ПЕРЕСЕЧЕНИЕ СТАЦИОНАРНЫХ ГАУССОВЫХ ПОСЛЕДОВАТЕЛЬНОСТЕЙ С НЕСЛУЧАЙНЫМИ УРОВНЯМИ

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  • 3, 2009 7

    519.2

    . . ,. . , . . ,-

    -. - . - - .

    , , , - .

    [1, 2], [3]. () . . [4], [5, 6].

    u(t): [7]. u(t) , . , ( ).

    X

    x(t) , M[x(t)] = 0. u(t) [0, T), u0 = = u(0) > 0 ( ). p0 = p{x0 > u0} t = 0. k

    ( ) ( ) ( )1

    0,

    k

    k k k i ii

    f t p x u x u

    =

    = > (1)

    t0 i = k xk > u, xi < u, i = 0, 1, , k 1 ( = 1), f(tk) = p(k) tk = k = k. (1) : X U pi = p{xi > ui}, x(t) u(t) .

    ( ), (1) :

  • 3, 20098

    ( ) ( )1

    01 .

    k

    k k ii

    f t p p

    == (2)

    (1) , .

    u

    0 01( ) ( ) ,k

    kf t p p= { }0 1 ( / ),p p x u u = > = (2) pi = p0 :

    0 0 0 0 0 0

    0 0 0

    10 1

    ( ) ( ) exp( )exp( ), , ,...,

    kkf t p p p p t

    p kp k

    = = == = (3)

    0 = ln(1 p0)/p0; () .

    [1] . (3) n, T >> 0, 0 ; T ( F = 1/). u n :

    0 1 ( / );p u =

    0 0d d... ( ,..., ) ... ,u u

    k k ku

    p f x x x x

    =

    1 2 1, ,..., .k n= (4)

    k = n,

    1 1( ) ( ) ( )exp( ( ) ),k k n n nf t p P p P k n p =

    1, ,...,k n n= + (5)1

    0 1; ln( ) /

    n

    i n n ni

    P p p p

    == =

    u [3] , 2.

    (T 0, ): .

    1.

    ( ) exp( ) cos sin ;R

    = + (6)

    ( ) exp( )cos ;R = (7)

    1 8

    ( ) exp( ) cos sin ,

    / ,

    R

    = = = (8)

    = 1/2 , T = 40. , , = 1/8 (0 20), . . . (4) u = 1,5 . (5) . 1 [ 1, 2, 3 (6), (7), (8) ]. [8].

    u = 1,5 , (7), . 2, ; . 2, ( N = 5000).

    2. . 3

    . 1.

  • 3, 2009 9

    u1 = 0,5 ( 1) u2 = 1,0 ( 2) ( = 1/2), (6), = 1/8, = p. u1 u2 u(t), 2 t 5 (. 4, ). . 4, ( N = 10 000) ( 1): t 4,5 , t 5 3, t = 3. 2 3 1 2, . 3. .

    k . , . [6]. . .

    . 3. u(t) = 5 t/4

    ( = 1/8, = p) (6)(8), = 1/2 (. 5, ). (. 5, , 1), PLOT, (N = 10 000). (. 5, , 2)

    1

    01 1 ( ) ( ), ( )

    k

    k k i i ii

    f t p p p u

    == =

    (9)

    xiN(0, 1). 16 15 , ,t =

    16 13 , ,t = 16 11 , ,t = t = 15,80.

    ( = p) . 6, : 1 = 1/16 ( ); 2 3 = 1/2 = 2 ( ); 4 . ,

    . 2. () - ()

    . 3. , (N = 10 000)

    . 4. , - (); - - ()

  • 3, 200910

    : 15 76 , ,t = 15 69 ,t = t0 = 15,80;

    2 5 49 , , = 2 5 46 , = 2 = 5,66; 16 29 , ,t = 2 6 00 , . =

    (6) = 1/4 . 6, : = p/2 ( 1), = p ( 2), = 2p ( 3). ( 4) = p:

    15 96 , ,t = 2 5 73 , ; = 16 62 , ,t = 16 44 , ;t = 2 6 80 , , = 2 5 46 , . =

    , . (9)

    pi = j(ui).

    .

    [7]. A h(t) = exp(t)sint.

    0

    2

    d( ) ( ) ( )

    exp( ) cos sin ,

    R h t h t t

    = + =

    = +

    (10)

    [9]1 0 ;( ( )),

    ( )( ) ( ), ,U R t t T

    s ts t s t T t T

    > 2 2 1. . 7, T = 5, U = 1 = 3/4, = p, t0 = 5. T

    c

    d d( ) ( ) ( ) .t T t

    t t T

    w t s s +

    = (11)

    1, 2, 3 (. 7, ) T = 3, 4, 5; 4 ( ) t0 = 7,5. (. . 7, ) t0.

    (11) . , u(t) w(t) = |w(t)|, s(t) = |s(t)| >> w(t).

    4. u(t) = 5 2t (11), . 7, , 4, 0 t 5 ( T = 5 t0 = 0). = 1/8 [ X (10), 2 = 1] . 8, ( ).

    . 5. () - ()

    . 6. - () ()

  • 3, 2009 11

    ( ){ } ( )1

    0 01

    1 ( ) ( )k

    k k i ki

    p g t g t h

    ==

    ( ) ( ){ }1

    0 011( ) ( ) .

    k

    k k k ki

    p g t g t h

    ==

    g0(t) = u0(t)/(t) (t) = 1 ( X0), :

    ( )1

    10 0

    11( ) / ( ) ,

    k

    k k ii

    u t h g t

    =

    = ; (12)

    ( ){ }1

    10 0

    11

    2 3

    ( ) / ( ) ,

    , ,...;

    k

    k k ki

    u t h g t

    k

    =

    = =

    (12)

    u0(t1) = 1(1 h1), u0(t1) =

    1(h1) (12) (12).

    5. (12) (. 9, , 2), u(t) 4 (. 9, , 1), .

    u0(t) at + b, a = 1,3630, b = 4,5551 (. 9, , 3) pk (. 10) .

    . [3, 4, 7].

    . 8. () - ()

    . 7. () - ()

    0 2 5( , )f t + (. 8, ) (MATLAB SPLINE). 2 5185 ,tm = , 0 4200 ,t = ( N = 1000).

    u(t), , u0(t), .

    u0(t), X0 . pk hk g(t) = u(t)/ ( )

    . 9. () ()

  • 3, 200912

    6. T = 5 A = 6 F = = 1 / [6]. (. 11, , 1) 2 = 1 m = 3,1686 ( t0 = 2,8937). (. 11, , 2), , (. 11, , 3)

    3 20 0,0705 0 5331 0 1899 3 7330( ) , , , .g t t t t= +

    m =

    . 10.

    = 3,1686 (. 11, , 1) , 5. , (. 11, , 2).

    n- , . . , . .

    . ., . . 1. . .: . , 1977. 488 . . ., . .2. . // . 1998. . 43. 5. . 501523. . .3. . .: . , 1966. 678 . . .4. . .: , 1986. 296 . . .5. // . 2004. 2. . 1620. . .6. // . 2004. 3. . 1216. . .7. . .: . , 1962. 199 . . ., . . 8. // . 2006. 4. . 49. . .9. / . ., 2003. 139 .

    . 11. : ; -

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