Maxima of Poisson-like variables and related triangular arrays

It is known that maxima of independent Poisson variables cannot be normalized to converge to a nondegenerate limit distribution. On the other hand, the Normal distribution approximates the Poisson distribution for large values of the Poisson mean, and maxima of random samples of Normal variables may be linearly scaled to converge to a classical extreme value distribution. We here explore the boundary between these two kinds of behavior. Motivation comes from the wish to construct models for the statistical analysis of extremes of background gamma radiation over the United Kingdom. The methods extend to row-wise maxima of certain triangular arrays, for which limiting distributions are also derived.