Skip to main content
SpringerLink
Log in

Waiting Times and Number of Appearances of Events in a Sequence of Discrete Random Variables

  • Chapter
Advances in Combinatorial Methods and Applications to Probability and Statistics

Part of the book series: Statistics for Industry and Technology ((SIT))

Abstract

In this article, the probability and moment generating functions of the number of appearances of a pattern ε in a sequence of discrete random variables (repeated trials) are expressed in terms of the generating function of the waiting time for the r-th occurrence of ε, The special case of delayed recurrent events is also examined in some detail. Finally, the general theory is employed for a systematic investigation of success runs enumeration problems in a sequence of binary outcomes arising from a first-order Markov chain.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Aki, S. (1992). Waiting time problems for a sequence of discrete random variables, Annals of the Institute of Statistical Mathematics, 44, 363–378.

    Article  MathSciNet  MATH  Google Scholar 

  2. Aki, S. and Hirano, K. (1993). Discrete distributions related to succession events in a two-state Markov chain, In Statistical Science and Data Analysis (Eds., K. Matusita, M. L. Puri and T. Hayakawa), pp. 467–474, Amsterdam: VSP International Science Publishers.

    Google Scholar 

  3. Aki, S., Balakrishnan, N. and Mohanty, S. G. (1996). Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials, Annals of the Institute of Statistical Mathematics, 48, 773–787.

    Article  MathSciNet  MATH  Google Scholar 

  4. Balakrishnan, N. and Koutras, M. V. (1997). Runs and Patterns with Applications, (under preparation).

    Google Scholar 

  5. Balasubramanian, K., Viveros, R. and Balakrishnan, N. (1993). Sooner and later waiting time problems for Markovian Bernoulli trials, Statistics&Probability Letters, 18, 153–161.

    MathSciNet  MATH  Google Scholar 

  6. Banjevic, D. (1988). On some statistics connected with runs in Markov chains, Journal of Applied Probability, 25, 815–821.

    Article  MathSciNet  MATH  Google Scholar 

  7. Barbour, A. D., Hoist, L. and Janson, S. (1992). Poisson Approximation, New York: Oxford University Press.

    MATH  Google Scholar 

  8. Benevento, R. (1984). The occurrence of sequence patterns in ergodic Markov chains, Stochastic Processes and their Applications, 17, 369–373.

    Article  MathSciNet  MATH  Google Scholar 

  9. Blom, G. and Thorburn, D. (1982). How many random digits are required until given sequences are obtained? Journal of Applied Probability, 19, 518–531.

    Article  MathSciNet  MATH  Google Scholar 

  10. Chao, M. T., Fu, J. C. and Koutras, M. V. (1995). Survey of the reliability studies of consecutive-k-out-of-n: F and related systems, IEEE Transactions on Reliability, 44, 120–127.

    Article  Google Scholar 

  11. Charalambides, Ch. A. (1994). Success runs in a circular sequence of independent Bernoulli trials, In Runs and Patterns in Probability (Eds., A. P. Godbole and S. G. Papastavridis), pp. 15–30, Amsterdam: Kluwer Academic Publishers.

    Google Scholar 

  12. Chen, J. and Glaz, J. (1997). Approximations and inequalities for the distribution of scan statistic for 0–1 Bernoulli trials, In Advances in the Theory and Practice of Statistics: A Volume in Honor of Samuel Kotz (Eds., N. L. Johnson and N. Balakrishnan), New York: John Wiley & Sons (to appear).

    Google Scholar 

  13. Chryssaphinou, O. and Papastavridis, S. (1988). A limit theorem for the number of non-overlapping occurrences of a pattern in a sequence of independent trials, Journal of Applied Probability, 25, 428–431.

    Article  MathSciNet  MATH  Google Scholar 

  14. Chryssaphinou, O. and Papastavridis, S. (1990). The occurrence of sequence patterns in repeated dependent experiments, Theory of Probability and its Applications, 35, 145–152.

    Article  MathSciNet  Google Scholar 

  15. Dembo, A. and Karlin, S. (1992). Poisson approximations for r-scan processes, Annals of Applied Probability, 2, 329–357.

    Article  MathSciNet  MATH  Google Scholar 

  16. Feller, W. (1968). An Introduction to Probability Theory and its Applications, Volume I, Third edition, New York: John Wiley & Sons.

    MATH  Google Scholar 

  17. Fu, J. C. (1996). Distribution theory of runs and patterns associated with a sequence of multi-state trials, Statistica Sinica, 6 (to appear).

    Google Scholar 

  18. Fu, J. C. and Koutras, M. V. (1994). Distribution theory of runs: a Markov chain approach, Journal of the American Statistical Association, 89, 1050–1058.

    Article  MathSciNet  MATH  Google Scholar 

  19. Gerber, H. U. and Li, S. R. (1981). The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain, Stochastic Processes and their Applications, 11, 101–108.

    Article  MathSciNet  MATH  Google Scholar 

  20. Glaz, J. (1983). Moving window detection for discrete data, IEEE Transactions on Information Theory, 29, 457–462.

    Article  MATH  Google Scholar 

  21. Glaz, J. (1989). Approximations and bounds for the distribution of the scan statistic, Journal of the American Statistical Association, 84, 560–566.

    Article  MathSciNet  MATH  Google Scholar 

  22. Glaz, J. and Naus, J. (1983). Multiple clusters on the line, Communications in Statistics-Theory and Methods, 12, 1961–1986.

    Article  MathSciNet  MATH  Google Scholar 

  23. Godbole, A. P. (1991). Poisson approximations for runs and patterns of rare events, Advances in Applied Probability, 23, 851–865.

    Article  MathSciNet  MATH  Google Scholar 

  24. Godbole, A. P. (1994). Applications of the Stein-Chen method to the theory of patterns and runs: An annotated bibliography, In Runs and Patterns in Probability (Eds., A. P. Godbole and S. G. Papastavridis), pp. 253–261, Amsterdam: Kluwer Academic Publishers.

    Google Scholar 

  25. Godbole, A. P. and Schaffner, A. A. (1993). Improved Poisson approximations for word patterns, Advances in Applied Probability, 25, 334–347.

    Article  MathSciNet  MATH  Google Scholar 

  26. Goldstein, L. (1990). Poisson approximation in DNA sequence matching, Communications in Statistics—Theory and Methods, 19, 4167–4179.

    Article  MathSciNet  MATH  Google Scholar 

  27. Guibas, L. J. and Odlyzko, A. M. (1980). Long repetitive patterns in random sequences, Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete, 53, 241–262.

    Article  MathSciNet  MATH  Google Scholar 

  28. Guibas, L. J. and Odlyzko, A. M. (1981). String overlaps, patterns matching and nontransitive games, Journal of Combinatorial Theory, 30, 183–208.

    Article  MathSciNet  MATH  Google Scholar 

  29. Hirano, K. and Aki, S. (1993). On number of occurrences of success runs of specified length in a two-state Markov chain, Statistica Sinica, 3, 313–320.

    MathSciNet  MATH  Google Scholar 

  30. Johnson, N. L., Kotz, S. and Kemp, A. W. (1992). Univariate Discrete Distributions, New York: John Wiley & Sons.

    MATH  Google Scholar 

  31. Koutras, M. V. (1996a). On a waiting time distribution in a sequence of Bernoulli trials, Annals of the Institute of Statistical Mathematics, 48 (to appear).

    Google Scholar 

  32. Koutras, M. V. (1996b). Waiting time distributions associated with runs of fixed length in two-state Markov Chains, Annals of the Institute of Statistical Mathematics, 48 (to appear).

    Google Scholar 

  33. Koutras, M. V. and Alexandrou, V. A. (1995). Runs, scans and urn model distributions: A unified Markov chain approach, Annals of the Institute of Statistical Mathematics, 47, 743–766.

    Article  MathSciNet  MATH  Google Scholar 

  34. Koutras, M. V. and Alexandrou, V. A. (1997). Sooner waiting time problems in a sequence of trinomial trials, Journal of Applied Probability, 33 (to appear).

    Google Scholar 

  35. Koutras, M. V., Papadopoulos, G. K. and Papastavridis, S. G. (1994). Circular overlapping success runs, In Runs and Patterns in Probability (Eds., A. P. Godbole and S. G. Papastavridis), pp. 287–305, Amsterdam: Kluwer Academic Publishers.

    Google Scholar 

  36. Koutras, M. V., Papadopoulos, G. K. and Papastavridis, S. G. (1995). Runs on a circle, Journal of Applied Probability, 32, 396–404.

    Article  MathSciNet  MATH  Google Scholar 

  37. Ling, K. D. (1988). On binomial distributions of order k, Statistics & Probability Letters, 6, 247–250.

    Article  MathSciNet  MATH  Google Scholar 

  38. Mohanty, S. G. (1994). Success runs of length-k in Markov dependent trials, Annals of the Institute of Statistical Mathematics, 46, 777–796.

    Article  MathSciNet  MATH  Google Scholar 

  39. Rajarshi, M. B. (1974). Success runs in a two-state Markov chain, Journal of Applied Probability, 11, 190–194.

    Article  MathSciNet  MATH  Google Scholar 

  40. Schwager, S. (1983). Run probabilities in sequences of Markov dependent trials, Journal of the American Statistical Association, 78, 168–175.

    Article  MathSciNet  MATH  Google Scholar 

  41. Solovev, A. D. (1966). A combinatorial identity and its application to the problems concerning the first occurrence of a rare event, Theory of Probability and its Applications, 11, 276–282.

    Article  Google Scholar 

  42. Uchida, M. and Aki, S. (1995). Sooner and later waiting time problems in a two-state Markov chain, Annals of the Institute of Statistical Mathematics, 47, 415–433.

    MathSciNet  MATH  Google Scholar 

  43. Viveros, R., Balasubramanian, K. and Balakrishnan, N. (1994). Binomial and negative binomial analogues under correlated Bernoulli trials, The American Statistician, 48, 243–247.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Athens, Athens, Greece

    Markos V. Koutras

Authors
  1. Markos V. Koutras
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, McMaster University, L8S 4Kl, Hamilton, Ontario, Canada

    N. Balakrishnan

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Birkhäuser Boston

About this chapter

Cite this chapter

Koutras, M.V. (1997). Waiting Times and Number of Appearances of Events in a Sequence of Discrete Random Variables. In: Balakrishnan, N. (eds) Advances in Combinatorial Methods and Applications to Probability and Statistics. Statistics for Industry and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4140-9_21

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-4140-9_21

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-8671-4

  • Online ISBN: 978-1-4612-4140-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation