Compound Poisson approximation for long increasing sequences

Consider a sequence X 1 ,…, 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.