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Drawing multisets of balls from tenable balanced linear urns. (English) Zbl 1275.60015
Summary: We investigate the evolution of an urn of balls of two colors, where one chooses a pair of balls and observes rules of ball-addition according to the outcome. A nonsquare ball-addition matrix of the form \(\begin{pmatrix} a&b\\c&d\\e&f\end{pmatrix}\) corresponds to such a scheme, in contrast to Pólya urn models that possess a square ball-addition matrix. We look into the case of constant row-sums (the so-called balanced urns) and identify a linear case therein. Two cases arise in linear urns, the nondegenerate and the degenerate one. Via martingales, in the nondegenerate case, one gets an asymptotic normal distribution for the number of balls of any color, in the degenerate case, a simpler probability structure underlies the process. We mention in passing a heuristic for the average-case analysis for the general case of constant row-sums.

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