Ladder Sets and Regenerative Phenomena: Further Remarks and Some Applications.
Abstract
Let X(t) be a Levy process with X(0) = 0, and M(t) = sup (X(s) (0 < or = s < or = t)), m(t) = inf(X(s) (0 < or = s < or = t)). Recent work of B. Fristedt, M. Rubinovitch and the author has revealed several interesting properties of the two processes M - X and X - m. The inverse of the local time at zero of each of these processes is a subordinator. It turns out that both these subordinators have nonzero drifts if and only if X is a compound Poisson process with zero drift. In all other cases (i) if one subordinator has a nonzero drift, then the other is a compound Poisson with zero drift, and (ii) if both subordinators have zero drifts, then neither can be a compound Poisson. In the case where one of them (say the one corresponding to M - X) has a nonzero drift, the family of events (M(t) - X(t) = 0) is a standard regenerative phenomenon. If it has zero drift then one speaks of a 'fictitious' regenerative phenomenon. Examples are given of storage models in which real as well as fictitious regenerative phenomena occur.
Document Details
- Document Type
- Technical Report
- Publication Date
- Dec 01, 1975
- Accession Number
- ADA020206
Entities
People
- N. U. Prabhu
Organizations
- University of Wisconsin–Madison