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

• Published in: Journal of statistical physics Vol. 148; no. 2; pp. 233 - 249
• Main Authors: Orsingher, Enzo, Polito, Federico
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01-08-2012; Springer
• Subjects: Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical; Mittag-Leffler functions; Mellin transforms; Fractional birth process; Linnik distribution; Bell polynomials; Discrete Mittag-Leffler distribution; Fibonacci numbers; Golden ratio; Continued fractions
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-012-0534-6

In this paper we consider the relation between random sums and compositions of different processes. In particular, for independent Poisson processes
N
α
(
t
),
N
β
(
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
,
ν
∈(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
Θ
(
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.