Some Characterizations of Mixed Renewal Processes

Abstract

Some characterizations of mixed renewal processes in terms of exchangeability and of different types of regular conditional probabilities are given. As a consequence, an existence result for mixed renewal processes, providing also a new construction for them, is obtained. As an application, some concrete examples of constructing such processes are presented and the corresponding regular conditional probabilities are explicitly computed.

Aging Renewal Point Processes and Exchangeability of Event Times

“…about lim n→∞ ∑(τ i ≤ t)/n i.e., through the empirical distribution F n . In what follows we recall the class of exchangeable mixed renewal processes as studied in [12][13][14]. Let us consider a counting process N(t) that counts the number of some type of events occurring during a time interval [0, t] and let us suppose 0 ≤ t 1 ≤ t 2 ≤ .…”

“…random variables. A recently introduced class of renewal processes, named mixed renewal processes, have been based on exchangeability, reflecting indifference among distinct events in a point process [12][13][14]. In addition, we study statistical aging as a form of aging where there is a latency in event detection -the occurrence of an event momentarily "freezes" the detection process for a time t a , the age.…”

Applications of a change of measures technique for compound mixed renewal processes to the ruin problem

“…(see also [17], Definition 3.1, or [19], Definition 3.2(b)). In particular, if the distribution P Θ of Θ is degenerate at some point θ 0 ∈ D, then the counting process N becomes a P -renewal process with interarrival time distribution K(θ 0 ) (written P -RP(K(θ 0 )) for short).…”

“…(b) Applying [17], Theorem 4.2, which holds true under some mild assumptions satisfied by the majority of the probability spaces appearing in applied Probability Theory, one gets that there exists a d-dimensional random vector Θ such that W is P -conditionally i.i.d. if and only if the sequence W is P -exchangeable (recall that a sequence of random variables is called exchangeable if the joint distribution of the sequence is invariant under the permutation of the indices); hence exchangeability seems to be an appealing way to introduce a dependence structure between the claim interarrival times of a counting process.…”

“…Mixed renewal processes (MRPs for short) serve as a proper generalization of MPPs, which has not been much considered in the literature, but they are interesting from both theoretical and applied point of view. Their study is mathematically challenging, as they are not, in general, Markov processes (see [14], Theorem 3, and [19], Proposition 3.2) and they have a close connection to exchangeable stochastic processes (see [14], Corollary on p. 20, and [17], Theorem 4.2). As far as practical models are considered, MRPs seem to have interesting applications in both life insurance (see, e.g., [8]) and nonlife insurance mathematics (see, e.g., [29] and [30]).…”