Distribution of the Length of the Longest Significance Run on a Bernoulli Net and Its Applications

Abstract

We consider the length of the longest significance run in a (two-dimensional) Bernoulli net and derive its asymptotic limit distribution. Our results can be considered as generalizations of known theorems in significance runs. We give three types of theoretical results: (1) reliability-style lower and upper bounds, (2) Erdös–Rényi law, and (3) the asymptotic limit distribution. To understand the rate of convergence to the asymptotic distributions, we carry out numerical simulations. The convergence rates in a variety of situations are presented. To understand the relation between the length of the longest significance run(s) and the success probability p , we propose a dynamic programming algorithm to implement simultaneous simulations. Insights from numerical studies are important for choosing the values of design parameters in a particular application, which motivates this article. The distribution of the length of the longest significance run in a Bernoulli net is critical in applying a multiscale methodology in image detection and computational vision. Approximation strategies to some critical quantities are discussed.