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Generalized Coverage Processes with Infinitely Divisible Finite Dimensional Distributions

George Makatis, Michael A. Zazanis

Abstract

In this paper we define a class of coverage processes with infinitely divisible finite dimensional distributions and a particular type of correlation structure that can be thought of as generalizations of the classical Ornstein--Uhlenbeck process and which include coverage processes such as the $M/GI/\infty$ process. We show how such processes arise naturally as limits of superpositions of independent ON/OFF Markov processes with different parameters by formulating an appropriate limit theorem. Various examples of processes of this type are given.

Generalized Coverage Processes with Infinitely Divisible Finite Dimensional Distributions

Abstract

In this paper we define a class of coverage processes with infinitely divisible finite dimensional distributions and a particular type of correlation structure that can be thought of as generalizations of the classical Ornstein--Uhlenbeck process and which include coverage processes such as the process. We show how such processes arise naturally as limits of superpositions of independent ON/OFF Markov processes with different parameters by formulating an appropriate limit theorem. Various examples of processes of this type are given.
Paper Structure (14 sections, 14 theorems, 153 equations)

This paper contains 14 sections, 14 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. The covariance structure of the classical Ornstein-Uhlenbeck process and an algebraic identity
  3. Infinitely divisible generalized coverage processes with a generalized OU covariance structure
  4. Processes with Poisson and compound Poisson marginals. The $M/GI/\infty$ process and its generalizations
  5. The number of customers in the system
  6. The marked $M/GI/\infty$ system
  7. Triangular arrays of ON/OFF sources and limiting theorems
  8. Part 1. Convergence of the finite dimensional distributions.
  9. Part 2. Tightness.
  10. Further examples of GCID processes
  11. Appendix
  12. Bounds for ON/OFF Markovian Processes
  13. A algebraic identity
  14. An example of a double array satisfying the assumptions of Theorem \ref{['th:array-convergence']}

Key Result

Proposition 1

Expression (key) is equal to the right hand side of (OU1).

Theorems & Definitions (33)

  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3: GCID
  • Remark 1
  • proof
  • Remark 2
  • Remark 3
  • Lemma 4
  • ...and 23 more