Compositions, Random Sums and Continued Random Fractions of Poisson and Fractional Poisson Processes

Abstract

In this paper we consider the relation between random sums and compositions of different processes. In particular, for independent Poisson processes N (alpha) (t), N (beta) (t), t > 0, we have that , where the X (j) s are Poisson random variables. We present a series of similar cases, where the outer process is Poisson with different inner processes. We highlight generalisations of these results where the external process is infinitely divisible. A section of the paper concerns compositions of the form , nu a(0,1], where is the inverse of the fractional Poisson process, and we show how these compositions can be represented as random sums. Furthermore we study compositions of the form I similar to(N(t)), t > 0, which can be represented as random products. The last section is devoted to studying continued fractions of Cauchy random variables with a Poisson number of levels. We evaluate the exact distribution and derive the scale parameter in terms of ratios of Fibonacci numbers

Population Processes Sampled at Random Times

“…We note that the probability (5.16) has the same structure of formula (36) in [11], which is related to the iterated Poisson process, despite the fact that the outer process D has decreasing paths.…”

“…The simplest case of the composition of two independent homogeneous Poisson processes (with different rates) has been already studied in [11]. More recently, the iterated Poisson process has been considered in [6].…”

Compound Poisson Process with a Poisson Subordinator

“…This special case corresponds to the assumption that X i = 1, i ≥ 1 a.s. so that F X (x) = 1 {x≥1} , F . We point out that various results on this process have been obtained in [12] and [13].…”

“…An example in queueing theory involving the iterated Poisson process is provided in Application 3. We remark that the probability law, the governing equations, its representation as a random sum, and various generalizations of the iterated Poisson process have been studied in [12] and [13]. For such a process, in Section 6 we express in series form the mean sojourn time in a fixed state.…”

“…Recalling (see [13]) that the mean and the variance of Z(t) are given by E{Z(t)} = λμt and var[Z(t)] = λμ(1 + μ)t, t ≥ 0, respectively, it follows that the iterated Poisson process is overdispersed, since its dispersion index var[Z(t)] E{Z(t)} = 1 + μ is larger than 1 and, in particular, is independent of time t.…”