Tightness of Synchronous Processes
Abstract
Let X = (X(t) : t > or = 0 be a positive recurrent synchronous process (PRS), that is, a process for which there exists an increasing sequence of random times tau=(tau(k)) such that for each k the distribution of theta sub tau(k) o X = X (t+tau(k)):t > or = 0) is the same and the cycle lengths (Tn = tau(n+1)-tau(n) have finite first moment. Whereas the ergodic properties of such processes are well known in the literature, the same is not so for the distributional properties of either the marginals X (t) or more generally the shifted processes theta sub s X = X (s+t) : t > or = 0) in function or space. The present paper shows that these distributions are in fact tight. In contrast to classical regenerative processes the standard types of regularity assumptions (non-lattice cycle length distribution, mixing) do not ensure weak convergence to steady-state for a PRS. Applications are given in the context of one- dependent regenerative (od-R) processes. These arise in the queueing models that motivated this paper.
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 01, 1989
- Accession Number
- ADA212612
Entities
People
- Karl Sigman
- Peter W. Glynn
Organizations
- Stanford University