Journal of Mathematical Sciences

, 91:2904 | Cite as

On waiting time in the Markov-Pólya scheme

  • A. V. Ivanov
  • G. I. Ivchenko
Article

Abstract

We consider the Markov-Pólya urn scheme. The urn contains a given number of balls of each of N different colors. The balls are sequentially drawn from the urn one at a time, independently of each other. and with equal probabilities to be drawn. After each draw, the ball drawn is returned into the urn together with c balls of the same color, c∈{−1, 0, 1, 2, ...}. The drawing process halts when, for the first time, the frequencies of k unspecified colors attain or exceed the corresponding (random) levels settled before the beginning of the trials. Limit distributions of the stopping time υc(N, K) are considered as N→∞, k=k(N); in particular, the dependence of the asymptotic properties of the waiting time on the parameter c is studied. Results concerning υc(N, K) are derived as consequences of general limit theorems for decomposable statistics
$$L_{NK} = \sum\limits_{j = 1}^N {g_j (\eta _j )} $$
where g1, ..., gN are given functions of integer argument, and η1...ηN are the frequencies, i.e., the amounts of balls of the corresponding colors, at the stopping time.

Keywords

Limit Theorem Limit Distribution Birth Process Random Level Integer Argument 

References

  1. 1.
    W. Feller,An Introduction to Probability Theory and Its Applications, Vol. 1, Wiley, New York (1957).MATHGoogle Scholar
  2. 2.
    F. Eggenberger and G. Pólya, “Über die statistik verketteter Vorgänge”,Z. Angew. Math. Mech.,3, 279–289 (1923).MATHGoogle Scholar
  3. 3.
    G. I. Ivchenko and Yu. I. Medvedev, “On the Markov-Pólya urn scheme from 1917 to the present,”Obozr. Prikl Prom. Mat.,3, 484–511 (1996).MATHGoogle Scholar
  4. 4.
    A. A. Markov, “On some limiting formulas of probability calculus,”Izv. Akad. Nauk,11, 177–186 (1917).MATHGoogle Scholar
  5. 5.
    A. A. Markov,Selected Works: Number Theory, Probability Theory [in Russian], Soviet Acad. Sci. Press, Moscow (1951).MATHGoogle Scholar
  6. 6.
    A. A. Chuprov and A. A. Markov,On Probability Theory and Mathematical Statistics [in Russian], Nauka, Moscow (1978).Google Scholar
  7. 7.
    G. I. Ivchenko, “Decomposable statistics in an inverse allocation problem,”Discrete Math. Appl.,1, 81–96 (1991).MATHMathSciNetGoogle Scholar
  8. 8.
    G. I. Ivchenko, “The waiting time and related statistics in the multinomial scheme: a survey,”Discrete Math. Appl.,3, 451–482 (1993).MATHMathSciNetGoogle Scholar
  9. 9.
    G. I. Ivchenko and A. V. Ivanov, “Decomposable statistics in inverse urn problems,”Discrete Math. Appl.,5, 159–172 (1995).MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    G. I. Ivchenko and A. V. Ivanov, “Decomposable statistics and stopping times in sampling without replacement,”Discrete Main. Appl.,7, 47–64 (1997).MATHMathSciNetGoogle Scholar
  11. 11.
    L. Holst and J. Husler, “Sequential urn schemes and birth processes,”Adv. Appl. Probab.,17, 257–259 (1985).MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    J. W. Pratt, “On interchanging limits and integrals,”Ann. Math. Statist.,31, 74–77 (1960).MATHMathSciNetGoogle Scholar
  13. 13.
    V. A. Ivanov, “Limit theorems in allocation scheme with random levels,”Math. Notes,31, 619–631 (1982).MATHGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • A. V. Ivanov
    • 1
  • G. I. Ivchenko
    • 1
  1. 1.Department of Probability Theory and Mathematical StatisticsMoscow Institute of Electronic MachineryMoscowRussia

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