Ukrainian Mathematical Journal, Vol. 65, No. 11, April, 2014 (Ukrainian Original Vol. 65, No. 11, November, 2013)

ASYMPTOTIC BEHAVIOR OF A COUNTING PROCESS IN THE MAXIMUM SCHEME

I. K. Matsak UDC 519.21

We determine the exact asymptotic behavior of the logarithm of a counting process in the maximumscheme.

Consider a sequence (ξn), n ≥ 1, of independent identically distributed random variables with distributionfunction F (x) = P(ξn < x). Let

zn = max1≤i≤n

ξi, N(t) = min(n ≥ 1: zn ≥ t).

By analogy with the renewal theory, the process N(t) is called a counting process for the sequence (zn). It is clearthat, at every point, the process N(t) has the geometric distribution

P�N(t) = k

�= q(1− q)k−1, k = 1, 2, . . . , q = 1− F (t).

It is known that this process has independent increments. It is studied in [1–3], where the other properties of thecounting process in the maximum scheme are discussed.

For a fairly complete review of the results and methods used for the investigation of generalized countingprocesses (or generalized renewal processes), see [4].

In the present paper, we establish a statement in the form of the law of iterated logarithm for a counting processin the maximum scheme. It is clear that this result is closely connected with the law of iterated logarithm for themaximum scheme.

The law of iterated logarithm was established for the first time for the sums of independent Bernoulli randomvariables by Khinchin [5]. Later, the law of iterated logarithm was extensively investigated for the sums of arbitraryindependent random variables (see, e.g., [6]).

For the maximum scheme, the law of iterated logarithm was studied in [7–11]. We now present one of themain results in this field (see [9]).

Assume that F has a positive derivative F �(x) for all sufficiently large x and that the functions f(x) andg(x) are defined by the equalities

f(x) =1− F (x)

F �(x), g(x) = f(x) ln ln

�1

1− F (x)

�.

If

limt→∞

g�(t) = 0, (1)

Shevchenko Kyiv National University, Kyiv, Ukraine.

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 11, pp. 1575–1579, November, 2013. Original article submitted Novem-ber 22, 2012; revision submitted March 18, 2013.

0041-5995/14/6511–1743 c� 2014 Springer Science+Business Media New York 1743

1744 I. K. MATSAK

then the following law of iterated logarithm is true for the maximum scheme:Almost surely,

lim supn→∞

zn − anf(an) ln lnn

= 1, (2)

lim infn→∞

zn − anf(an) ln lnn

= 0, (3)

where

an = F−1

�1− 1

n

�and F−1(y) = inf {x : F (x) ≥ y} is the function inverse to F (x).

We set

R(x) = − ln�1− F (x)

�or F (x) = 1− exp

�−R(x)

�.

In [11], it has been recently shown that, under condition that at least one of the functions f(x) and h(x) =

f�R−1(x)

�is regularly varying , equality (3) can be generalized as follows:

lim infn→∞

zn − anf(an) ln ln lnn

= −1. (4)

Our aim is to investigate the asymptotic behavior of the counting process on the basis of equalities of the form(2)–(4).

We now formulate the main result of the paper in the following form:

Theorem 1. Let the distribution function F (x) be continuous, strictly monotonically increasing, and such

that F (x) < 1 for all x ∈ R. Then, almost surely,

lim supt→∞

lnN(t)−R(t)

ln lnR(t)= 1, (5)

lim inft→∞

lnN(t)−R(t)

lnR(t)= −1. (6)

Proof. First, we consider the case F (x) = 1 − exp(−x), x > 0. In this case, equalities (5) and (6) can berewritten in the form: almost surely,

lim supt→∞

lnN(t)− t

ln ln t= 1, (7)

lim inft→∞

lnN(t)− t

ln t= −1. (8)

To prove equalities (7) and (8), we need the following auxiliary assertion established in [11]:

ASYMPTOTIC BEHAVIOR OF A COUNTING PROCESS IN THE MAXIMUM SCHEME 1745

Lemma. Let (ξi) be a sequence of independent random variables with distribution function F (x) = 1 −exp(−x), x > 0. Then, almost surely,

lim supn→∞

zn − lnn

ln lnn= 1, (9)

lim infn→∞

zn − lnn

ln ln lnn= −1. (10)

We first prove equality (8). The equality will be established if we show that, for any sufficiently small � > 0,

P

�lim inft→∞

lnN(t)− t

ln t≤ −1 + �

�= 1, (11)

P

�lim inft→∞

lnN(t)− t

ln t≤ −1− �

�= 0. (12)

We introduce the notation�g1(t) ≤ g2(t), infinitely many times, t ↑ ∞

�.

This means that there exists a sequence (tn), tn ↑ ∞, such that g1(tn) ≤ g2(tn) ∀n ≥ 1.

We set

A(c) =

�lnN(t)− t

ln t≤ c, infinitely many times, t ↑ ∞

�.

Let δ > 0. Thus, we get

A(c− δ) ⊂�lim inft→∞

lnN(t)− t

ln t≤ c

�⊂ A(c+ δ).

Hence, equalities (11) and (12) are true for any � > 0 if and only if

P�A(−1 + �)

�= 1 ∀� > 0, (13)

P�A(−1− �)

�= 0 ∀� > 0. (14)

It is clear that

A(−1 + �) =�lnN(t) ≤ t+ (−1 + �) ln t, infinitely many times, t ↑ ∞

�

=�N(t) ≤ r̂(t), infinitely many times, t ↑ ∞

�,

where r̂(t) = exp�t+ (−1 + �) ln t

�.

Let r = r(t) =�r̂(t)

�be the integer part of the number r̂(t). Since

�N(t) ≤ r̂(t)

�=

�N(t) ≤ r(t)

�

1746 I. K. MATSAK

and, for integer r,�N(t) ≤ r

�= {Zr ≥ t},

we get

A(−1 + �) =�N(t) ≤ r(t), infinitely many times, t ↑ ∞

�

=�Zr(t) ≥ t, infinitely many times, t ↑ ∞

�

=

�Zr(t) − ln r(t)

ln ln r(t)≥ t− ln r(t)

ln ln r(t), infinitely many times, t ↑ ∞

�.

It is easy to see that, as t → ∞,

ln r(t) = t+ (−1 + �) ln t+ o(1),

ln ln r(t) = ln t+ o(1).

Therefore,

t− ln r(t)

ln ln r(t)= 1− �+ o(1), t → ∞,

and, hence,

A(−1 + �) =

�Zr(t) − ln r(t)

ln ln r(t)≥ 1− �+ o(1), infinitely many times, t ↑ ∞

�.

If t varies from 1 to ∞, then r(t) runs through all natural numbers greater than 1. This result and equality(9) yield (13).

In exactly the same way, for the event A(−1− �), we get

A(−1− �) =

�Zr(t) − ln r(t)

ln ln r(t)≥ 1 + �+ o(1), infinitely many times, t ↑ ∞

�.

As above, together with (9), this leads to equality (14).We now prove equality (7). By analogy with A(c), we introduce an event

B(c) =

�lnN(t)− t

ln ln t> c, infinitely many times, t ↑ ∞

�.

Thus, for 1 > � > 0, we obtain

B(1− �) =�N(t) > m̂(t), infinitely many times, t ↑ ∞

�,

where

m̂(t) = exp�t+ (1− �) ln ln t

�.

ASYMPTOTIC BEHAVIOR OF A COUNTING PROCESS IN THE MAXIMUM SCHEME 1747

Let m = m(t) =�m̂(t)

�. Further, by using the equalities

�N(t) > m̂(t)

�=

�N(t) > m(t)

�

and

{N(t) > m} = {Zm < t},

we rewrite the event B(1− �) as follows:

B(1− �) =�N(t) > m(t), infinitely many times, t ↑ ∞

�

= {Zm(t) < t, infinitely many times, t ↑ ∞}

=

�Zm(t) − lnm(t)

ln ln lnm(t)<

t− lnm(t)

ln ln lnm(t), infinitely many times, t ↑ ∞

�. (15)

Since

lnm(t) = t+ (1− �) ln ln t+ o(1),

ln ln lnm(t) = ln ln t+ o(1)

as t → ∞, we find

t− lnm(t)

ln ln lnm(t)= −1 + �+ o(1).

By using this result and equalities (15), we get

B(1− �) =

�Zm(t) − lnm(t)

ln ln lnm(t)< −1 + �+ o(1), infinitely many times, t ↑ ∞

�.

This representation and relation (10) enable us to conclude that

P�B(1− �)

�= 1 ∀� > 0,

and, hence,

P

�lim supt→∞

lnN(t)− t

ln ln t> 1− �

�= 1 ∀� > 0. (16)

Similarly, in view of relation (10), we arrive at the equalities

P

�lim supt→∞

lnN(t)− t

ln ln t> 1 + �

�= P

�B(1 + �)

�= 0 ∀� > 0. (17)

By using (16) and (17), we obtain (7).

1748 I. K. MATSAK

One can easily pass from equalities (7) and (8) to similar relations in the general case. Indeed, it is known(see, e.g., [9]) that, under the conditions of Theorem 1, the random variables τ ei = R(ξi), i ≥ 1, have the standardexponential distribution

P(τ ei < x) = 1− exp(−x).

Let

zen = max1≤i≤n

τ ei , N e(t) = min(n ≥ 1: zen ≥ t).

Then

N(t) = min(n ≥ 1: zn ≥ t) = min�n ≥ 1: R(zn) ≥ R(t)

�

= min�n ≥ 1: zen ≥ R(t)

�= N e

�R(t)

�.

By using this relation and relations (7) and (8), we immediately obtain equalities (5) and (6).The theorem is proved.

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