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Joint distributions of numbers of occurrences of a discrete pattern and weak convergence of an empirical process for the pattern

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  • Aki, Sigeo

Abstract

For a {0, 1}-pattern of finite length, an empirical process is introduced in order to describe the number of overlapping occurrences of the pattern at each level t[set membership, variant][0,1] in a sequence of the corresponding indicators of i.i.d. [0, 1]-valued observations of length n. A method for obtaining the exact finite-dimensional distributions of the empirical process is given. The weak convergence of the process to a Gaussian process in D[0,1] as n tends to infinity is also established. The limiting process depends on the given pattern. The exact covariance function is compared with the asymptotic covariance function in a numerical example.

Suggested Citation

  • Aki, Sigeo, 2008. "Joint distributions of numbers of occurrences of a discrete pattern and weak convergence of an empirical process for the pattern," Journal of Multivariate Analysis, Elsevier, vol. 99(7), pages 1460-1473, August.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:7:p:1460-1473
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    References listed on IDEAS

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    1. Philippou, Andreas N. & Georghiou, Costas & Philippou, George N., 1983. "A generalized geometric distribution and some of its properties," Statistics & Probability Letters, Elsevier, vol. 1(4), pages 171-175, June.
    2. M. Koutras & V. Alexandrou, 1995. "Runs, scans and URN model distributions: A unified Markov chain approach," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(4), pages 743-766, December.
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    Cited by:

    1. Sigeo Aki, 2012. "Statistical modeling for discrete patterns in a sequence of exchangeable trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(3), pages 633-655, June.

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