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Limiting distributions for a class of diminishing urn models. (English) Zbl 1300.60023
Let \(m, n, a, b, c, d, p\) be fixed integers \(\geq 0\). An urn contains \(md\) white balls and \(na\) black balls. A ball is chosen at random from the urn, its color noted, and it is returned to the urn. Also, if the color is white, \(a\) white balls and \(b\) black balls are added to the urn; similarly, if the color is black, \(c\) white balls and \(d\) black balls are added to the urn. This process is continued. This model is denoted by \(M=\left(\begin{matrix} a &b \\ c & d\end{matrix}\right)\).
The present paper studies the model \(M=\left(\begin{matrix} -a&0\\ c& -d\end{matrix}\right)\), where \(a\geq 1\), \(d\geq 1\), and \(c=pa\) (negative value denotes removal of balls as opposed to adding balls) and the distribution of the r.v. \(X_{md,na}\) representing the number of white balls in the urn when all the black balls have been removed. As motivation for this study, the paper cites the pills problem [D. Knuth, “Problem E3429: big pills and little pills”, Am. Math. Monthly 98, No. 3, 264 (1991), http://www.jstor.org/stable/2325034; “Solution: E3429”, Am. Math. Monthly 99, No. 7, 684 (1992), http://www.jstor.org/stable/2325015] and sampling without replacement defined by the model \(M=\left(\begin{matrix} -a & 0\\ 0& -d\end{matrix}\right)\).
For convenience, denote by \(Y\) the r.v. \(X_{md,na}\). Also, for reference we include here (a) the Kumaraswamy r.v. \(K(r, s)\) with distribution function \(F(x)=1-(1-xr)s\), \(x \in [0, 1]\), (b) the Weibull r.v. \(W(r, s)\) with distribution function \(F(x)=1-\exp[-(x/s)r]\), \(x\geq 0\). Note that \(W(1,s)\) is the exponential r.v. \(E(s)\) and \(W(2, s)\) is the Raleigh r.v. \(R(s)\).
The following limiting distributions are the major results in the paper.
Theorem 1. \((c=0)\). For the model \(M=\left(\begin{matrix} -a& 0\\ 0& -d\end{matrix}\right)\),
(i)
for fixed \(m\) and \(n\to\infty\), \(Y/na\) converges in distribution to \(K(d/a, m)\);
(ii)
for \(m, n \to \infty\) such that \(m^{a/d} = o(n)\), \(m^{a/d} Y/na\) converges in distribution to \(W(d/a, 1)\);
(iii)
for \(m, n \to \infty\) such that \(n\sim pm^{a/d}\), where \(p >0\), \(Y/a\) converges in distribution to \(U\), where the moment generating function of \(U\) is given by \(\varphi (p (\exp(z)-1)\), with \(\varphi(z)\) being the moment generating function of the Weibull distribution;
(iv)
for \(m\to \infty\) and \(n=o(m^{a/d} )\), \(Y\) converges in distribution to the point mass at \(0\).
Theorem 2. (\(c>0\) and \(a\leq d\)). For the model \(M=\left(\begin{matrix}-a& 0\\ c& -d\end{matrix}\right)\),
(i)
for fixed \(m\) and \(n\to \infty\), \(Y/na\) converges in distribution to \(K(d/a, m)\);
(ii)
for \(m\to \infty\) and arbitrary \(n\), \(Y/g(m,n)\) converges in distribution to \(W(d/a, 1)\).
The normalizing constants \(g(m,n)\) are explicitly given in the paper. Note that the above Theorem 2 contains the known special cases \(a=c=d=1\) and \(a=c=1\) and \(d=2\) [H. K. Hwang, M. Kuba and A. Panholzer, “Analysis of some exactly solvable diminishing urn models”, in: Proc. 19th Int. Conf. Formal Power Series and Algebraic Combinatorics, Nankai University, Tianjin (2007), http://www.fpsac.cn/PDF-Proc./Posters/43.pdf].
Theorem 3. (\(c>0\) and \(a>d\)). For the model \(M=\left(\begin{matrix}-a& 0\\ c& -d\end{matrix}\right)\),
(i)
for fixed \(m\) and \(n\to \infty\), \(Y/na\) converges in distribution to \(K(d/a, m)\);
(ii)
for \(m, n \to \infty\) such that \(m^{a/d} = o(n)\), the moments of the r.v. \(m^{a/d} Y/na\) converge to the moments of \(W(d/a, 1)\);
(iii)
for \(m, n \to \infty\) such that \(n \sim pm^{a/d}\), where \(p>0\), the moments of \(Y\) converge to a complicated expression which is explicitly given in the paper;
(iv)
for \(m\to \infty\) and \(n=o(m^{a/d})\), the moments of \(Y\) converge to a less complicated expression which is explicitly given in the paper.
Detailed derivations of the structure of the moments of the r.v.’s mentioned in the theorems above are obtained in Section 4 of the paper; these are too technical to reproduce here.
The paper ends with a section on the study of a biased Pólya-Eggenberger urn model.

MSC:
60C05 Combinatorial probability
60F05 Central limit and other weak theorems
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