A negative binomial approximation to the distribution of the sum of maxima of indicator random variables

Abstract

In a sequence of independent indicator random variables, we consider the sum of maxima of k consecutive indicator random variables that represent a specific type of k-runs in the theory of runs and patterns. Intriguingly, it can be expressed as the sum of locally dependent random variables. Our focus is on employing Stein’s method to provide a negative binomial approximation to the distribution of this specific random variable under certain conditions on moments. The derived bounds exhibit an optimal order, reflecting the precision and efficiency of the proposed approximation.