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Analytic urns. (English) Zbl 1073.60007
The article considers the “generalized” (or “extended) Pólya-Eggenberger urn model with two types of balls, say “black” (B) and “white” (W). At time $$0$$ the urn contains $$a_0$$ B-balls and $$b_0$$ W-balls. At time $$n$$ a ball in the urn is randomly chosen and its color is observed (thus the ball is placed back in the urn). If it is B, then $$\alpha_{BB}$$ B-balls and $$\alpha_{BW}$$ W-balls are inserted, while if the chosen ball is W, then $$\alpha_{WB}$$ B-balls and $$\alpha_{WW}$$ W-balls are inserted. This “urn evolution” rule is conveniently represented by the matrix $$M = \{ (\alpha_{BB}, \alpha_{BW}); (\alpha_{WB}, \alpha_{WW}) \}$$. It is, furthermore, assumed that the urn is “balanced”, namely the total number of balls inserted (during each step) is always $$s$$. Thus, $$s = \alpha_{BB} + \alpha_{BW} = \alpha_{WB} + \alpha_{WW}$$, and the total number of balls in the urn at time $$n$$ is $$t_0 + s n$$, where $$t_0 = a_0 + b_0$$ is the initial size. The authors focus attention in the case involving subtraction (although, as they explain, their method can be applied to all other balanced cases), namely when $$\alpha_{BB} = -a$$, $$\alpha_{BW} = a + s$$, $$\alpha_{WB} = b + s$$, and $$\alpha_{WW} = -b$$, where $$a,b > 0$$, and $$s > 0$$. In order for the urn not to be blocked by an infeasible request, one needs to impose the so-called “tenability” conditions: ($${\text T}_0$$) $$a$$ divides $$a_0$$ and $$b$$ divides $$b_0$$; ($${\text T}_1$$) $$a$$ divides $$b + s$$ and $$b$$ divides $$a + s$$. For $$n = 0, 1, 2, \dots$$ let $$X_n$$ be the number of B-balls at time $$n$$, $$p_n(u) = E[u^{X_n}]$$ its PGF (probability generating function), and $$h_n(u) = \sum_{k \geq 0} c(k; n) u^k$$ the counting generating function (meaning that $$c(k; n)$$ is the number of ways of getting exactly $$k$$ B-balls in the urn at time $$n$$ – in fact, for each $$n$$, the polynomials $$p_n(u)$$ and $$h_n(u)$$ differ by a simple constant factor).
The authors introduce the exponential generating function of $$h_n(u)$$, namely the BGF (bivariate generating function) $$H(z, u) = \sum_{n \geq 0} h_n(u) z^n / n!$$ and notice that $$H(z, 1) = (1 - s z)^{-t_0/s}$$. Then, after a series of elegant steps, where certain partial differential operators are interpreted combinatorially, they arrive to the “fundamental equation” (14) for $$H(z, u)$$, which is a first-order linear PDE. Thus, the old method of characteristics gives $$H(z, u)$$ explicitly in terms of the Abelian integral $$I(u) = \int u^{a - 1} v^{- a - b}$$ over the (complex) “Fermat curve” $$u^h + v^h = 1$$, where $$h = a + b + s$$ (Theorem 1), and it turns out that the major characteristics of the associated urn model are determined by the (global) analytic function $$I(u)$$. The authors, then, determine the fundamental polygon of $$I(u)$$, which is a quadrilateral, the “elementary kite”. In the last part of the paper, the form of $$H(z, u)$$ is used in order to obtain some deep probabilistic properties of the urn model (e.g. speed of convergence to the Gaussian limit, as $$n \to \infty$$, large deviations, etc). Naturally, particular attention is given to the special case where $$H(z, u)$$ can be expressed in terms of elliptic functions. The authors show that there are exactly six (non-equivalent) “elliptic” urn models. The paper ends with some final comments and plans of future research.

##### MSC:
 60C05 Combinatorial probability 33E05 Elliptic functions and integrals 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 60K99 Special processes 60Fxx Limit theorems in probability theory
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