Convergence results for compound Poisson distributions and applications to the standard Luria-Delbrück distribution. (English) Zbl 1086.60014

Möhle, M.

J. Appl. Probab. 42, No. 3, 620-631 (2005).

If the random variables \((X_j,j\geq 1)\) are independent and identically distributed and \(N_m\sim\text{Po}(m)\) is independent of them, then the random variable \(Y_m:= \sum^{N_m}_{j=1} X_j\) has a compound Poisson distribution. If the characteristic function \(\varphi\) of \(X_1\) is such that \(\varphi'(t)+ ia\log|t|\sim- ia-c\text{\,sgn\,}t\) as \(t\to 0\), then it is shown, by considering characteristic function asymptotics, that \(m^{-1}Y_m- a\log m\) converges in distribution as \(m\to\infty\) to a stable distribution explicitly determined by \(a\) and \(c\). The Luria-Delbrück family of distributions provides such an example; see also A. G. Pakes [J. Appl. Probab. 30, No. 4, 991–994 (1993; Zbl 0801.60005)].

Reviewer: Andrew D. Barbour (Zürich)

MSC:

60F05	Central limit and other weak theorems
92D15	Problems related to evolution
60E10	Characteristic functions; other transforms
92D10	Genetics and epigenetics

Citations:

Zbl 0801.60005

References:

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