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2 results

  • Compound Poisson approximation for the distribution of extremes

  • Part of
  • A. D. Barbour, S. Y. Novak, A. Xia
  • Journal:
    Advances in Applied Probability / Volume 34 / Issue 1 / March 2002
    Published online by Cambridge University Press:
    01 July 2016, pp. 223-240
    Print publication:
    March 2002

  • Empirical point processes of exceedances play an important role in extreme value theory, and their limiting behaviour has been extensively studied. Here, we provide explicit bounds on the accuracy of approximating an exceedance process by a compound Poisson or Poisson cluster process, in terms of a Wasserstein metric that is generally more suitable for the purpose than the total variation metric. The bounds only involve properties of the finite, empirical sequence that is under consideration, and not of any limiting process. The argument uses Bernstein blocks and Lindeberg's method of compositions.

  • Approximating kth-order two-state Markov chains

  • Part of
  • Y. H. Wang
  • Journal:
    Journal of Applied Probability / Volume 29 / Issue 4 / December 1992
    Published online by Cambridge University Press:
    14 July 2016, pp. 861-868
    Print publication:
    December 1992

  • In this paper, we consider kth-order two-state Markov chains {Xi} with stationary transition probabilities. For k = 1, we construct in detail an upper bound for the total variation d(Sn, Y) = Σx |𝐏(Sn = x) − 𝐏(Y = x)|, where Sn = X1 + · ··+ Xn and Y is a compound Poisson random variable. We also show that, under certain conditions, d(Sn, Y) converges to 0 as n tends to ∞. For k = 2, the corresponding results are given without derivation. For general k ≧ 3, a conjecture is proposed.