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A new urn model. (English) Zbl 1093.60007
From the authors’ summary: A single urn contains \(b\) black balls and \(w\) white balls. For each observation, we randomly draw \(m\) balls and note their colors, say \(k\) black balls and \(m-k\) white balls. We return the drawn balls to the urn with additional \(ck\) black balls and \(c(m-k)\) white balls. This procedure is repeated \(n\) times and let \(X_n\) be the fraction of black balls after the \(n\)th draw. The asymptotic properties of \(X_n\) are investigated and it is shown that \((X_n)\) is a martingale which converges a.s. to a r.v. \(X\) whose distribution is absolutely continuous.

MSC:
60C05 Combinatorial probability
60G42 Martingales with discrete parameter
60F15 Strong limit theorems
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[1] Bagchi, A. and Pal, A. K. (1985). Asymptoic normality in the generalized Pólya–Eggenberger urn model, with an application to computer data structures. SIAM J. Algebraic Discrete Meth. 6 , 394–405. · Zbl 0568.60010 · doi:10.1137/0606041
[2] Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York. · Zbl 0822.60002
[3] Eggenberger, F. and Pólya, G. (1923). Über die statistik verketteter vorgänge. Z. Angewandte Math. Mech. 1 , 279–289. · JFM 49.0382.01
[4] Feller, W. (1971). An Introduction to Probability Theory and Its Applications , Vol. II, 2nd edn. John Wiley, New York. · Zbl 0219.60003
[5] Gouet, R. (1989). A martingale approach to strong convergence in a generalized Pólya–Eggenberger urn model. Statist. Prob. Lett. 8 , 225–228. · Zbl 0686.60023 · doi:10.1016/0167-7152(89)90126-0
[6] Gouet, R. (1993). Martingale functional central limit theorems for a generalized Pólya urn. Ann. Prob. 21 , 1624–1639. JSTOR: · Zbl 0788.60044 · doi:10.1214/aop/1176989134 · links.jstor.org
[7] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application . Academic Press, New York. · Zbl 0462.60045
[8] Hill, B., Lane, D. and Sudderth, W. (1980). A strong law for some generalized urn processes. Ann. Prob. 8 , 214–226. JSTOR: · Zbl 0429.60021 · doi:10.1214/aop/1176994772 · links.jstor.org
[9] Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application . John Wiley, New York. · Zbl 0352.60001
[10] Kotz, S. and Balakrishnan, N. (1997). Advances in urn models during the past two decades. In Advances in Combinatorial Methods and Applications to Probability and Statistics , Birkhäuser, Boston, MA, pp. 203–257. · Zbl 0888.60014
[11] Maistrov, L. E. (1974). Probability Theory: a Historical Sketch . Academic Press, New York. · Zbl 0308.01001
[12] Pemantle, R. (1990). A time-dependent version of Pólya’s urn. J. Theoret. Prob. 3 , 627–637. · Zbl 0708.60015 · doi:10.1007/BF01046101
[13] Schreiber, S. J. (2001). Urn models, replicator processes, and random genetic drift. SIAM J. Appl. Math. 61 , 2148–2167. · Zbl 0993.92026 · doi:10.1137/S0036139999352857
[14] Stewart, I. (1989). Galois Theory , 2nd edn. Chapman and Hall, New York. · Zbl 0269.12104
[15] Wei, C. Z. (1993). Martingale transforms with non-atomic limits and stochastic approximation. Prob. Theory Relat. Fields 95 , 103–114. · Zbl 0792.60034 · doi:10.1007/BF01197340
[16] Wheeden, R. L. and Zygmund, A. (1977). Measure and Integral . Marcel Dekker, New York. · Zbl 0362.26004
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