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Дата изменения: Mon Sep 6 15:20:44 2010
Дата индексирования: Sun Apr 10 07:41:58 2016
Кодировка:

In = hn (u, x) =

-

gi (u)P{

k,n-i-k+1

< hn (u, x)}du, (1 - F (u)),

F u+

x - F (u) nf ( )
(n)

gi (u)- Xi , r,s - , - (r, s). - r,s , r , s . sr,s r , r - (1, r). hn (u, x) u = F -1 (p) hn (u, x) = In
-

x 1 . +o n(1 - p) n

gi (u)P{k < x}du = P{k < x}.

21. X1 , X2 , ... 2 , EX1 = 0, EX1 = 1, i = 1, 2, ..., cov (Xi , Xj ) = > 0 i = j. ,
( lim P{Xnn) < n

2(1 - ) ln n + x} = (x/ ).

( Xnn)

Tn = - 2(1 - ) ln n . . Y , Y1 , Y2 , ..., EY = EYi = 0, EY 2 = , EYi2 = 1 - , Xi = Y + Yi , i = 1, 2, .... , Y (. 1.1).
(n) n

-

2(1 - ) ln n - 0 n .

P

22. (Xi , Yi ), i = 1, ..., n, - (n) (n) . X1 X2 ( ... Xnn) (n) (n) (n) (n) ( Y<1> , Y<2> , ..., Y . , Xi = Xr n) , Yi = Y . (Xi , Yi ) N2 (0, 0, 1, 1, ), i = 1, ..., n. , lim P{Y n
(n)

< 2 ln n + x} = ((1 - 2 )-1/2 x),

(2 ln n)-

1/2

Y

(n)

-

P

n . Tn = (n) Y - 2 ln n . (n) n = (2 ln n)-1/2 Y . 25

. Yi = Xi + 1 - 2 Zi , i = 1, ..., n,

Xi Zi , Y
(n)

= X

(n) n

+

1 - 2 Z

(n)

,

X

(n) n

-

2 ln n - 0,

P

Z

(n)

N (0, 1).

23. X1 , ..., Xn .. F (x), 0 b.

.. () 0 0 bn , Tn = (Xnn) - bn )/an n . .. Gm (x) Tn y = 1 - ln Gm (x) y = x .

25. X1 , ..., Xn - .. F (x),
n

Rn =
i=1

X1 / ln n,

(i)

X

(i) 1

= min {Xj }, i = 1, ..., n.
1j i

,

, F (x) = x , 1,
n

0

x 0, 0 < x < 1, x 1,

1) lim ERn = 1; 2)cov (X1 , X
(n) (n+k) 1

)=

n , k = 1, 2, ... (n + 1)(n + k + 1)(n + k + 2) 26

3) nlim DRn = 0. , .. F (x), F (0) = 0, Rn - I
P

n ,

I = lim

t+0

t . F (t)

, I . Rn , n > 1. 25.1. F (x) = 1 - e
-x

, x > 0;

25.2. 2 F (x) = 1 - e-x , x > 0. .
n

lim n - .

k=1

1 - ln n = = 0, 577..., k

26. X1 , ..., Xn , P{Xk < x} = F (k x), k > 0, k = 1, 2, ...n, F = 0, F = , r > 0
t+0

lim F (tx)/F (t) = xr ,

k=1

r = , k

n

n = o
k=1

r k

n .

, an > 0 ,
n

lim P{X

(n) 1

< an x} = 1 - exp(-xr ), x > 0,

. .. -x , x > 0, k = k , k = 1, 2, ..., F (x) = 1 - e (n) Tn = X1 /an . 27. X1 , ..., Xn , P{Xk < x} = F (k x), k > 0, k = 1, ..., n, F = , x F (x) 1 - ax- , a, > 0,
- n n

= o(
k=1

- ) n , k

k=1

- = . k

, an > 0,
n ( lim P{Xnn) < an x} = exp(-x -

), x > 0.

1 F (x) = arctg x + 1 , - < x < 2 1 ( , k = k , k = 1, 2, ..., Tn = Xnn) /an .

27

28. 1). X1 , X2 , ... - , k 1 n P{Xij bn + an x,
1i1
1 j k}
n

exp(-k x), an > 0 b . ,
n

- - < x < .

( lim P{Xnn) < bn + an x} = 1/(1 + exp(-x)),

2). P{Yi < y } = 1 - exp(-y ), y > 0, i = 1, 2, ..., P{Z < z } = 1 - exp(- exp(-z )), - < z < , Z, Y1 , Y2 , ... . Xi = Yi - Z, i = 1, 2, .... ,
n

lim P{X

(n) n

< ln n + x} = 1/(1 + exp(-x)), - < x < ,

) ; ) .1). 3). .2) ,
n

lim P{nX

(n) 1

< x + ln n!} = 1 - exp(-x), x > 0.

Tn = (n) - ln n Kn = nX1 - ln n! . . 2).
( Xnn) ( P{Xnn) < ln n + x} = P{ n

(Yi < ln n + x + Z )} = E(1 - exp(- ln n - x - Z ))n

i=1

E exp(- exp(-x - Z )). 29. Z, Y1 , Y2 , ... - , P (0, 1), Xi = Yi - Z, i = 1, 2, .... , 1). nlim P{nX
(n) 1

< x - ln n} = 1 - exp(-x), x > 0; - < x < .

( 2). nlim P{Xnn) < x + ln n} = exp(x)(1 - exp(- exp(-x))),

Tn = (n) ( nX1 + ln n Kn = Xnn) - ln n . . 2).
( P{Xnn) < ln n + x} = P{ n

(Yi < ln n + x + Z )} = E(1 - exp(- ln n - x - Z ))n

i=1

E exp(- exp(-x - Z )). 28

30. (X1 , Y1 ), (X2 , Y2 ), ... .. F (x, y ) f (x, y ). Z2n = max(X1 , Y1 , ..., Xn , Yn ). an > 0 bn , n Tn = ( Z
2n

- bn )/an

.. Tn . n = Z2n /bn . 30.1. f (x, y ) = e- 30.2. f (x, y ) = 30.3. f (x, y ) = exp(-x - y + xy )((1 - x)(1 - y ) + ), 30.4. F (x, y ) = 1 - exp(-x) - exp(-y ) + exp(- x2 + y 2 ), x, y > 0. x, y > 0, -1 0. 2ex ey , (ex + ey - 1)3 x, y > 0.
x-y

(1 + (1 - 2e-x )(1 - 2e-y )),

x, y > 0,

|| 1.

30.5. 1 f (x, y ) = (1 (x, y ) + 2 (x, y )), 2 (x, y ) - N2 (0, 0, 1, 1, ). . , (x, x) 1 x 2 (x) : an = (2 ln n)-1/2 , (an x + bn ) 1 - bn = (2 ln n)1/2 - e-x n ln ln n + ln 4 , 2(2 ln n)1/2

n .

29

(x) - (0, 1), - N2 (0, 0, 1, 1, ).

(x, y )

31. (X1 , Y1 ), (X2 , Y2 ), ... .. F (x) = P{X1 < x} = P{Y1 < x} .. F (x, y ) = P{X1 < x, Y1 < y }. , an > 0 bn , n nF (bn + an x) ex , W ,
n 2n

nF (bn + an x, bn + an x) 0.

= min{X1 , Y1 , ..., Xn , Yn }.

lim P{W

2n

< bn + an x} = 1 - exp(-2ex ),

|x | < .

Tn = (W2n -bn )/an . 1 + e-x + e-y F (x, y ) = , |x| < , |y | < . (1 + e-x )2 (1 + e-y )2 32. X1 , X2 , ... . X1 F (x) f (x) .. , (Xi , Xi+1 ) F (x, y ) f (x, y ) .. , G(x, y ) = P{Xi x, Xi+1 y }, i = 1, 2, ... , ,
( P{Xnn) < x} = (F (x, x))n-1 /F n-2

(x),

P{X1

(n)

x} = (G(x, x))n-1 /(1 - F (x))n-2 .

32.1. X1 , X2 , ..., (Xi , Xi+1 ) N2 (0, 0, 1, 1, ), || < 1, yn (t) t 1 - (yn (t)) = , t > 0. n ,
n ( lim P{(Xnn) - bn )/an < x} = exp(-e-x ),

an = (2 ln n)-1/2 ,

bn = (2 ln n)1/2 -

ln ln n + ln 4 , 2(2 ln n)1/2

an bn , ( (Xnn) - bn )/an n . 30

, , ,
n

nX

n(S - 1) n

( lim P{((Xnn) - X )/S - bn )/an < x} = exp(-e-x ),

n

lim P{(X

(n) n

- X )/S < yn (t)} = e-t .

( ( P{Xnn) < yn (t)} P{(Xnn) - X )/S < yn (t)} -t e . . 1 - (x) (x)/x, x . 32.2. , f (x) = e-x , x, y > 0, || 1, ,
n

1 - (x, x) 2(x)/x,

x > 0,

f (x, y ) = e

-x-y

(1 + (1 - 2e-x )(1 - 2e-y )),

yn (t) = ln(n/t),

t > 0.
(n) n

( lim P{Xnn) - ln n < x} = exp(-e-x ), lim P{X n

< yn (t)} = e-t ,

n

lim P{nX

(n) 1

< x} = 1 - e-x , x > 0. n(X -

, n (n) X1 - 1) , ,
n ( lim P{(Xnn) - X1 )/(X - X (n) (n) 1

) < ln n + x} = exp(-e-x ).

Tn = nX
(n) 1 ( ( , Ln = Xnn) - ln n Hn = (Xnn) - X1 )/(X - X1 ) - ln n (n) (n)

. . F (x, x) 1 - 2f (x) x , G(x, x) 1 - 2x x 0.

32.3. , f (x) = C e-x , x > 0, (1 + x) C = yn (t) = f (x, y ) = C exp(-x - y - xy ),
0

x, y > 0, 0,

e

-u

-1

/(1 + u)du

, > 0, = 0, t > 0.

ln(n/t) - ln ln(n/t) - ln(/C ) , ln(n/t) , 31

,
n ( lim P{Xnn) - yn (1) < x} = exp(-e-x ), n

lim P{X

(n) n

< yn (t)} = e-t ,

n

lim P{nC X1

(n)

< x} = 1 - e-x , x > 0.

, n (n) n(X - X1 - (C - 1)/) , ,
(C - 1)(X (n) - X (n) ) 1 n - yn (1) < x = exp(-e-x ). lim P (n) n (X - X1 )

= 1, C1 = 1, 677 Tn = nC1 X1
(n)

Hn = (C1 - 1)(X

(n) n

- X1 )/(X - X1 ) - yn (1)

(n)

(n)

. . F (x) 1 - f (x), F (x, x) 1 - 2f (x) x , F (x) C x, G(x, x) 1 - 2C x x 0. 32.4. 1 Xn = Xn-1 + n , r r 2- , { n }- , {0, 1/r, 2/r, ..., (r - 1)/r},
n

X ,

n-1

, Xn U (0, 1). exp( 1
r-1 r

n

lim P{(X

(n) n

- 1)n < x} =

x) , , x) , ,

x 0, x > 0, x 0, x < 0.

n

lim P{nX1

(n)

< x} =

1 - exp(- 0

r -1 r

Tn = (n) - 1)n Hn = nX1 . Xn , n = 1, 2, ... .
( (Xnn)

F (x, x) =

x/r (1 +

r-1 r

)x -

r-1 r

x (0; 1 ], r x ( r-1 ; 1], r

32

G(x, x) = F (x, x) + 1 - 2F (x). 32.5. X1 , X2 , ..., F (x, y ) = P{Xi < x, Xi+1 < y }, i = 1, 2, .... an > 0, cn > 0, bn , dn ( .. H (x) L(x), Tn = (Xnn) - bn )/an (n) Kn = (X1 - dn )/cn n .. H (x) L(x) . a).F (x, y ) = 1 - e
-x

-e

-y

+ (ex + ey - 1)-1 ,

x, y > 0; x, y > 0; x, y > 0; |x| < , |y | <

b).F (x, y ) = 1 - e-x - e-y + exp(- x2 + y 2 ), c).F (x, y ) = 1 - d).F (x, y ) = 1-e x
-x

-

1-e y

-y

+

1 - e-x-y , (x + y )

1 xy (arctan + arctan x + arctan y + /2), 2 1 + x2 + y 2

( ). T Kn .

n

33. N (n), X1 , X2 , ... , P{Xi < x} = F (x), i = 1, 2, ..., N (n)- , an > 0 bn ,
n

lim n(1 - F (bn + an x)) = h(x),

0 < h(x) < ,

N (n)/n -

P

n ,

- . ,
n

lim P{(XN XN

(N (n)) (n)

- bn )/an < x} = exp(- h(x)), }.

(N (n)) (n)

= max{X1 , X2 , ..., X

N (n)

(N (n)) (XN (n)

Tn = - bn )/an . 33.1.

1 - e-x , x > 0, x Zn b(n, p), F (x) = 1 - 33.2. F (x) =

N (n) = Zn + 1, 0 0,
P

N (n) - , N (n)/n - n , - , XN
(N (n)) (n)

= max{X1 , X2 , ..., X

N (n)

},

, n
(N (n)) N (n) (ln n)1/

X

-

P

1 b
1/

.

F (x) = 1 - exp(-4x2 ), x > 0, N (n) = Zn + 1, Zn (n). (N (n)) n = XN (n) / ln n, n > 1. 35. N , X1 , ..., Xn , X1 , ..., Xn .. F (x) = 1 - k k+x

,

k , > 0,

x > 0,

N - , MN = max{X1 , ..., XN }. an > 0 bn ,
n

lim P{MN < an x + bn } = exp(-

1 ), x

x > 0.

(MN - bn )/an . 35.1. N (n); 35.2. N Ї(n, p). b 36. N , X1 , ..., Xn , X1 , ..., Xn , F (x) = ( P{X1 < x}, Xnn) = max{X1 , ..., Xn }, N - , Nd - n n , P{ > 0} = 1.

( n n(1 - F (Xnn) )) Tn = n(1 - (N ) F (XN )). .. Tn ..

34

36.1. U (1; 2); 36.2. 1 N G( n ); 36.3. N Ї(2, n ). b1 38. N , X1 , ..., Xn , X1 , ..., Xn - .. F (x), N - d , N/n - n , P{ > 0} = 1. an > 0,
n

lim P{X[
(N ) [N ] 2

(n) n ] 2

< an x} = (x).

,
n (N )

lim P{X

< an x} = E(x ),

X[ N ] - [ N ], 2 2 N . Tn = (N ) X[ N ] /an . 2 . P{Xk
(n)

< x} = P{Zn k },

Z

n

b(n, F (x)),

- . 38.1. F (x) = 1 - 1e 2 1x e 2
-x

, x 0, , x < 0, 0 < x < 1;
1/2

P{ < x} = x 38.2. F (x) = (x), 38.3.

3/2

,

P{ < x} = x

,

0 < x < 1;

1 N G( n ),

F (x) =

1 1 arctan x + , 2

- < x < ;

38.4. N Ї(2, n ), b1 X1 . (-1, 1); 38.5. P{ < x} = x
1/3

,

0 < x < 1,

f (x) = F (x) = 35

1 exp(-|x|/2), 4

- < x < .

39. X1 , ..., Xn .. (n) F (x). Xk (n) (n) ( X1 X2 ... Xnn) , k = [n ], 0 < < 1, n . , (n) P{Xk < x} = P{Zn k }, Zn b(n, F (x)). (n) an > 0 bn , (Xk - bn )/an n . , an bn k k+ k F (bn ) = , F (an + bn ) = . n n 39.1. , limn

F (x) = 1 - e-x , P{X
(n) k

x > 0,

> 0.

< an x + bn } = (x), an bn ,

-(1 - ) ln n ^ n = (n) ln Xk . . ,
P{X1 < x} = 1 - e-x ,

x > 0.

39.2. F (x) = 1 - ,
n

1 , x

x > 1,

> 0.

lim P{Xk

(n)

< an x + bn } = (x), 1 (X
(n) k

an bn , ^ n = - 1)n1-

. . , 1 P{X1 < x} = 1 - , x x > 1.

39.1 39.2 ^ n . 40. X1 , ..., Xn .. F (x), F (x) = (ln |x|)-2(1- 1
)/

, x -e, , x > -e,

0 < < 1.

36

[n ]. an , F (an ) = , lim P{X n
(n) [n ]

1 n
1-

,

< a n x} =

(- 1

2(1-)

ln |x|) , x < 0, , x 0.

Tn = X[n ] /an . 41. X1 .. F (x), 1 F (x) = 1 - , x x an > 0 (n) Tn = (X[ln n] - bn ) ln n , n , X2 , ... > 1. bn , n /an
(n)

, an bn F (bn ) =
n

F (an + bn ) =

ln n 1 1+ . n ln n

lim P{Tn < x}.

(n) n = X[ln n] /bn , n 2. . P{X[ln
(n) n]

< an x + bn } = P{Zn [ln n]},

Zn b(n, F (an x + bn )). - . 42. X1 , X2 , ... .. F (x) = C , ln x 1 0 (n)

Xk , 2 , k = [n ], 0 < < 1. , 3 < < 1,
n

k, n

lim P{X

(n) [n ]

< an x + bn } = (x). 37

an bn k F (bn ) = , n k+ k F (an + bn ) = . n

= 2 , 3 Tn = (X[n ] - bn )/an . . 5 Tn = 6 = 2 . 3 43. N , X1 , ..., Xn , X1 , ..., Xn , EX1 = a, DX1 = ,
2 n (n)

0 < < ,

2

Xn =
i=1

Xi /n,

N - , Nd - n 43.1. , (X n - a) N x P < x - E n . n , P{ > 0} = 1.

1 P{ < x} = 1 - , x > 1. x Tn = (X n - a) N /. 43.2. , P
N i=1

x (Xi - a) < x - E n P{ < x} = exp -

n .

1 , x > 0. x
N

Tn =
i=1

(Xi - a)/( n).

43.3. , (X N - a) n < x - E(x ) n . P 38

a). P{ < x} = - e-x , 1 x > 0; b). P{ < x} = x, 0 < x < 1; -x c). P{ < x} = 1 - (1 + x)e , x > 0. Tn = (X N - a) n/. 43.1 - 43.3 Tn . 44. X1 , ..., Xn , P{X1 = k } = pk , k = 0, 1, ..., F (x) = P{X1 x}, Mn = max{X1 , ..., Xn }. (Mn -bn )/an n an > 0 bn . 1.
k

lim

P{X1 = k } = lim 1 - F (k - 1) k

pk j =k

pj

= 0,

an > 0 bn , (Mn - bn )/an n . 2. an > 0 bn ,
n

lim n{1 - F (an x + bn )} = - ln H (x), lim P{Mn an x + bn } = H (x).

n

44.1. X1 , ..., Xn {1, 2, ..., N }. 1). bn )/an 2). n , an > 0 bn , (Mn - n . Gn (x) = P{Mn nx + n n}. , N [n n] , lim Gn (x) = ex , x < 0.
n

y = ln Gn (x) y = x. n = Mn /(n n). 44.2. {Ni }, {Xi }, i = 1, 2, ... , Ni e (n2 ), Xi {1, 2, ..., Ni }, i = 1, 2, ... 39

Gn (x) = P{Mn nx + n2 }. ,
n

lim Gn (x) = ex ,

x < 0.

y = ln Gn (x) y = x. n = Mn /n2 . . n (Ni - n2 )/n = i + op (1), i N (0, 1), Ni = n2 + ni + op (n), i = 1, 2, ..., n. 44.3. X1 , ..., Xn G(p), P{X1 = k } = p(1 - p)k , k = 0, 1, ..., 0 0, bn , (Mn - bn )/an n . 2). Gn (x) = P{Mn nx + n ln n}. , 1 p n
n

n , |x| < .

lim Gn (x) = exp(-e-x ), y = - ln(- ln Gn (x)) y = x. n = Mn /( n ln n). 44.4. X1 , ..., Xn , P{X1 = k } = (k + 1)p2 (1 - p)k , k = 0, 1, ..., 0 0 bn , (Mn - bn )/an n . 2). Gn (x) = P{Mn n , p n-
n 1/4 1/4

(x + ln n + ln ln n)}.

n ,

lim Gn (x) = exp(-e-x ), |x| < .

y = - ln(- ln Gn (x)) y = x. n = Mn /(n1/4 ln n). 40

44.5. X1 , ..., Xn , P{X1 = k } = k 2 (1 - p)k /C (p),

k = 1, 2, ...,

C (p) =
k=1

k 2 (1 - p)k ,

0 0 bn , (Mn - bn )/an n . 2). Gn (x) = P{Mn ln2 n(x + ln((n ln2 n)/2))}. , p
n

1 ln2 n

n , |x| < .

lim Gn (x) = exp(-e-x ),

y = - ln(- ln Gn (x)) y = x. n = Mn /(ln2 n ln((n ln2 n)/2)). 44.6. {Ni }, {Xi }, i = 1, 2, ... , Ni b(n2 , p), 0 < p < 1, Xi {1, 2, ..., Ni }, i = 1, 2, ... Gn (x) = P{Mn nx + n2 p}. ,
n

lim Gn (x) = exp(x/p),

x < 0.

y = ln Gn (x) y = x/p. n = Mn /(n2 p). . n (Ni - n2 p)/(n p(1 - p)) = i + op (1), i N (0, 1), Ni = n2 p + n p(1 - p)i + op (n), i = 1, ..., n. 45. X1 , X2 , ... .. F (x), Mk = max{X1 , ..., Xk }. Xk , Xk > Mk-1 , k 2, X1 . , {Mn }, n = 1, 2, ..., . L1 < L2 < ..., , . 45.1. Yk = I {Xk > M N1 = 1 ,
k-1

}, k 2, Y1 = 1.
n

Nn = 1 +
k=2

Yk ,

n 2.

41

,
n

ENn =
k=1

1 , k

DN

n k

1 1 ( - 2 ), k =1 k

n

Nn P - 1 n . ln n

n = Nn , ln n n 2.

F (x) = (x). . , Y1 , Y2 , ... ,
n k=1

1 ln n + k

n ,

= 0, 5772...- . 45.2. 45.1 , Y1 , Y2 , ... lim P n Nn - ln n < x = (x). ln n

.. Tn = Nn - ln n ln n

.. F (x) = 1 1+e
-x

,

- < x < .

. ( ). 45.3. , P{Ln > m} = P{Nm < n} Nn ( . 45.2 ), ,
n

lim P

ln Ln - n < x = (x), n

ln Ln P - 1 n . n

n = ln Ln . n

F (x) = 1 - x-2 , x > 1.

42

45.4. X1 , ..., Xn .. F (x) = 1 - exp(-x x > 0, > 0. X (n) n- . , n Zn = ln X (n) P - ; ln n

1/

),

1 ln X (n) d n ln n - - N (0, 1). ln n Zn . . . e1 , e2 , ... , P{e1 < x} = 1 - e-x , x > 0, e(n)- n- , d e(n) = n=1 ei . i d e(n) X 1/ (n) = e(n). 45.5. X1 , ..., Xn .. F (x) = 1 - x 1, > 0. X (n) n- . , n Rn = ln X (n) P 1 - ; n
-

,

x>

1 d - N (0, 1). Rn . d . 45.4 ln X (n) = e(n). n Rn - 46. M N , , 1/N , . µr , r . 46.1. , 1) M , N , M /N = + ln N , > 0, µ0 d X, X (e- ), µ0 - X ; 2) M , N , M /N = , µ0 /N P - e , µ0 /N - e- . 3) M - (), µ0 b(N , e-/N ). .1) .. µ0 .. X. N = µ0 /N .2). 46.2. , r 1 , 1) M , N , M /N = + (1 - r) ln N , > 0, µr d X, X (e- ), µr - X ; P 2) M , N , M /N = + r ln N , µr /N - e- . .. µr .. N = µr /N .2). 43

.
N

µr =
i=1

i , r ,

i =

1, 0,

i - .

Eµ
[k ] r

= Eµr (µr - 1)...(µr - k + 1) = k !
1i1
P{i1 = ... = ik = 1}

. , X (), EX [k] = k , k = 0, 1, ... . 1 , 2 , ... k -
n [ lim Enk] = k , k = 0, 1, 2, ....

n () n . 47. N N , N . t = 0, 1, ... t . . , t , t = 0, 1, ... , ,
t k N lim P{t = k |0 = m} = k = (CN )2 /C2N ,

k , m = 0, 1, ..., N .

t k .
n

n =
i=0

i /(n + 1).

48. , : e0 - , e1 - , e2 - . n n - . , n 0, 8 0, 1 0, 1 P = 0, 3 0, 1 0, 6 0, 7 0, 01 0, 29 (0 , 1 , 2 ). n , n 1, . 44

49. N1 , N2 , ... - ,
d-1

P{N1 = j } = qj , X0 = 0, X 1 = N1 -

j = 0, 1, ..., d - 1,
j =0

qj = 1.

N1 d, d

X 2 = X 1 + N2 -

X1 + N d

2

d

.., Xn d Xn-1 + Nn . X1 , X2 , ... . , pij = P{Xn = j |Xn
-1

= i} =

q q

j -i d+j -i

, j i, , j < i.

pj = nlim pij (n),
d-1 d-1

j = 0, 1, ..., d - 1,

pi pij = pj ,
i=0

j = 0, 1, ..., d - 1,
j =0

pj = 1.

n n =
i=0

Xi /n, n 1.

45

1. . . ., " 1984. 2. ., . . ., " 1988. 3. . . ., " 1979. 4. ., ., . . ., " 1989. 5. .. . . .. , 1949, .25, 5-59. 6. . . . . . ., " 1979. 7. .. . . ., 1964, .9, 1, 159-165. 8. .., .. . ., - , 1990. 9. Berred M. On record values and the exponent of a distribution with regularly varying upper tail. Journal of Applied Probability, 1992, 29(3), 575-586. 10. Embrechts P., Kluppelberg K., Mikosch T. Modelling of Extremal Events for Finance and Insurance. Springer, Berlin - New York, 1997. 11. Nadara jah S., Mitov K. Extremal Limit Laws for Discrete Random Variables. 12. Bening V.E., Korolev V.Yu. The Student Distribution as an Asymptotic Approximation in Statistics. In: Kolmogorov and Contemporary Mathematics, Abstracts, 2003, 398-399. 13. .. . ., - " 1999. 14. .. . M., " 1972.

54

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. . . . .

. . . .

. . . .

. . . . .

. . . . .

. . .

. . .

. . .

3 4 6 8

.

. . . 11 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 16 46 54

, . . . .

55