We analyze almost sure asymptotic behavior of extreme values of a regenerative process. We show that under certain conditions a properly centered and normalized running maximum of a regenerative process satisfies a law of the iterated logarithm for the lim sup and a law of the triple logarithm for the lim inf. This complements a previously known result of Glasserman and Kou [Ann. Appl. Probab. 5(2) (1995), 424–445]. We apply our results to several queuing systems and a birth and death process.

Various problems related to asymptotic behavior of extreme values of regenerative processes is of considerable practical interest and has attracted a lot of attention in probabilistic community. For example, extremes in queuing systems and of birth and death processes have been investigated in [

The aforementioned works were mostly aimed at the derivation of

Before formulating the results we introduce necessary definitions. Let us recall, see [

Given a function

The following definition is of crucial importance for our main results.

We say that a function

the function

the derivative

Note that the assumption of regular variation of

Let

For

We are ready to formulate our first result.

Our next result is a counterpart of Theorem

In the discrete setting we assume that there exist extensions of the sequences

The article is organized as follows. In Section

Let us consider a sequence

The following result was proved in [

The proof of Lemma

We need the following generalization of Lemma

To prove Lemma

Fix a sequence of standard exponential random variables

It is known, see [

Similarly, from Lemma

It remains to note that the asymptotics of

The next lemma is a counterpart of Lemma

Similarly to Lemma

Let us consider

Assuming (

The next simple lemma is probably known, however we prefer to give an elementary few lines proof.

By the Stolz–Cesáro theorem we have

Let us start with a proof of equality (

Let us recall that under the assumptions of Theorem

The derivation of Theorem

Suppose that under the assumptions of Theorem

Put

In what follows the next proposition will be useful.

A proof given below is similar to the proof of Theorem

Let us consider a single-channel queuing system with customers arriving at

Let

Thus, if we set

Suppose that

Suppose that

Let us now consider a queuing system with

We impose the following assumption on the parameters

Relations (

Let

We assume that

Further,

Using equalities (

Since

It remains to find a simple formula for the function

It is clear from the above calculations that the condition

Let us finally mention without a proof a statement which follows easily from equations (

We thank anonymous referees for a number of useful suggestions and remarks.