Asymptotic time averages and frequency distributions
type: Item
instance of: scholarly article
title: Asymptotic time averages and frequency distributions
zbMATH Open document ID: 1384.60074
DOI: 10.1155/2016/2741214
author: Muhammad El-Taha
published in: International Journal of Stochastic Analysis
publication date: 2017-02-07T00:00:00Z
review text: Summary: Consider an arbitrary nonnegative deterministic process (in a stochastic setting \(\{X(t), \, t \geq 0 \}\) is a fixed realization, i.e., sample-path of the underlying stochastic process) with state space \(S = (- \infty, \infty)\). 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.
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60G07
zbMATH DE Number: 6681139
zbMATH Keywords: nonnegative deterministic process
zbMATH Keywords: sample-path
zbMATH Keywords: long-run frequency distribution
Wikidata QID: Q59125089
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2016/2741214
OpenAlex ID: W2516030786
cites work: Sample-path analysis of processes with imbedded point processes
cites work: On Arrivals That See Time Averages
cites work: On arrivals that see time averages: a martingale approach
cites work: Poisson Arrivals See Time Averages
cites work: Characteristics of queueing systems observed at events and the connection between stochastic intensity and Palm probability
cites work: Event and time averages: a review
cites work: Q5560061
cites work: Semi-stationary processes
cites work: Functional limit theorems for stochastic processes based on embedded processes
cites work: Q4742085
cites work: Stationary random processes associated with point processes
cites work: Q3998392
cites work: Semi-stationary clearing processes
cites work: Regenerative processes in the theory of queues, with applications to the alternating-priority queue
cites work: Asymptotic Properties of Cumulative Processes
cites work: Existence of Limits in Regenerative Processes
cites work: Sample path analysis of contribution and reward in cooperative groups
cites work: Population models at stochastic times
cites work: Technical Note—A Last Word on L = λW
cites work: Q3780228
cites work: Stationary deterministic flows in discrete systems. I
cites work: Stationary deterministic flows: II. The Single-server queue
cites work: Pathwise comparison results for stochastic fluid networks
cites work: Q4114551
cites work: A Unifying Conservation Law for Single-Server Queues
cites work: Q3970132
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description: scientific article; zbMATH DE number 6681139