A Poisson Limit Theorem for Triangular Arrays

This chapter deals with triangular arrays of row-wise independent non-negative integer-valued random variables. The main result is a Poisson limit theorem for the row totals of such an array, provided that this array is asymptotically negligible in a certain sense. This result, which is a far-reaching generalization of the law of rare events, is remarkable inasmuch it provides a condition for convergence in distribution to a Poisson distribution that is both necessary and sufficient. The method of proof is the continuity theorem for probability generating functions. The chapter concludes with highlighting a connection between Poisson convergence and extreme value stochastics. There, Poisson convergence is a key feature whenever one counts, for example, the number of exceedances over large thresholds among n independent and identically distributed random variables, as n tends to infinity.