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

Authors
Journal
Journal of Multivariate Analysis
0047-259X
Publisher
Elsevier
Publication Date
Volume
99
Issue
7
Identifiers
DOI: 10.1016/j.jmva.2008.01.019
Keywords
  • Primary
  • Secondary
  • Run
  • Pattern
  • Type Iii Binomial Distribution Of Order [Formula Omitted]
  • Empirical Process
  • Weak Convergence
  • Distribution Theory

Abstract

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 ∈ [ 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.

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