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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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##### References:
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