On Occurrences of F-S Strings in Linearly and Circularly Ordered Binary Sequences

Abstract

Consider a sequence of exchangeable or independent binary trials ordered on a line or on a circle. The statistics denoting the number of times an F-S string of length (at least) k 1 + k 2, that is, (at least) k 1 failures followed by (at least) k 2 successes in n such trials, are studied. The associated waiting time for the rth occurrence of an F-S string of length (at least) k 1 + k 2 in linearly ordered trials is also examined. Exact formulae, lower/upper bounds and approximations are derived for their distributions. Mean values and variances of the number of occurrences of F-S strings are given in exact formulae too. Particular exchangeable and independent sequences of binary random variables, used in applied research, combined with numerical examples clarify further the theoretical results.