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Compound Poisson approximation for long increasing sequences

Part of: Distribution theory Limit theorems Nonparametric inference Applications

Published online by Cambridge University Press:  14 July 2016

Ourania Chryssaphinou  and
Eutichia Vaggelatou
Ourania Chryssaphinou*
Affiliation:
University of Athens
Eutichia Vaggelatou*
Affiliation:
University of Athens
*
∗ Postal address: Section of Statistics and Operations Research, Department of Mathematics, University of Athens, Panepistemiopolis, Athens 15784, Greece.
∗ Postal address: Section of Statistics and Operations Research, Department of Mathematics, University of Athens, Panepistemiopolis, Athens 15784, Greece.
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Abstract

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.

Keywords

increasing sequenceslongest increasing sequenceStein-Chencompound Poissonnon-parametric randomness test

MSC classification

Primary: 60F05: Central limit and other weak theorems 62E20: Asymptotic distribution theory
Secondary: 62G10: Hypothesis testing 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 2 , June 2001 , pp. 449 - 463
Copyright
Copyright © Applied Probability Trust 2001 

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