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Compound Poisson approximation for long increasing sequences

Part of: Distribution theory Applications Limit theorems Nonparametric inference

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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References

Arratia, R., Goldstein, L., and Gordon, L. (1990). Poisson approximation and the Chen–Stein method. Statist. Sci. 5, 403–434.Google Scholar
Barbour, A. D., and Chryssaphinou, O. (2001). Compound Poisson approximation: a user's guide. To appear in Ann. Appl. Prob.Google Scholar
Barbour, A. D., and Xia, A. (1999). Poisson perturbations. ESAIM Prob. Statist. 3, 131–150.Google Scholar
Barbour, A. D., and Xia, A. (2000). Estimating Stein's constants for compound Poisson approximation. Bernoulli 6, 581–590.Google Scholar
Brockwell, P. J., and Davis, R. A. (1996). Introduction to Time Series and Forecasting. Springer, New York.Google Scholar
David, F. N., and Barton, D. E. (1962). Combinatorial Chance. Hafner, New York.Google Scholar
Csáki, E. and Földes, A. (1996). On the length of the longest monotone block. Stud. Sci. Math. Hungar. 31, 35–46.Google Scholar
Frolov, A. N., and Martikainen, A. I. (1999). On the length of the longest increasing run in Rd. Statist. Prob. Lett. 41, 153–161.Google Scholar
Pittel, B. G. (1981). Limiting behavior of a process of runs. Ann. Prob. 9, 119–129.Google Scholar
Pittenger, A. (1994). Length of the longest non-decreasing subsequence on two symbols. In Runs and Patterns in Probability: Selected Papers (Math. Appl. 283). Kluwer, Dordrecht, pp. 83–89.Google Scholar
Révész, P. (1983). Three problems on the lengths of increasing runs. Stoch. Proc. Appl. 15, 169–179.Google Scholar
Schensted, C. (1961). Longest increasing and decreasing subsequences. Canad. J. Math. 13, 179–191.Google Scholar
Wolfowitz, J. (1944). Asymptotic distribution of runs up and down. Ann. Math. Statist. 15, 163–172.CrossRefGoogle Scholar