Npi(s)pij (t) =Xi6Npi pij (t):pi pij (t)(8.27)P j t > 0.
0 (8.22) N i pj =j 6NP p (t) 6 1, "= tlimij!1j 6NXjpj 6 1:*, (8.27) , Xjpj >XXjipi pij (t) =(8.28)X Xipijpij (t) =Xipi :+ . 2% 8.9. # i j 2 X (8.22). P pj = 1,j. . pj # , , pj = 0, . . pj = 0.jP2 DP p 6= 0 ( (8.28)), p (0) = p= P p,jijijjpij (t),i 2 X , 8.1. @ 8.8 t j 2 XP pp (t)i ijXpjiPPpj (t) = pi (0)pij (t) =(8.29)pj = pj = pj (0):ij0 (8.25) pj = pj (0).
0,152jP p = 1. 2jj@ " " ) .+ fXt t 2 T Rg % ( % % ), n 2 N, t1 : : : tn 2 T h 2 R, t1 + h : : : tn + h 2 T L(Xh+t1 : : : Xh+tn ) = L(Xt1 : : : Xtn ):* , ' %, ( ) T Q " ), , , T = Z). 0A 8.10. % X = (Xt t > 0) % fpj g. Y = (Yt t > 0) { () %, #) , X , fpj g. Y { .2 + Y ) 8.1. * (8.7), Y X , 0 6 t1 < : : : < tn, Bk X , k = 1 : : : n, n 2 NP (Yt1 2 B1 : : : Ytn 2 Bn) =Xj1 2B1pj1 (t1)Xj2 2B2pj1 j2 (t1 t2) : : :Xjn 2Bnpjn;1 jn (tn;1 tn) pj (t) = P (Yt = j ) = pj j 2 X , t > 0 (8.29). = , pij (s t) = pij (t ; s) = pij (s + h t + h) i j 2 X , 0 6 s 6 t, h > ;s. 2+, , (8.22), (8.29), %..
% , , % % (.. P (Xt = j ) = P (X0 = j ) j 2 X t > 0). C , .C . & & %, -%< %< % & % % () 6, . 6.42). + (E F P ) k , k 2 N, , , .. ! 2 E0, P (E0) = 1, 0 < 1 < 2 < : : : n ! 1 n ! 1:(8.30)=# # (B !) =1Xk=1 : B(R+) E ! N+ = f0 1 : : : g f1g,k (!)(B ) B 2 B(R+) ! 2 E(8.31) R+ = O0 1), x { C. + ! 2 E0, , ( !) B(R+). 9 (8.30) , ( !) ! 2 E0 +, .. , R+, , , ( !) - B(R+) "!.E.
8.1. =7, (B ) 2 FjA B 2 B(R+), A {- N+, .. A N+.153M% -% % % . E ! = (t1 t2 : : : ), 0 < t1 < t2 < : : : tn ! 1 n ! 1. = fn n 2 Ng { E, .. n(!) = tn ! = (t1 t2 : : : ). E- F = fn n 2 Ng (. 1.3) (B !) =1Xk=1tk (B ) B 2 B(R+) ! 2 E:(8.32) " ( B ) M ' ! 2 E. C F , n n , (8.30).', , .. (ftg !) 6 1 t 2 R+ ! 2 E0: %Y = fY (t !) = ((0 t] !) t > 0 Y (0 !) = 0g:@Y (t !) =1Xk=11 (0t](k (!)) t > 0:(8.33)=, fY (t) > ng = fn 6 tg t 2 R+ n 2 N+ = f0 1 : : : g, "Y (t ) 2 FjA t 2 R+.E.
8.2. = m(B ) = E (B ) B 2 B(R+). 1 m() B(R+)? , ) N (t) t > 0, (m((s t]) = E(N (t);N (s)), 0 6 s < t < 1).E. 8.3. ( C6.10). C, (8.33) , Y ), Y R+ (. (2.35)).E. 8.4. + 7.7 (, T = N) k = 1+: : :+k ,k 2 N 4, . 1 (8.33), fk k 2 Ng { 4, ) (8.30)?E. 8.5. C, > 0 N (t)=t ! .. t ! 1.E. 8.6.
O ] + ft t > 0g (8.12).( t > 0 P+1t t = j=1 jP; t, .. "" ) Sk = kj=1 j , k 2 N. M " . + j { " . 0, , t?E. 8.7. ( 8.6). C, t > 0 t t +2 t+3 : : : " .154@ 8.4 % ( % >). + S(t ! ) { (E F P ),) ) ( ) t > 0, { ). + = 1. - Z (t !) = (t!)(!).E.
8.8. 1 - ?= % .X (t !) =1Xk=1(t ; k (!)) t > 0(8.34) : R ! R , k(k 2 N) (8.30) ( (8.34) t > 0 E0).E. 8.9. ( (8.34). C (8.34) t > 0 , 3?M %< %. + k k 2 N, ) (8.30), (E F P ) k k 2 N. (, k k , .. k ("") k .
= (C Q !) =1Xk=1(k(!)k(!))(C ) C 2 B(R+) B(R) ! 2 E:0 (8.31) (B !) = (B RQ !). #)# $%#Y (t DQ !) = ((0 t] DQ !) t > 0 D 2 B(R):@ 0Y (t DQ !) = P1 1 6 t 2 Dg(!)kk=1 f k ! 2 fY (t) = 0g ! 2 fY (t) > 1g Y (t) (8.33). C ,0Y (t D) = PY (t) 1 ( )D kk=1 fY (t) = 0g fY (t) > 1g:(8.35)E. 8.10. ( Y (t D), t > 0, D 2 B(R).E. 8.11. + Y (t) t > 0 { ) m. +fk gk2N { ... L(k ) = Q. D 2 B(R) , Q(D) > 0. C, " %"Y (t D), t > 0, (8.35), ) m()Q(D).155@ &% '< , ' % %, . 0 fTt t > 0g, ) B ( k k) , T0 = I Ts+t = TsTt s t > 0:(8.36)+ , sup kTtk 6 M < 1:t>0(8.37)C L : B ! B kLk = supfkLf k=kf k f 2 B f 6= 0g(8.38)( " B , .. , 7 ).
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