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Maxima of Poisson-like variables and related triangular arrays. (English) Zbl 0897.60052

The paper deals with the extremal behavior of a sequence of independent Poisson random variables. It is well known that maxima of independent, identically distributed Poisson random variables cannot be normalized to converge to a nondegenerate limit distribution. The authors study \(T_n= \max_{1\leq i\leq n} R_{n,j}\) as \(n\to\infty\), where, for each positive integer \(n\), \(R_{n,i}\), \(i=1,\dots,n\), are independent Poisson random variables with mean \(\lambda_n\) growing with \(n\). Using the normal approximation to the Poisson distribution for large values of the Poisson mean \(\lambda_n\), they obtain a Gumbel approximation to the distribution of Poisson maxima \(T_n\). It is also shown that if the normal convergence is too slow, then no limiting distribution is possible for Poisson maxima. Motivation comes from the wish to construct models for the statistical analysis of extremes of background gamma radiation over the United Kingdom. The authors also present some further results about row-wise maxima of certain triangular arrays. Note that a similar problem was considered by G. I. Ivchenko [Theory Probab. Appl. 20, 545-559 (1975); translation from Teor. Veroyatn. Primen. 20, 557-570 (1975; Zbl 0358.60054)].

MSC:

60G70 Extreme value theory; extremal stochastic processes
60F10 Large deviations

Citations:

Zbl 0358.60054
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References:

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