International Journal of Stochastic Analysis
Volume 2016 (2016), Article ID 2741214, 10 pages
http://dx.doi.org/10.1155/2016/2741214

Research Article

Asymptotic Time Averages and Frequency Distributions

Muhammad El-Taha

Department of Mathematics and Statistics, University of Southern Maine, 96 Falmouth Street, Portland, ME 04104-9300, USA

Received 7 March 2016; Accepted 17 July 2016

Academic Editor: MJ Lopez-Herrero

Copyright © 2016 Muhammad El-Taha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Consider an arbitrary nonnegative deterministic process (in a stochastic setting is a fixed realization, i.e., sample-path of the underlying stochastic process) with state space . Using a sample-path approach, we give necessary and sufficient conditions for the long-run time average of a measurable function of process to be equal to the expectation taken with respect to the same measurable function of its long-run frequency distribution. The results are further extended to allow unrestricted parameter (time) space. Examples are provided to show that our condition is not superfluous and that it is weaker than uniform integrability. The case of discrete-time processes is also considered. The relationship to previously known sufficient conditions, usually given in stochastic settings, will also be discussed. Our approach is applied to regenerative processes and an extension of a well-known result is given. For researchers interested in sample-path analysis, our results will give them the choice to work with the time average of a process or its frequency distribution function and go back and forth between the two under a mild condition.

1. Introduction

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 [1–3]) in the sense that we restrict attention to one realization (sample-path) of the process of interest. Our approach reveals that the stochastic assumptions (regenerative, semistationary, etc.) in the most part are not needed for this result to hold, but rather to ensure that the process itself is ergodic and that a stationary distribution exists. In other words, the long-run time averages for different sample-paths of the given stochastic process converge to a common limiting value with probability one (see Example 17 in Section 4). Our approach is intuitive and the proofs are rather elementary; nothing beyond Riemann-Stieltjes integration theory is needed.

Let be an arbitrary real-valued right-continuous deterministic process (in a stochastic setting can be a fixed realization, that is, sample-path of the underlying stochastic process) with state space . Let where is an indicator function and is a real-valued measurable function. It is also assumed that is integrable with respect to in the Riemann-Stieltjes sense. Define the following limits when they exist:

Let and be the complements of and , respectively (it follows that ).

Note that and are the long-run time average of the processes and , respectively, and is the long-run frequency distribution of . In a queueing system may represent the long-run average number of customers in the system, , and is the “stationary distribution.”

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:It turns out that in our sample-path setting, when , there is a necessary and sufficient condition for relation (3) to hold. Relation (3) may also be valid even when .

In a stochastic setting relation (3) may have the following interpretation: for each , let be a process such that, for each , has as its distribution function. Then the process represents the status of the original process as seen by a random observer that arrives at a random time uniformly distributed between and . If we let be a random variable with as its distribution function, then describes the behavior of the process in steady state, and relation (3) may be written as , where is the expected value of .

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 , where represents the number of customers present in the system (both in queue and service) at instant . However, in applications, is usually calculated as the expectation of the stationary distribution , of the process , provided it exists; that is, . The question arises as to when the above two quantities are equal, particularly when is not stationary and ergodic. Similar questions arise also when calculating other performance measures for queueing systems such as , the long-run average waiting time in the system per customer. Another example arises when dealing with the (Arrivals that See Time Averages) property (see [1, 4–8]). Following [4, 5] the ASTA problem is posed as follows: Given two stochastic processes and defined on a common probability space, where is intended to represent a queue or a network of queues, and as an arrival point process, let , be an imbedded point process, , and , where denotes weak convergence [9]. Then the ASTA problem is to find conditions under which , where denotes equality in distribution. However, rather than working to verify directly, Melamed and Whitt [4, 5] find conditions such that , where and is a bounded measurable real-valued function. In general and (equivalently and )) are not necessarily equal unless is stationary and ergodic. On the other hand, Stidham Jr. and El-Taha [1] prove by working directly with sample-path versions of and . In this paper we establish weak conditions (necessary and sufficient) to guarantee the equality of time averages of a process and the expectation taken with respect to its long-run frequency distribution without explicitly assuming stationarity and ergodicity.

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 [10, 11]. Franken et al. [12] derive similar results for processes with imbedded marked point processes. Rolski [13] introduces and exploits the notion of ergodically stable processes to prove a similar result. Wolff [14] proves a similar result for processes that are regenerative or nonnegative and stochastically increasing. The case for stochastic clearing processes is considered by [15]. Relevant also are [16–20]. For references on sample-path analysis, the reader is referred to [1, 2, 16, 21–25]. This article should be of interest to readers with interest in establishing relationships that involve time averages and frequency distributions in a deterministic framework.

El-Taha and Stidham Jr. [2] provide a result in a sample-path setting for the special case of a nonnegative deterministic process. In this article the results of [2] are extended to any function of the process; thus all moments of a process can be treated within one framework. We also extend the state space to allow for the process to take negative as well as positive values. Moreover we consider the case where the parameter space is extended from to .

The article is organized as follows. In Section 2 we concentrate on the case when is a general unrestricted process and give necessary and sufficient conditions (see conditions and in Section 2) when a measurable function, satisfying a weak regularity condition, of the process is considered. We also provide some insight into the relationship between our condition and uniform integrability. We also extend our results to allow the parameter to be unrestricted in sign. Section 3 provides similar results when is a discrete-time process. In Section 4 three examples are given to help clarify the need for condition , the relation of condition to uniform integrability, and the relationship between our pure sample-path setting and stochastic settings. In Section 5 we close by looking at regenerative processes. A process with an imbedded sequence is investigated and then used to extend a well-known result for regenerative processes.

2. Main Results and Related Analysis

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 (5) reduces to (3) when is an identity function. First we establish the following time dependent relationships in Lemmas 1–3, key to proving the main result.

Lemma 1. Let be any real-valued measurable function and a distribution function defined on . Then for all ,
(i) (ii) (iii) provided all integrals are well defined (i.e., is integrable with respect to in the Riemann-Stieltjes sense).

Proof. Note that which proves part (i). The proof of (ii) is similar. Part (iii) follows by taking in (i) and (ii) and combining both results.

The results in Lemma 1 remain valid if we replace by when the limit exists. Now we give the second partial result.

Lemma 2. For all

Proof. Note that Similarly Then the result follows by appealing to Lemma 1 (iii).

By taking limits, as of both sides of Lemma 2, showing relation (5) holds is seen to be equivalent to a problem of interchanging limits and integrals. The next result is a key lemma.

Lemma 3. For all and

Proof. The result will follow if we show that
(i) (ii)Now Therefore, using Lemma 1 (i), we obtain which proves part (i). Part (ii) is proved similarly using Lemma 1 (ii).

Because Lemmas 1–3 are time dependent, they can be useful in time dependent analysis of nonstationary stochastic systems. Now we give the main result.

Theorem 4. Consider the deterministic process , with state space , and let be any real-valued measurable function. Then, the following are equivalent.
(i) Condition A1:(ii) Condition A2:(iii)

Proof. Taking the limits as on both sides of Lemma 3 proves the equivalence of (i) and (ii). Now, using Lemmas 3 and 2, we writeSuppose (equivalently ) holds. Now, taking limits as and assuming they exist giveThen part (iii) of the theorem follows by taking limits of both sides in (22) and (23) as . Conversely if part (iii) holds, it is straightforward to see that and hold.

It follows from Theorem 4 that the following conditions are sufficient for (5) to hold.

Corollary 5. Consider the process . Suppose uniformly in as . Suppose also that condition (i) (equivalently (ii)) of Theorem 4 holds. Then for any measurable real-valued function , is well defined if and only if is well defined, in which case relation (5) holds; that is,

Proof. The proof follows by taking limits, as , in (22) and (23) and using a similar argument as in Theorem 4.

Corollary 6. Consider the process and suppose that the conditions of Corollary 5 hold. Additionally, let be differentiable and . Then

Proof. By Corollary 5, it suffices to show thatSo, the result follows by noting that and using Lemma 1 (iii).

In Corollary 6, let , then (26) gives the moment of a distribution function which is familiar when is defined only on .

Remarks 7. (i) It follows immediately from Corollary 5 that when (recall ) condition (equivalently ) is necessary and sufficient for relation (5) to hold. The requirement that is not needed to prove the sufficiency part of the assertion.
(ii) From (23), it is clear that if is well defined and finite for all , then is well defined iff is well defined for all . Now take limits in (23) as to obtainIf any two of the three terms in (29) are well defined and at least one of them is finite then the third term exists and Corollary 5 holds.
(iii) Assume that all the relevant limits in Corollary 5 are well defined. From (23) one concludes that, for all , Moreover, if and condition is not satisfied, we can distinguish to possibilities(a); then relation (5) holds;(b); then relation (5) does not hold and condition takes the value . A similar argument applies when . The above discussion should not imply that if condition does not hold it should be infinite. Example 15 given below shows that relation (5) fails with condition assuming a finite value.(iv) One can construct sufficient conditions for condition to hold. Following an argument by Billingsley [26, page 186], one can easily see that is sufficient for (5) to hold.
Next we explore the relationship between condition and uniform integrability of the process .
Relation to Uniform Integrability. We discuss the connection between condition and uniform integrability when is an identity function. Condition of Theorem 4 requires, roughly speaking, that the area, for , between and minus the area between and goes to zero as approaches infinity. We point out here that there is a difference between condition and uniform integrability (of ) which requires that the above two areas add up to zero as approaches infinity. Wolff [14] suggests that proving relation (5), in a stochastic setting, is equivalent to showing that the process and are uniformly integrable (u.i.) in . Condition is weaker than being u.i.; it only coincides with uniform integrability of the process when the process is nonnegative.
To shed more light on this difference, we show that the following modified condition is the equivalent to uniform integrability.
Condition  A3.
Note that in we take the absolute value of . In our sample-path setting the definition of uniform integrability [26, 27] of the process is equivalent toNow, using an argument similar to that used in proving Lemma 3, we obtain Thus condition and (31) are equivalent. We note that condition (equivalently (31)) is sufficient for relation (5) to hold. Example 16 given below shows the existence of a process that obeys relation (5), yet condition is not satisfied.

2.1. Moments

An important special case is when , , that provides a relation between time averages of a process moments and the moments obtained by using its asymptotic distribution function. For , we have the following result.

Corollary 8. Let . Then the following are equivalent:
(i)(ii)(iii)where .

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

2.2. Extension

Here, a generalization of Theorem 4 will be considered. We allow the parameter space to extend to and show that, for any measurable function of the new process, relation (5) remains valid under conditions similar to and . Theorem 4 is extended to the case where the parameter can also be negative.

Let be an arbitrary (deterministic) process with state space , and let Here we seek weak conditions under which the following relation holds:

Theorem 9. Consider the deterministic process , , and let be any real-valued measurable function. Then, the following are equivalent.
(i) Condition B1:(ii) Condition B2:(iii)

Proof. Similar to Lemma 3 we can show that for all and Now, let in (41); then we have for all Now an argument similar to that of the proof of Theorem 4 will yield this result.

Note that proof of Theorem 9 follows the same lines of argument as that of Theorem 4. Note also that conditions and are similar to conditions and , but with in the denominator in .

3. Discrete-Time Processes

In this section we consider a discrete-time process; specifically let be any deterministic discrete-time process, that is, an infinite sequence of real numbers, and let be a real-valued measurable function. Moreover, letand define the following limits when they exist: Similar to the continuous time model, we have the following results.

Theorem 10. Let be any discrete-time process, and let be any a real-valued measurable function. Then, the following are equivalent: (i);(ii);(iii).

Note that when is an identity function, condition (i) of Theorem 10 is sometimes referred to in the literature as the condition for uniform integrability.

Corollary 11. Consider the process . Suppose uniformly in as . Suppose also that condition (i) (equivalently (ii)) of Theorem 10 holds. Then for any measurable real-valued function , is well defined if and only if is well defined, in which case

This corollary has been found useful in the literature.

Corollary 12. Let be any nonnegative deterministic discrete-time process, that is, infinite sequence of nonnegative real numbers. Suppose that any of the conditions (i), (ii), and (iii) of Theorem 10 is satisfied and is a proper distribution function. Now, suppose there exists such that if is a nondecreasing real-valued measurable function on thenand if is a nonincreasing real-valued measurable function on , then

Proof. Note that by condition (ii) of Theorem 10. The proof of the second statement is similar.

The results in Corollary 12 apply to the continuous time process as well. When the sequence represents, say, service times in queueing model, let in Theorem 10 to, immediately, obtain the following useful results.

Corollary 13. Let . Then the following are equivalent: (i);(ii);(iii).

Corollary 14. Suppose that either (i) or(ii). Then

Corollary 14 follows from Corollary 12 by letting . This result is used in Ayesta [28].

4. Examples and Discussion

In this section, we give three examples. The first example shows that condition is not superfluous. The second example, a modification of the first one, shows that the new modified process does not satisfy the uniform integrability condition (condition ), yet conditions and are satisfied and therefore relation (5) holds. In the third example, we verify that when condition is satisfied, for a stationary nonergodic stochastic process, even though relation (5) is not satisfied in a stochastic setting, it remains valid for every individual sample-path of the process.

Example 15. Consider a process in discrete time, and let the sequence represent one sample-path such that

For example if , then , , ; if , then , , , ; if , then , , , , , , , ; if , then , and so on. Figure 1 plots the cumulative sequence for all , showing the jumps at positive terms of .

Figure 1: Cumulative sequence for .

Now, consider the sum of the sequence at positive terms; then Now, consider the sum just before positive terms to obtain Therefore , and hence . We also conclude, using Lemma 2, that The next step is to calculate the r.h.s. of relation (5) and show that it differs from . Now Successive values of determine successive cycles. Now consider at the beginning of such cycles to obtain Since decreases in as long as , it suffices to consider at a subsequence of positive terms. Then Therefore as for all and as for all ;  , is well defined and for all . Thus , so relation (5) fails. In this example condition takes the value .

The next example gives a sequence that does not satisfy the uniform integrability condition , yet it satisfies relation (5).

Example 16. Now, we modify Example 15 as follows: letwith .

Using Example 15 and the fact that , we conclude thus the sequence does not satisfy the uniform integrability condition (). Now, using the fact that , we conclude that condition is satisfied, and therefore relation (5) holds.

Example 17. Let the stationary stochastic process be defined aswhere ,  , and for all .

The sequence is an example of a stationary nonergodic process (see, e.g., [29]). The sample space is given by , where represents the sequence . It is clear that and . It should be clear that none of the time average values , except possibly one, is equal to . So relation (5) is true for every realization of the process. On the other hand, is a random variable for all . Therefore is a random variable, while ; thus relation (5) does not hold with probability one.

To extend our results, that is, relation (5), to stochastic settings an ergodicity assumption is needed; more precisely, we need to insure that the time average in relation (5) assumes a constant value, say, , with probability one. To illustrate, let the event relation (5) fails, then contains the following realizations: (i) sample-paths for which the limits in relation (5) do not exist; (ii) sample-paths for which the limits exist—but relation (5) does not hold (as in Example 15); (iii) sample-paths for which relation (5) holds, but .

Relation (5) holds in a stochastic setting if . In contrast, in Example 17, . Thus the following corollary follows:

Corollary 18. Let be a stochastic process defined on a probability space . If there exists a constant such that with probability one, then (with probability one) Condition is satisfied iff , and relation (5) holds.

5. Application to Regenerative Processes

In this section, we give a simple proof of relation (5) for regenerative processes. Consider a general process with an imbedded point process. Here we need only to consider the limit in condition at the imbedded sequence of time points and then specialize to regenerative processes and extend a result by Wolff [14]. Let be any process with , be an imbedded point process. Define Assume that iff . Let be the long-run frequency distribution of the process at the imbedded points , and assume that exists. Using the relation [1], Corollary 5 remains valid when condition is replaced by the following equivalent condition:Now, we shift attention to stochastic settings and assume that is a stochastic regenerative process defined on a probability space . Let the sequence , where , be such that for all . Then the conditionis sufficient for relation (5) to hold with probability one. Note that with probability one. Moreover the following result can be easily derived.

Theorem 19. Let be a regenerative stochastic process with regeneration points such that , and ; then with probability onewhere has distribution function ; that is, relation (5) holds with probability one.

Proof. It follows from the regenerative processes theory, for example, Theorem B.4 in [2], that condition can be written as So, we need only to verify that condition (62) is satisfied. Using the dominated convergence theorem [26] and the hypothesis of the theorem, it follows that

When is an identity function, Theorem 19 slightly extends Theorem of Wolff [14, page 92], in the sense that the absolute value of is not needed in the hypothesis of the theorem.

Competing Interests

The author declares that they have no competing interests.

Acknowledgments

The author wishes to thank Professor Shaler Stidham, Jr., for helpful comments, in particular for suggesting Example 15.

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