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1 results

  • Compound Poisson approximation for long increasing sequences

  • Part of
  • Ourania Chryssaphinou, Eutichia Vaggelatou
  • Journal:
    Journal of Applied Probability / Volume 38 / Issue 2 / June 2001
    Published online by Cambridge University Press:
    14 July 2016, pp. 449-463
    Print publication:
    June 2001

  • Consider a sequence X1,…,Xn of independent random variables with the same continuous distribution and the event Xi-r+1 < ⋯ < Xi 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.