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

被引:14
|
作者
Orsingher, Enzo [2 ]
Polito, Federico [1 ]
机构
[1] Univ Turin, Dipartimento Matemat, I-10123 Turin, Italy
[2] Univ Roma La Sapienza, Dipartimento Sci Stat, Rome, Italy
关键词
Fractional birth process; Bell polynomials; Mittag-Leffler functions; Fibonacci numbers; Continued fractions; Golden ratio; Linnik distribution; Discrete Mittag-Leffler distribution; Mellin transforms;
D O I
10.1007/s10955-012-0534-6
中图分类号
O4 [物理学];
学科分类号
0702 ;
摘要
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.
引用
收藏
页码:233 / 249
页数:17
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