# 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. : ; -