Compound Poisson approximation for long increasing sequences

Abstract

Consider a sequence X₁,...,X_n of independent random variables with the same continuous distribution and the event {X_i-r+1 < ⋯ < X_i} of the appearance of an increasing sequence with length r, for i=r,...,n. Denote by W the number of overlapping appearances of the above event in the sequence of n trials. In this work, we derive bounds for the total variation and Kolmogorov distances between the distribution of W and a suitable compound Poisson distribution. Via these bounds, an associated theorem concerning the limit distribution of W is obtained. Moreover, using the previous results we study the asymptotic behaviour of the length of the longest increasing sequence. Finally, we suggest a non-parametric test based on W for checking randomness against local increasing trend.