Advertisement

Markov Chains

  • László Lakatos
  • László Szeidl
  • Miklós Telek
Chapter

Abstract

In the early twentieth century, Markov (1856–1922) introduced in Markov (Izvestiya Fiziko-matematicheskogo Obschestva pri Kazanskom Universitete 15:135–156, 1906) a new class of models called Markov chains, applying sequences of dependent random variables that enable one to capture dependencies over time.

References

  1. 1.
    Apostol, T.: Calculus II. Wiley, London (1969)zbMATHGoogle Scholar
  2. 2.
    Bellman, R.: Introduction to Matrix Analysis. SIAM, Philadelphia (1997)zbMATHGoogle Scholar
  3. 3.
    Bernstein, S.N.: The Theory of Probabilities. Leningrad, Moscow (1946)Google Scholar
  4. 4.
    Erdős, P., Feller, W., Pollard, H.: A theorem on power series. Bull. Amer. Math. Soc. 55, 201–203 (1949)Google Scholar
  5. 5.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. I. Wiley, New York (1968)zbMATHGoogle Scholar
  6. 6.
    Foster, F.G.: On the stochastic matrices associated with certain queuing processes. Ann. Math. Stat. 24, 355–360 (1953)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gikhman, I., Skorokhod, A.: The Theory of Stochastic Processes, vol. I. Springer, Berlin (1974)Google Scholar
  8. 8.
    Gikhman, I.I., Skorokhod, A.V.: The Theory of Stochastic Processes, vol. II. Springer, Berlin (1975)zbMATHGoogle Scholar
  9. 9.
    Haccou, P., Jagers, P., Vatutin, V.A.: Branching Processes: Variation, Growth, and Extinction of Populations, vol. 5. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  10. 10.
    Harris, T.E.: The Theory of Branching Processes. Courier Corporation, Chelmsford (2002)zbMATHGoogle Scholar
  11. 11.
    Izsák, J., Szeidl, L.: Population dynamic models leading to logarithmic and Yule distribution. Acta Inform. Hung. 15(1), 149–162 (2018)Google Scholar
  12. 12.
    Johnson, N.L., Kemp, A.M., Kotz, S.: Univariate Discrete Distributions, 3rd edn. Wiley, London (2005)CrossRefGoogle Scholar
  13. 13.
    Kamke, E.: Differencialgleichungen. I. Gewoehnliche Differencialgleichungen, 10th edn. Springer, Berlin (1977)Google Scholar
  14. 14.
    Kendall, D.G.: On the generalized birth-and-death process. Ann. Math. Stat. 19(1), 1–15 (1948)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kendall, D.G.: Stochastic processes and population growth. J. R. Stat. Soc. Ser. B 11(2), 230–282 (1949)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Klimov, G.P.: Extremal Problems in Queueing Theory. Energia, Moscow (1964)Google Scholar
  17. 17.
    Lange, K.: Applied Probability, 2nd edn. Springer, Berlin (2010)CrossRefGoogle Scholar
  18. 18.
    Markov, A.A.: Rasprostranenie zakona bol’shih chisel na velichiny, zavisyaschie drug ot druga. Izvestiya Fiziko-matematicheskogo Obschestva pri Kazanskom Universitete 15, 135–156 (1906)Google Scholar
  19. 19.
    Matveev, V.F., Ushakov, V.G.: Queueing Systems. MGU, Moscow (1984)Google Scholar
  20. 20.
    Meyn, S., Tweedie, R.: Markov Chains and Stochastic Stability. Springer, Berlin (1993)CrossRefGoogle Scholar
  21. 21.
    Olver, P.J.: Introduction to Partial Differential Equations. Springer, Berlin (2014)CrossRefGoogle Scholar
  22. 22.
    Seneta, E.: Non-negative Matrices and Markov Chains. Springer, Berlin (2006)zbMATHGoogle Scholar
  23. 23.
    Serfozo, R.: Basics of Applied Stochastic Processes. Springer, Berlin (2009)CrossRefGoogle Scholar
  24. 24.
    Shiryaev, A.N.: Probability. Springer, Berlin (1994)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • László Lakatos
    • 1
  • László Szeidl
    • 2
  • Miklós Telek
    • 3
  1. 1.Eotvos Lorant UniversityBudapestHungary
  2. 2.Obuda UniversityBudapestHungary
  3. 3.Technical University of BudapestBudapestHungary

Personalised recommendations