Journal of Applied Probability

On a 'replicating character string' model

Richard C. Bradley

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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.

Article information

J. Appl. Probab., Volume 51, Number 2 (2014), 512-527.

First available in Project Euclid: 12 June 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G10: Stationary processes

Replicating character string stationarity absolute regularity


Bradley, Richard C. On a 'replicating character string' model. J. Appl. Probab. 51 (2014), no. 2, 512--527. doi:10.1239/jap/1402578640.

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