: i = 1, . . . , n, n = 1, 2, . . .} be an array of independent random variables with a common distribution function F

J. D U D K I E W I C Z (Kielce)

COMPOUND POISSON APPROXIMATION FOR EXTREMES OF MOVING MINIMA IN ARRAYS

OF INDEPENDENT RANDOM VARIABLES

Abstract. We present conditions sufficient for the weak convergence to a compound Poisson distribution of the distributions of the kth order statistics for extremes of moving minima in arrays of independent random variables.

1. Introduction. Let {X

: i = 1, . . . , n, n = 1, 2, . . .} be an array of independent random variables with a common distribution function F

for fixed n. We define

(1) V

= min

j≤i<j+mn

X

, j = 1, . . . , n − m

+ 1,

where m

is a sequence of positive integers. The array {V

: j = 1, . . . , n − m

+ 1, n = 1, 2, . . .} is stationary and (m

− 1)-dependent in each row.

Denote by

(2) min(V

: j = 1, . . . , n − m

+ 1) = M

≤ M

n

≤ . . . ≤ M

= max(V

: j = 1, . . . , n − m

+ 1) the order statistics of the sequence V

, . . . , V

. In [2] E. R. Canfield and W. P. McCormick have obtained a limit law for M

. They showed that if

(3) m

ln n → d ≥ 0 as n → ∞, then

(4) P {M

≤ u

} → e

as n → ∞,

1991 Mathematics Subject Classification: 60F, 60E.

Key words and phrases: compound Poisson distribution, order statistics, moving min- ima, consecutive-m-out-of-n system.

[19]

where θ = 1 − exp(−1/d), while λ > 0 and the sequence {u

: n = 1, 2, . . .}

of real numbers are related by

(5) nP

{X

> u

} = λ.

In this paper we extend (4) to the case of any kth order statistic. The limit law will be represented in terms of a compound Poisson distribution.

Our result is also a generalization of [4] where Zubkov’s method (see [7]) was used to obtain weak convergence of the distributions of the kth order statistics (2) to the Poisson law under the condition

m

/ ln n → 0 as n → ∞.

The proofs of the main result of this paper are based on Stein’s method (see [1]).

The problems considered have a connection with reliability theory. The random variables M

can be interpreted as lifetimes of consecutive-m- out-of-n systems. Such a system fails if and only if at least m consecutive components out of n linearly ordered components fail. Some examples of applications to telecommunication and oil pipelines modelling may be found in [3] and [5].

2. Definitions and preliminary results. We say that a discrete random variable W has a compound Poisson distribution if

(6) M (t) = E exp(−tW ) = exp



−

∞

X

n=1

c

(1 − e

)



for all t > 0, where c

≥ 0, n = 1, 2, . . . , are such that 0 < P

n=1

c

< ∞.

Note that the corresponding distribution function is

(7) G(x, {c

}) = X

s≤x

p

({c

}), x ∈ R, where

(8) p

({c

})

=



 

 

 



 

 

 

 exp 

−

∞

X

n=1

c



, s = 0,

exp 

−

∞

X

n=1

c

 X

k1+2k2+...+sks=s kj≥0, j=1,...,s

c

c

. . . c

k

!k

! . . . k

! , s = 1, 2, . . . The total variation distance between two probability measures F and G is defined by

d(F, G) = sup

E

|F (E) − G(E)|,

where the supremum is taken over all measurable subsets E of the real line.

Denote by L(X) the law of a random variable X and recall (see [6]) that if d(L(X

), L(X)) → 0 as n → ∞ then X

→ X (weak convergence; see [6]).

The following lemma will be used in the next section.

Lemma 1. Let {X

: i = 1, . . . , n, n = 1, 2, . . .} be an array of indepen- dent random variables with a common distribution function F

for fixed n.

If the sequence {m

: n = 1, 2, . . .} of positive integers is such that

(9) lim

n→∞

m

/ ln n = d, d ≥ 0, and

(10) lim

n→∞

n[1 − F

(u

)]

= λ where {u

} is a sequence of real numbers, then

(11) lim

n→∞

F

(u

) = 1 − e

. P r o o f. From (10) we obtain

n→∞

lim ln[n(1 − F

(u

))

] = ln λ Since

n→∞

lim

ln n + m

ln[1 − F

(u

)]

m

= lim

n→∞

ln λ m

= 0, we conclude from (9) that

n→∞

lim [1 − F

(u

)] = e

.

3. The main results. Let {X

: i = 1, . . . , n, n = 1, 2, . . .} be an array of independent random variables with a common distribution function F

for fixed n, and let {V

: j = 1, . . . , n − m

+ 1, n = 1, 2, . . .} be defined by (1). Consider an array {I

: j = 1, . . . , n − m

+ 1, n = 1, 2, . . .} of zero-one random variables I

= I

, where u

is a sequence of real numbers and I

denotes the indicator function of the set A. This last array is stationary and (m

− 1)-dependent in each row and

P {I

= 1} = P {I

= 1} = P {V

> u

} (12)

= P {X

> u

, X

> u

, . . . , X

> u

}

= [1 − F

(u

)]

.

Let us observe that (m − 1)-dependence is a special case of local dependence defined in [1] with

A

= {β ∈ I : |α − β| < m},

B

= {β ∈ I : |α − β| ≤ 2(m − 1)}, I = {1, . . . , n}.

Set

S

=

n−mn+1

X

i=1

I

. We define, as in [1],

Y

= X

|α−β|<mn

α6=β

I

, α = 1, . . . , n − m

+ 1,

and

λ

= 1 i

n−mn+1

X

α=1

P {I

= 1, Y

= i − 1}, i = 1, . . . , 2m

− 1.

Let M

≤ . . . ≤ M

be the order statistics of the sequence V

, . . . , V

defined by (2).

Lemma 2. For k = 1, 2, . . . ,

(13) |P {M

≤ u

} − G(k − 1, {λ

})|

≤ 2(1 ∧ λ

) exp 

−

∞

X

i=1

λ



X

α=1

X

β∈Bn,α

P

P

, where

a ∧ b = min(a, b),

P

= P {I

= 1} = [1 − F

(u

)]

,

B

= {β ∈ {1, . . . , n − m

+ 1} : |α − β| ≤ 2(m

− 1)}.

P r o o f. This follows from the equality P {M

≤ u

} = P {S

< k} = P {S

≤ k − 1} and Theorem 8 of [1].

Lemma 3. If

(14) lim

n→∞

n[1 − F

(u

)]

= λ, λ > 0, and

(15) lim

n→∞

m

/ ln n = d ≥ 0 then

n→∞

lim λ

= λ

, i = 1, 2, . . . , where

λ

= λ, λ

= 0, i = 2, 3, . . . , for d = 0, and

λ

= λθ

e

, i = 1, 2, . . . , for d > 0.

P r o o f. We fix i, i = 1, . . . , 2m

− 1. For each n we divide the integers 1, . . . , n − m

+ 1 in three parts:

J

= {1, . . . , m

− 1},

J

= {m

, . . . , n − 2m

+ 2},

J

= {n − 2m

+ 3, . . . , n − m

+ 1}.

Because the array {I

} is stationary, we have

λ

= 1 i



X

α=1

P {I

= 1, Y

= i − 1}

+ (n − 3m

+ 3)

i−1

X

j=0

P n

X

k=1

I

= j, I

= 1,

2mn−1

X

k=mn+1

I

= i − 1 − j o

+

n−mn+1

X

α=n−2mn+3

P {I

= 1, Y

= i − 1}

 .

Define

R

= n

X

k=1

I

= j, I

= 1,

2mn−1

X

k=mn+1

I

= i − 1 − j o ,

j = 0, . . . , i − 1.

Observe that events of the form {. . . , V

> u

, V

≤ u

, V

>

u

, . . .} are impossible because {X

≤ u

} and {X

> u

} are mutually exclusive. Thus

R

= {V

≤ u

, . . . , V

≤ u

,

V

> u

, . . . , V

> u

, . . . , V

> u

, V

≤ u

, . . . , V

≤ u

}.

We fix j = 0, . . . , i − 1. By the definition of {I

} and {V

}, and the assumptions on {X

}, we have

P {R

} =

mn−j−2

X

l=0

mn−i+j−1

X

p=0

P n

mn−j−2

X

k=1

I

= l,

I

= 0,

2mn+i−j−2

X

k=mn−j

I

= m

+ i − 1, I

= 0,

3mn−2

X

k=2mn+i−j

I

= p o

= [1 − F

(u

)]

F

(u

).

Because P {R

} does not depend on j, for each 0 ≤ j, k ≤ i − 1 we have

(16) P {R

} = P {R

}.

Next, set

K

(i) = (n − 3m

+ 3)

i−1

X

j=0

P {R

}.

From (16) we obtain

K

(i) = i(n − 3m

+ 3)P {R

}

= i(n − 3m

+ 3)[1 − F

(u

)]

F

(u

).

Hence

1

i K

(i) = (n − 3m

+ 3)[1 − F

(u

)]

(17)

× F

(u

)[1 − F

(u

)]

.

Using Lemma 1 and (14) we have the following result: for d = 0,

n→∞

lim 1

i K

(i) =  0, i > 1, λ, i = 1, and

n→∞

lim 1

i K

(i) = λθ

e

for d > 0.

Now let

L

(i) =

mn−1

X

α=1

P {I

= 1, Y

= i − 1},

L

(i) =

n−mn+1

X

α=n−2mn+3

P {I

= 1, Y

= i − 1}.

Our purpose is to show that

n→∞

lim L

(i) = 0, r = 1, 2.

Set

A

(j) = n

X

k=1

I

= j, I

= 1,

α+mn−1

X

k=α+1

I

= i − j − 1 o ,

α = 1, . . . , m

− 1, 0 ≤ j ≤ i − 1.

Then

L

(i) = X

α<i i−1

X

j=0

P {A

(j)} + X

i≤α≤mn−1 i−1

X

j=0

P {A

(j)}.

Note that

A

(0) = {V

> u

, . . . , V

> u

, V

≤ u

, . . . , V

≤ u

}, A

(j) = {V

≤ u

, . . . , V

≤ u

,

V

> u

, . . . , V

> u

, . . . , V

> u

, V

≤ u

, . . . , V

≤ u

}.

Now it is easy to see that A

(j) ⊂ C

(j) where

C

(j) = {V

> u

, . . . , V

> u

, . . . , V

> u

,

V

≤ u

, . . . , V

≤ u

}.

From the stationarity of the array {V

},

P {C

(j)} = P {D

(j)}, where

D

(j) = {V

> u

, . . . , V

> u

, . . . , V

> u

,

V

≤ u

, . . . , V

≤ u

}.

It is obvious that D

(j) ⊂ A

(0) so

(18) P {A

(j)} ≤ P {A

(0)}

for j = 0, . . . , α − 1 if α < i, and for j = 0, . . . , i − 1 otherwise. From (18) and the assumed properties of {X

} and {V

} we obtain

L

(i) ≤ (m

− 1)i[1 − F

(u

)]

F

(u

) (19)

×

mn−i−1

X

s=0

m

− i − 1 s



[1 − F

(u

)]

F

(u

)

= m

− 1

n · i · n[1 − F

(u

)]

F

(u

)[1 − F

(u

)]

.

Note that in view of (15), m

= o(n), which together with the assumption

(14) and Lemma 1 implies that the right-hand side of (19) tends to zero as

n → ∞. The same is true for L

(i).

Finally, for d = 0,

n→∞

lim λ

=  λ, i = 1,

0, i = 2, . . . , 2m

− 1, and for d > 0,

n→∞

lim λ

= λθ

e

, i = 1, . . . , 2m

− 1.

Lemma 4. We have

(20) lim

n→∞

∞

X

i=1

λ

=

∞

X

i=1

λ

= λθ.

P r o o f. The second equality of (20) follows simply, since

∞

X

i=1

λ

= λθ

1

1 − e

= λθ.

Next, for fixed n, because of (17) and (19), we obtain

λ

≤ n[1 − F

(u

)]

F

(u

)[1 − F

(u

)]

. Hence, from Lemma 1,

λ

≤ λ(1 − e

)e

. Set

a

= λθe

and note that

∞

X

i=1

a

= λθ 1

1 − e

= λ < ∞.

Hence the series P

i=1

λ

is uniformly convergent and thus in view of Lem- ma 3 we have (20).

Lemma 5. If (14) and (15) hold , then

(21) lim

n→∞

n−mn+1

X

α=1

X

β∈Bn,α

P

P

= 0, where P

and B

are as in Lemma 2.

P r o o f. Since P

= [1 − F

(u

)]

, we have

n−mn+1

X

α=1

X

β∈Bn,α

P

P

≤ 2(n − m

+ 1)(m

− 1)[1 − F

(u

)]

= 2 n − m

+ 1

n · m

− 1

n n

[1 − F

(u

)]

.

The right side converges to zero as n → ∞ by (14) and (15).

The main result of this paper may now be readily proved.

Theorem 1. Let {m

} be a sequence of positive integers satisfying (15) and {u

} a sequence of real numbers satisfying (14). Then for k = 1, 2, . . . ,

(22) lim

n→∞

P {M

≤ u

} = G(k − 1, {λ

}) where the distribution function G is given by (7)–(8).

P r o o f. We have

(23) |P {M

≤ u

} − G(k − 1, {λ

})|

≤ |P {M

≤ u

} − G(k − 1, {λ

})|

+ |G(k − 1, {λ

}) − G(k − 1, {λ

})|, k = 1, 2, . . . From Lemma 2,

(24) |P {M

≤ u

} − G(k − 1, {λ

})|

≤ 2(1 ∧ λ

) exp 

−

∞

X

i=1

λ



X

α=1

X

β∈Bn,α

P

P

.

Note that lim

λ

= λθ

and since lim

P

i=1

λ

= λθ we have

(25) lim

n→∞

exp



−

∞

X

i=1

λ



= exp(−λθ).

Thus from (21) the right side of (24) tends to zero as n → ∞. For k = 1, 2, . . . we also have

|G(k − 1, {λ

}) − G(k − 1, {λ

})|

≤ X

s<k

exp



−

∞

X

i=1

λ

 X

k1+2k2+...+sks=s kj≥0, j=1,...,s

λ

λ

. . . λ

k

!k

! . . . k

!

− exp(−λθ) X

k1+2k2+...+sks=s kj≥0, j=1,...,s

λ

λ

. . . λ

k

!k

! . . . k

!

    .

Note that for fixed k we have a finite number of terms in the last two sums.

Hence by (25) and Lemma 3 the right side of the inequality (23) converges to zero as n → ∞.

As an immediate corollary of Theorem 1 we easily obtain the result of

Canfield and McCormick [2].

Corollary 1. Let {m

} and {u

} be sequences satisfying (15) and (14) respectively. Then

n→∞

lim P {M

≤ u

} = e

. P r o o f. Using Theorem 1 for k = 1 we obtain

n→∞

lim P {M

≤ u

} = G(0, {λ

}) where

G(0, {λ

}) = p

({λ

}) = exp 

−

∞

X

i=1

λ



= exp(−λθ).

References

[1] A. D. B a r b o u r, L. H. Y. C h e n and W. L. L o h, Compound Poisson approximation for nonnegative random variables via Stein’s method , Ann. Probab. 20 (1992), 1843–

1866.

[2] E. R. C a n f i e l d and W. P. M c C o r m i c k, Asymptotic reliability of consecutive k- out-of-n systems, J. Appl. Probab. 29 (1992), 142–155.

[3] O. C h r y s s a p h i n o u and S. G. P a p a s t a v r i d i s, Limit distribution for a consecutive k-out-of-n:F system, Adv. Appl. Probab. 22 (1990), 491–493.

[4] J. D u d k i e w i c z, Asymptotic of extremes of moving minima in arrays of independent random variables, Demonstratio Math. 29 (1996), 715–721.

[5] S. G. P a p a s t a v r i d i s, A limit theorem for the reliability of a consecutive-k-out-of-n system, Adv. Appl. Probab. 19 (1987), 746–748.

[6] R. J. S e r f l i n g, A general Poisson approximation theorem, Ann. Probab. 3 (1975), 726–731.

[7] A. M. Z u b k o v, Estimates for sums of finitely dependent indicators and for the time of first occurrence of a rare event , Probabilistic Problems of Discrete Mathematics, Trudy Mat. Inst. Steklov. 177 (1986), 33–46, 207 (in Russian).

Jadwiga Dudkiewicz Institute of Mathematics Technical University of Kielce Tysi¸aclecia PP 7

25-314 Kielce, Poland

Received on 11.7.1996;

revised version on 25.1.1997