А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 4
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+, ht Xt ht(Xt) O0 1]1,t 2 T.E. 2.2. +, Zt = Oht(Xt)] (O0 1]1 d), O] .31+ . 2.1, 2.2 1.30, QJ (XJ BJ ) PJ (ZJ AJ ) J 2 F (T ), AU { Q- ZU = t2U Zt U T .I 2. C P (ZT AT ) PJ J 2 F (T ),%% %%, PU AU U 2 N (T ), %.. , % '% U T , PU R;UV1 = PV(2.23) V U ( V 2 F (T ) ), RUV { " "" ZU ZV .
C, 1.5 B 2 AT U 2 N (T ) BU 2 AU , B = R;TU1 BU . +" P (B ) = PU (BU ):(2.24)E. 2.3. +, " , .. B = R;TV1 BV BV 2 AV V 2 N (T ) PV (BV ) = PU (BU ) (2.23).E. 2.4. C, (2.24) AT , ) PJ J 2 F (T ).I 3. F %%, PU U 2 N (T ), %< <% (2.23).+ U = ft1 t2 : : : g, .. U 2 N (T ) . ZU dU (y z) =1Xk=12;k d(ytk ztk )(2.25) y = (yt t 2 U ), z = (zt t 2 U ), (2.25) , .. d( ) 6 1.E. 2.5.
+, (ZU dU ) { U 2 N (T ).E. 2.6. C, ZU (U 2 N (T )) U (2.25).E. 2.7. C, (ZU dU ) U 2 N (T ), -B(ZU ) AU .M C (ZU Q R) , (ZU dU ). + HU { C (ZU Q R), ) H , ) J = J (H ) 2 F (T ) hJ (ZJ dJ ) ( J dJ = PJ (2.1), t = d t 2 J ) , H (z()) = hJ (zJ ) zJ = zjJ = (z(t) t 2 J ):(2.26)0 HU C (ZU R), .. , HG 2 HU , H G 2 HU . ' OHU ] HU C (ZU Q R), sup-, ZU , ..
z y 2 ZU H 2 OHU ] , H (z) 6= H (y) ( ), " ( { :. (., ., O?, . 119])OHU ] = C (ZU Q R).324 HU FU (H ) =ZZJhJ (zJ )dPJ (2.27) H 2 HU (2.26). * (2.27) , hJ 2 B(ZJ )jB(R) B(ZJ ) = AJ . 2.7, , %.E. 2.8. C, (2.27) , .. RH (z()) = hI (zI ) I 2 F (T ) hI 2 C (ZI Q R), ZI hJ (zI )dPI (2.27).= FU , (2.27), HU . +" FU '% C (ZU Q R). + "FU (f ) > 0 f > 0 f 2 C (ZU R) FU (1 ZU ) = 1( f 2 HU " ). 0, + { ;(., ., O?, . 124])FU (f ) =ZZUfdPU f 2 C (ZU R)(2.28) PU { , &- - ZU .+ " - (., ., O?, .
122]) , % % %% &- &'% <% (. O?, . 123]). (, , , .I 4. ', % % PU U 2 N (T ), .+ V U 2 N (T ). @ HV HU . D f 2 HV , ) J 2 F (T ) fU 2 C (ZJ Q R) , f (zU ) = f (zV ) = fJ (zJ ) zU 2 ZU zV 2 ZV zJ = zU jJ = zV jJ : (2.27) (2.28) FU (f ) =FV (f ) =ZZUZZVf (zU )dPU =f (zV )dPV =ZZJZZJfJ (zJ )dPJ (2.29)fJ (zJ )dPJ :(2.30)+ 3 (. (2.10)), , (2.29) (2.30),ZZVf (zV )dPV =ZZVf (zV )dPU R;UV1 :(2.31)0 0 { , , (2.31) f 2 C (ZV Q R).
= 33E. 2.9. + Q1 Q2 { B(X ), X { -. C, ZXf (x)dQ1 =ZXf (x)dQ2(2.32) f : X ! R, Q1 = Q2 B(X ).E. 2.10. C Q BT , ) QJ J 2 F (T ).@ - . 2M % %% Xt, t 2 T , % O0 1]% % %%, %,% % 1.7 ' . 22, " Xt : : : " K , Cn 6= Bn 6= ,.. Q(Cn) = QJn (Bn) > 0 n 2 NQ , () = 0 -;10. + U = 1n=1 Jn ( ) Cn = UJn Bn , n 2 N.00@ Cn Cm , ;1 C 0 Cm = ;1 C 0 (n > m):Cn = TUnTU m- ,;1 C 0 = ;1 \1 C 0 = \1n=1Cn = \1n=1 TUnTU n=1 n0" \1n=1 Cn = . @ , UJn { U O0 1] ( (2.22)) O0 1]Jn ( Jn ).0, fCn0 gn2N { O0 1]U . (0 \1n=1 Cn 6= . + .@ 92.11. * (X B) , ) X , )) X , B = B(X ) ..
B ( ). , E { X , E { E , (E E ) { () , ) E " , X ). =,% 0 % %% & %% Xt t 2 T .+ (X B) (Y A) { . ' (X B) (Y A) , D 2 A , (X B) (D AjD ) ( "" 1.29, AjD = A \ D = fAD : A 2 A). - K = f0 1gN - B.E. 2.12. +, (K B), d (. (2.22)) B = B(K ). + " (K d) { (, ,(K d) { ). - , (K S B(K )S ) (K B(K )) S =6 .( % & %%.A 92.13 (., ., O?, .
98]). (X B) (Y B(Y )), (Y B(Y )) { . 8 (K B(K ))8 ,<<(X B) : (N A(N)) X,:((1 : : : jXj) A(1 : : : jXj)),34 A(M ) { - M , j j { . 4 %%, (X B ) { %% &%% X %% % %, % X { %, B = B(X ).* ) O?].@ % %% . , BT (. 1.5), C %, , , , .+ X = fXt t 2 T g, (E F P ) t 2 T ( ) ) (X ), .# T0 T , T0 { T ) N 2 F , P (N ) = 0 t 2 T ! 2 E n NX (t !) 2 \G3tG2JfX (s !) s 2 G \ T0g (2.33) OB ] B , J { (X ).A 92.14 (.
O?, . 128]). (X ) { , (T ) { . % X = fXt t 2 T g, (E F P ) t 2 T #) (X ) # $%#. " X { , $% X . (2.33) , O?, . 111], T = O0 1] X = R. = . O?]. ) . + (T A) { .0 X = fXt t 2 T g, (E F P ) ) t 2 T (X B), , (t !) 2 T E 7! Xt(!) 2 X(2.34) A FjB . - , X { , B = B(X ). J T { % '% ( ) A {- T , % X, , .A 92.15 (O?]). C2.14 % X T , ..
t 2 T # " > 0limP ((Xs Xt ) > ") = 0:(2.35)s!tX ) $%.E. 2.16. C, X { , (2.35) , .. F . 35E. 2.17. + , .. , -) .E. 2.18. C, , AjB- . ?E. 2.19. + X = fXt t 2 T g { : E ! T , 2 FjA.C, Y (!) = X (!)(!) FjB- .E. 2.20. +, , ) .E. 2.21. + C2.14 (T ). , ?@ & % , < % %-% . G ,%% 0.A 92.22 (. O?, . 124]). X = fX (t) t 2 Oa b]g { % , " > 0 C = C ( ") > 0EjX (t) ; X (s)j 6 C jt ; sj1+" s t 2 Oa b]: %X ) ..
$%.(2.36)E. 2.23. +, (2.36) " = 0, C2.22, ) , ( ).@ - ). M .(, U T (U T ), (T ) { , "-# S" T , t 2 U s 2 S" , (s t) 6 ". C , U B Os "] = ft 2 T : (s t) 6 "g s 2 S":(2.37)D U "-, "-# "- S"min(U ), jS"min(U )j , jV j " V . + N (" U ) = jS"min(U )j, . TH (" U ) = log N (" U )(2.38) "- U ( (2.38) 2).0 O?], (T ). + ) m 2 N a > 0 , U T , ) DU := supf(s t) : s t 2 U g < 1, " > 0N (" U ) 6 maxfa(DU =")m 1g:(2.39)+ ) b > 0, "- S"min(T )(2.40)jS"min(T ) \ B Ot 5"]j 6 b t 2 S"min(T ) B O cdot] (2.37).36A 92.24 (.
O?, . 134] ). X = fX (t) t 2 T g { .$. (X d) t 2 T , (E F P ) (T ), #) (2.39) (2.40). , 2 (0 1) > m= > 0E(d(X (t) X (s)) 6 ((t s)) s t 2 T:(2.41) ) $% Y = fY (t) t 2 T g, #) , , # 2 (0 ; m=) t0 2 Tlimsup d(Y (t) Y (s))=(t s) = 0 .., #0(st)6nE sup d(Y (t) Y (t0at2T$ ))o6 a(2DT ) =(2;m= ; 1)(2.42)(2.43)(2.39).@ 92.25.
D D { Rm Rm+1( Rq ), (2.39) (2.40) . +" C2.24 C2.22, .C , %& % & & % .. X, % '%,<, % %< <, < < ,< R+ = O0 1) %<, % (0) 6 1.A 92.26 (. O?]). X = fX (t) t 2 T g { .$. (X d) t 2 T , (E F P ) (T ), , M > 0!d(X(t)X(s))E6 M s t 2 T (s t) 6= 0(2.44)(s t)d(X (t) X (s)) = 0 .., (s t) = 0: Z(2.45)1Y(N (T "))d" < (2.46)+0 Y { $%, , (2.46) 0. .$.) .. $%.XE.
2.27. C2.26, X = fX (t),t 2 O0 1]mg t 2 T (X ) ) f R+ p > 1 E(d(X (t) X (s))p 6 f p(ks ; tk)(2.47)Z +1f (x;p=m)dx < +1(2.48).. +1, ) .. .37@ 92.28 (. O?]). 3%, (2.48) % %,, .. , ) , (2.48), , .. .C . . X = fX (t) t 2 T g , n 2 N t1 : : : tn 2 T (X (t1) : : : X (tn)) . 2 3.* C2.26 ) .A 92.29 (. O?]).
X = fX (t) t 2 T g { % T , (.(1.28)). Z p1log N (T ")d" < + +0 ) $%, #) .. .(2.49)+ " , % , %#) . 2 T . C , " " " "" . , - (T ), #), supZ1t2T 0j log (B Ot "])j1=2d" < 1(2.50) B O ] (2.37).+ (B Ot "]) = 1 " > DT , (2.50) 0 DT , (2.50) " 0.E. 2.30. (. O?, . 182]) C, ) ). 0) q > 1 C = fCk k 2 Ng) T - , sup DC 6 2q;k(2.51)C 2Cksup1Xt2T k=1q;k j log (Ck (t))j1=2d" < 1(2.52) Ck (t) { () Ck , t.A 92.31 (. O?, .
193]). X = fX (t) t 2 T g { .$. (T ). - )#) , limsup #0Zj log (B Ot "])j1=2d" = 0:(2.53)t2T 0+, $% #) (..).38C J, {@.', ) , ) Z !1Y(B Ot "]) d"+0 Y , ., ., O?]. 0 O?], . ( , , ) .39 3. " .4 # . # .% . 5#- . 3 ! ( ). &# !. $1 6. $ # . 70 1], 70 1). 4 !.4 , 2, % %' &= . (, Y Rn ( Y N (a C )), 2 Rn'Y () = exp i(a ) ; 21 (C )XnnX1= exp i ak k ;ckm k m2k=1km=1(3.1) a 2 Rn, C = (ckm )nkm=1 | ". =% C ( C > 0), (C ) > 0 2 Rn.
3 , : C > 0 C = C ( ) :nXkl=1ckl zk zZl > 0(3.2) z1 : : : zn ( z ).C (., ., O?, . 175]), C a 2 Rn, , ) (3.1), . . Y . D C > 0, . . (C ) > 0 6= 0 2 Rn, Y ( 3)PY (x) = (2); n2 jC j; 21 expf;(C ;1(x ; a) x ; a)g jC j | C .C Y N (a C ) % a C :ak = EYk ckm = cov(Yk Ym ) k m = 1 : : : n (n > 1):(3.3)C . . X (t), T (E F P ), , .-.. . C , (X (t1) : : : X (tn)) n 2 N t1 : : : tn 2 T ( " (t1 : : : tn), ) , , , ) " , ).
3.1. : Y =n(Y1 : : : Yn) Rn P Y , ( Y ) =k kk=1 = (1 : : : n) 2 Rn.402 + Y N (a C ). @ 2 R (3.1)Eei(Y )= Eei(Y )= exp i(a ) ; 21 (C ) = exp i(a ) ; 12 (C ) 2 . . ( Y ) N ((a ) (C )) ( , = (1 : : : n )).=. + ( Y ) N (a 2). @ (. (3.3) n = 1)a = E( Y ) =Xn2 = D( Y ) = Dk Yk =k=1nXkm=1nXk=1k EYk = ( EY )k m cov(Yk Ym ) =(3.4)nXkm=1k m ckm = (C )( j = 1 k = 0 k 6= j , , Yj { , " EYj2 < 1, j = 1 : : : n). 0, 2R122i(Y)(3.5)Ee= exp ia ; 2 :+ (3.5) = 1 a 2 (3.4), (3.1). 2C r(s t), T T , %, (r(tk tm))nkm=1 > 0 n 2 N t1 : : : tn 2 T .A 3.2.
$% a(t), t 2 T , % $% r(s t),s t 2 T . T (E F P ) ) .$. X (t !) 2 R, a(t) = EX (t) r(s t) = cov(X (s) X (t)) s t 2 T .2 C n 2 N = (t1 : : : tn) 2 T n (Rn B(Rn)) Q , ) . . (3.1), a =(a(t1) : : : a(tn)) ckm = r(tk tm)( ). @ (a) (b) 2.9. 2@ , .., %, (RT BT ), % 7: . =, . . X (t), ) t 2 T , n 2 N, t1 : : : tn 2 T 1 : : : n 2 R nXkm=1cov(X (tk ) X (tm))k m = covXnk=1k X (tk )nXm=1m X (tm) > 0:=, cov(X (s) X (t)) = cov(X (t) X (s)).
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