Free Access
Issue
ESAIM: PS
Volume 13, January 2009
Page(s) 15 - 37
DOI https://doi.org/10.1051/ps:2007043
Published online 21 February 2009
  1. M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series 55. For sale by the superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (1964). [Google Scholar]
  2. D. Aldous and P. Shields, A diffusion limit for a class of randomly-growing binary search trees. Probab. Theory Related Fields 79 (1998) 509–542. [CrossRef] [Google Scholar]
  3. J.S. Almeida, J.A. Carriço, A. Maretzek, P.A. Noble and M. Fletcher, Analysis of genomic sequences by Chaos Game Representation. Bioinformatics 17 (2001) 429–437. [CrossRef] [PubMed] [Google Scholar]
  4. G. Blom and D. Thorburn, How many random digits are required until given sequences are obtained? J. Appl. Probab. 19 (1982) 518–531. [CrossRef] [MathSciNet] [Google Scholar]
  5. P. Cénac, Test on the structure of biological sequences via chaos game representation. Stat. Appl. Genet. Mol. Biol. 4 (2005) 36 (electronic). [MathSciNet] [Google Scholar]
  6. P. Cénac, G. Fayolle and J.M. Lasgouttes, Dynamical systems in the analysis of biological sequences. Technical Report 5351, INRIA (2004). [Google Scholar]
  7. M. Drmota, The variance of the height of digital search trees. Acta Informatica 38 (2002) 261–276. [CrossRef] [MathSciNet] [Google Scholar]
  8. M. Duflo, Random Iterative Models. Springer (1997). [Google Scholar]
  9. P. Erdős and P. Révész, On the length of the longest head run, in Topics in Information Theory 16 (1975) 219–228, I. Csizàr and P. Elias, Eds. North-Holland, Amsterdam Colloq. Math. Soc. Jànos Bolyai. [Google Scholar]
  10. P. Erdős and P. Révész, On the length of the longest head-run. In Topics in information theory (Second Colloq., Keszthely, 1975). Colloq. Math. Soc. János Bolyai 16 (1977) 219–228. [Google Scholar]
  11. J. Fayolle, Compression de données sans perte et combinatoire analytique. Thèse de l'université Paris VI (2006), available at http://www.lri.fr/ fayolle/these.pdf. [Google Scholar]
  12. J.C. Fu, Bounds for reliability of large consecutive-k-out-of-n:f system. IEEE Trans. Reliability 35 (1986) 316–319. [CrossRef] [Google Scholar]
  13. J.C. Fu and M.V. Koutras, Distribution theory of runs: a markov chain approach. J. Amer. Statist. Soc. 89 (1994) 1050–1058. [Google Scholar]
  14. H. Gerber and S. Li, The occurence of sequence patterns in repeated experiments and hitting times in a markov chain. Stochastic Process. Appl. 11 (1981) 101–108. [CrossRef] [MathSciNet] [Google Scholar]
  15. N. Goldman, Nucleotide, dinucleotide and trinucleotide frequencies explain patterns observed in chaos game representations of DNA sequences. Nucleic Acids Res. 21 (1993) 2487–2491. [CrossRef] [PubMed] [Google Scholar]
  16. L. Gordon, M.F. Schilling and M.S. Waterman, An extreme value theory for long head runs. Probab. Theory Related Fields 72 (1986) 279–287. [CrossRef] [Google Scholar]
  17. H.J. Jeffrey, Chaos Game Representation of gene structure. Nucleic Acid. Res. 18 (1990) 2163–2170. [CrossRef] [Google Scholar]
  18. M.V. Koutras, Waiting times and number of appearances of events in a sequence of discrete random variables, in Advances in combinatorial methods and applications to probability and statistics, Stat. Ind. Technol., Birkhäuser Boston, Boston, MA (1997) 363–384. [Google Scholar]
  19. Shuo-Yen Robert Li, A martingale approach to the study of occurrence of sequence patterns in repeated experiments. Ann. Probab. 8 (1980) 1171–1176. [CrossRef] [MathSciNet] [Google Scholar]
  20. H.M. Mahmoud, Evolution of random search trees. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons Inc., New York (1992). [Google Scholar]
  21. W. Penney, Problem: Penney-ante. J. Recreational Math. 2 (1969) 241. [Google Scholar]
  22. V. Petrov, On the probabilities of large deviations for sums of independent random variables. Theory Prob. Appl. (1965) 287–298. [Google Scholar]
  23. B. Pittel, Asymptotic growth of a class of random trees. Annals Probab. 13 (1985) 414–427. [CrossRef] [Google Scholar]
  24. V. Pozdnyakov, J. Glaz, M. Kulldorff and J.M. Steele, A martingale approach to scan statistics. Ann. Inst. Statist. Math. 57 (2005) 21–37. [CrossRef] [MathSciNet] [Google Scholar]
  25. M. Régnier, A unified approach to word occurence probabilities. Discrete Appl. Math. 104 (2000) 259–280. [CrossRef] [MathSciNet] [Google Scholar]
  26. G. Reinert, S. Schbath and M.S. Waterman, Probabilistic and statistical properties of words: An overview. J. Comput. Biology 7 (2000) 1–46. [CrossRef] [Google Scholar]
  27. S. Robin and J.J. Daudin, Exact distribution of word occurences in a random sequence of letters. J. Appl. Prob. 36 (1999) 179–193. [CrossRef] [MathSciNet] [Google Scholar]
  28. A. Roy, C. Raychaudhury and A. Nandy, Novel techniques of graphical representation and analysis of DNA sequences – A review. J. Biosci. 23 (1998) 55–71. [CrossRef] [Google Scholar]
  29. S.S. Samarova, On the length of the longest head-run for a markov chain with two states. Theory Probab. Appl. 26 (1981) 498–509. [CrossRef] [Google Scholar]
  30. V. Stefanov and A.G. Pakes, Explicit distributional results in pattern formation. Ann. Appl. Probab. 7 (1997) 666–678. [CrossRef] [MathSciNet] [Google Scholar]
  31. W. Szpankowski, Average Case Analysis of Algorithms on Sequences. John Wiley & Sons, New York (2001). [Google Scholar]
  32. D. Williams, Probability with martingales. Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge (1991). [Google Scholar]
  33. * Partially supported by the French Agence Nationale de la Recherche, project SADA ANR-05-BLAN-0372. [Google Scholar]

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