Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-21T19:38:28.209Z Has data issue: false hasContentIssue false
  • English
  • Français

On a ‘Replicating Character String’ Model

Part of: Stochastic processes

Published online by Cambridge University Press:  19 February 2016

Richard C. Bradley
Richard C. Bradley*
Affiliation:
Indiana University
*
∗ Postal address: Department of Mathematics, Indiana University, Bloomington, Indiana 47405, USA. Email address: bradleyr@indiana.edu.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In Chaudhuri and Dasgupta's 2006 paper a certain stochastic model for ‘replicating character strings’ (such as in DNA sequences) was studied. In their model, a random ‘input’ sequence was subjected to random mutations, insertions, and deletions, resulting in a random ‘output’ sequence. In this paper their model will be set up in a slightly different way, in an effort to facilitate further development of the theory for their model. In their 2006 paper, Chaudhuri and Dasgupta showed that, under certain conditions, strict stationarity of the ‘input’ sequence would be preserved by the ‘output’ sequence, and they proved a similar ‘preservation’ result for the property of strong mixing with exponential mixing rate. In our setup, we will in spirit slightly extend their ‘preservation of stationarity’ result, and also prove a ‘preservation’ result for the property of absolute regularity with summable mixing rate.

Keywords

Replicating character stringstationarityabsolute regularity

MSC classification

Primary: 60G10: Stationary processes
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 2 , June 2014 , pp. 512 - 527
Copyright
© Applied Probability Trust 

References

Berbee, H. C. P. (1979). Random Walks with Stationary Increments and Renewal Theory. Mathematical Centre, Amsterdam.Google Scholar
Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.Google Scholar
Bradley, R. C. (1980). A remark on the central limit question for dependent random variables. J. Appl. Prob. 17, 94–101.Google Scholar
Bradley, R. C. (1989). A stationary, pairwise independent, absolutely regular sequence for which the central limit theorem fails. Prob. Theory Relat. Fields 81, 1–10.Google Scholar
Bradley, R. C. (2007). Introduction to Strong Mixing Conditions, Vol. 1. Kendrick Press, Heber City, UT.Google Scholar
Bradley, R. C. (2007). Introduction to Strong Mixing Conditions, Vol. 2. Kendrick Press, Heber City, UT.Google Scholar
Bradley, R. C. (2007). Introduction to Strong Mixing Conditions, Vol. 3. Kendrick Press, Heber City, UT.Google Scholar
Chaudhuri, P. and Dasgupta, A. (2006). Stationarity and mixing properties of replicating character strings. Statistica Sinica 16, 29–43.Google Scholar
Davydov, Y. A. (1973). Mixing conditions for Markov chains. Theory Prob. Appl. 18, 312–328.Google Scholar
Dudley, R. M. and Philipp, W. (1983). Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrscheinlichkeitsth 62, 509–552.Google Scholar
Goldstein, S. (1979). Maximal coupling. Z. Wahrscheinlichkeitsth 46, 193–204.Google Scholar
Ibragimov, I. A. and Linnik, Y. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.Google Scholar
Merlevède, F. and Peligrad, M. (2000). The functional central limit theorem under the strong mixing condition. Ann. Prob. 28, 1336–1352.Google Scholar
Pinsker, M. S. (1964). Information and Information Stability of Random Variables and Processes. Holden-Day, San Francisco.Google Scholar
Rio, E. (2000). Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants (Math. Appl. 31). Springer, Berlin.Google Scholar
Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42, 43–47.Google Scholar
Skorohod, A. V. (1976). On a representation of random variables. Theory Prob. Appl. 21, 628–632.Google Scholar
Volkonskii, V. A. and Rozanov, Y. A. (1959). Some limit theorems for random functions. I. Theory Prob. Appl. 4, 178–197.Google Scholar
Waterman, M. S. (1995). Introduction to Computational Biology. Chapman and Hall, New York.Google Scholar