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Martingale Methods

Living reference work entry

Abstract

We survey the ways that martingales and the method of gambling teams can be used to obtain otherwise hard-to-get information for the moments and distributions of waiting times for the occurrence of simple or compound patterns in an independent or a Markov sequence. We also survey how such methods can be used to provide moments and distribution approximations for a variety of scan statistics, including variable length scan statistics. Each of the general problems considered here is accompanied by one or more concrete examples that illustrate the computational tractability of the methods.

Keywords

Scan Pattern Martingale Waiting time Gambling teams Shifted exponential distribution 

References

  1. Aldous D (1989) Probability approximations via the Poisson clumping heuristic. Springer Publishing, New YorkCrossRefMATHGoogle Scholar
  2. Benevento RV (1984) The occurrence of sequence patterns in ergodic Markov chains. Stoch Process Appl 17:369–373MathSciNetCrossRefMATHGoogle Scholar
  3. Fisher E, Cui S (2010) Patterns generated by m-order Markov chains. Stat Probab Lett 80: 1157–1166MathSciNetCrossRefMATHGoogle Scholar
  4. Fu JC, Chang Y (2002) On probability generating functions for waiting time distribution of compound patterns in a sequence of multistate trials. J Appl Probab 39:70–80MathSciNetCrossRefMATHGoogle Scholar
  5. Fu JC, Lou WYW (2006) Waiting time distributions of simple and compound patterns in a sequence of r-th order Markov dependent multi-state trials. Ann Inst Stat Math 58:291–310MathSciNetCrossRefMATHGoogle Scholar
  6. Gava RJ, Salotti D (2014) Stopping probabilities for patterns in Markov chains. J Appl Probab 51:287–292MathSciNetCrossRefMATHGoogle Scholar
  7. Gerber H, Li S (1981) The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain. Stoch Process Appl 11:101–108MathSciNetCrossRefMATHGoogle Scholar
  8. Glaz J, Naus JI (1991) Tight bounds for scan statistics probabilities for discrete data. Ann Appl Probab 1:306–318MathSciNetCrossRefMATHGoogle Scholar
  9. Glaz J, Naus JI, Wallenstein S (2001) Scan statistics. Springer, New YorkCrossRefMATHGoogle Scholar
  10. Glaz J, Zhang Z (2006) Maximum scan score-type statistics. Stat Probab Lett 76:1316–1322MathSciNetCrossRefMATHGoogle Scholar
  11. Glaz J, Kulldorff M, Pozdnyakov V, Steele JM (2006) Gambling teams and waiting times for patterns in two-state Markov chains. J Appl Probab 43:127–140MathSciNetCrossRefMATHGoogle Scholar
  12. Graham RL Knuth DE, Patashnik O (1994) Concrete mathematics. A foundation for computer science, 2nd edn. Addison-Wesley Publishing Company, ReadingGoogle Scholar
  13. Guibas LJ, Odlyzko AM (1981) String overlaps, pattern matching, and nontransitive games. J Comb Theory Ser A 30:183–208MathSciNetCrossRefMATHGoogle Scholar
  14. Li S (1980) A martingale approach to the study of occurrence of sequence patterns in repeated experiments. Ann Probab 8:1171–1176MathSciNetCrossRefMATHGoogle Scholar
  15. Naus JI (1965) The distribution of the size of the maximum cluster of points on a line. J Am Stat Assoc 60:532–538MathSciNetCrossRefGoogle Scholar
  16. Naus JI, Stefanov VT (2002) Double-scan statistics. Methodol Comput Appl Probab 4:163–180MathSciNetCrossRefMATHGoogle Scholar
  17. Naus JI, Wartenberg DA (1997) A double-scan statistic for clusters of two types of events. J Am Stat Assoc 92:1105–1113MathSciNetCrossRefMATHGoogle Scholar
  18. Pozdnyakov V (2008) On occurrence of patterns in Markov chains: method of gambling teams. Stat Probab Lett 78:2762–2767MathSciNetCrossRefMATHGoogle Scholar
  19. Pozdnyakov V, Kulldorff M (2006) Waiting times for patterns and a method of gambling teams. Am Math Mon 113:134–143MathSciNetCrossRefMATHGoogle Scholar
  20. Pozdnyakov V, Steele JM (2009) Martingale methods for patterns and scan statistics. In: Glaz J, Pozdnyakov V, Wallenstein S (eds) Scan statistics: methods and applications. Birkhauser, Boston, pp 289–318CrossRefGoogle Scholar
  21. Pozdnyakov V, Glaz J, Kulldorff M, Steele JM (2005) A martingale approach to scan statistics. Ann Inst Stat Math 57:21–37MathSciNetCrossRefMATHGoogle Scholar
  22. Shiryaev AN (1995) Probability. Springer, New YorkMATHGoogle Scholar
  23. Solov’ev AD (1966) A combinatorial identity and its application to the problem on the first occurrence of a rare event. Teorija Verojatnostei i ee Primenenija 11:313–320MathSciNetGoogle Scholar
  24. Williams D (1991) Probability with martingales. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  25. Zajkowski K (2014) A note on the gambling team method. Stat Probab Lett 85:45–50MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of ConnecticutStorrsUSA
  2. 2.Department of Statistics, The Wharton SchoolUniversity of PennsylvaniaPhiladelphiaUSA

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