Consider an arbitrary nonnegative deterministic process (in a stochastic setting

In this article we seek weak conditions, necessary and sufficient, for the long-run time average of a process or any measurable function of it to be equal to the expectation taken with respect to its long-run frequency distribution. Throughout the paper we use a sample-path approach (see [

Let

Let

Note that

At an elementary level the problem can be posed as follows. It is of interest to establish conditions under which the long-run time average of a given process is equal to the expectation taken with respect to its long-run frequency distribution; that is, the following relation holds:

In a stochastic setting relation (

This problem has theoretical as well as practical significance. For example, in queueing theory the long-run average number of customers in a queueing system is, sometimes, defined as

In a stochastic framework this problem has been treated by many authors, mostly to establish conditions that guarantee ergodicity. For example, the case of a process with an imbedded stationary sequence (the semistationary process) is proved by [

El-Taha and Stidham Jr. [

The article is organized as follows. In Section

In this section we prove the main result, provide several applications, study the connection of the conditions needed to uniform integrability, and point out an extension of the parameter space to allow negative time. Given any deterministic process, we show that for any measurable function of the process the asymptotic time average of a function of a given process is equal to the expectation taken with respect to its asymptotic frequency distribution function under weak conditions. Then several special cases of interest will be stated. Our objective is to seek weak conditions under which the asymptotic time average of a function of a given process is equal to the expectation taken with respect to it asymptotic frequency distribution function; that is,

Note that (

Let

(i)

(ii)

(iii)

Note that

The results in Lemma

For all

Note that

By taking limits, as

For all

The result will follow if we show that

(i)

(ii)

Because Lemmas

Consider the deterministic process

(i) Condition A1:

(ii) Condition A2:

(iii)

Taking the limits as

It follows from Theorem

Consider the process

The proof follows by taking limits, as

Consider the process

By Corollary

In Corollary

(i) It follows immediately from Corollary

(ii) From (

(iii) Assume that all the relevant limits in Corollary

(iv) One can construct sufficient conditions for condition

Next we explore the relationship between condition

To shed more light on this difference, we show that the following modified condition is the equivalent to uniform integrability.

Note that in

An important special case is when

Let

(i)

(ii)

(iii)

Another important special case is when considering absolute moments, that is, when

Here, a generalization of Theorem

Let

Consider the deterministic process

(i) Condition B1:

(ii) Condition B2:

(iii)

Similar to Lemma

Now, let

Note that proof of Theorem

In this section we consider a discrete-time process; specifically let

Let

Note that when

Consider the process

This corollary has been found useful in the literature.

Let

Note that

The results in Corollary

Let

Suppose that either

Corollary

In this section, we give three examples. The first example shows that condition

Consider a process

For example if

Cumulative sequence for

Now, consider the sum of the sequence

The next example gives a sequence that does not satisfy the uniform integrability condition

Now, we modify Example

Using Example

Let the stationary stochastic process

The sequence

To extend our results, that is, relation (

Relation (

Let

In this section, we give a simple proof of relation (

Let

It follows from the regenerative processes theory, for example, Theorem B.4 in [

When

The author declares that they have no competing interests.

The author wishes to thank Professor Shaler Stidham, Jr., for helpful comments, in particular for suggesting Example