Almost certain behavior of row sums of double arrays

Abstract

The almost certain (probability one) behavior of row sums of double arrays of rowwise independent random variables X_{n, i} is investigated primarily when the X_{n, i} assume only the values zero and one. If k_np_n, the expected number of successes, grows more rapidly than C log n, then converges almost certainly to one. However, when the expected number of successes grows less rapidly than C log n, it no longer determines the asymptotic behavior of and the outcome depends upon additional stochastic assumptions. Two alternative sets of stochastic constraints are imposed and the behavior of under each is analyzed and contrasted. Finally, the case of exponential random variables is considered briefly.

Research supported by the National Science Foundation under Grant NSF-MCS-8005481.