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Periodic Pólya urns, the density method and asymptotics of Young tableaux. (English) Zbl 1453.60010
This paper introduces a balanced periodic Pólya urn model of period \(p\) and establishes some explicit enumeration results and links with hypergeometric functions as well as the limit law using a product of generalized gamma distributions. For example, the re-normalized distribution of black balls in a Young-Pólya urn of period \(p\) and parameter \(l\) is shown to be given by the following product of distributions as \(n\) goes to infinity: \(p^{\delta}B_n/[(p+l)n^{\delta}]\) tends in distribution to \(\mathrm{Beta}(b_0,w_0)\prod_{i=0}^{l-1}\mathrm{GenGamma}(b_0+w_0+p+i,p+l)\), where \(B_n\) is the number of black balls after \(n\) steps, \(\delta=p/(p+l)\) and \(\mathrm{Beta}(b_0,w_0)=1\) when initial white ball number \(w_0=0\) or \(\mathrm{Beta}(b_0,w_0)\) is the beta distribution with support \([0,1]\) and density \([\Gamma(b_0+w_0)/[\Gamma(b_0)\Gamma(w_0)]]x^{b_0-1}(1-x)^{w_0-1}\) otherwise. A relation between the southeast and the northwest corners of triangular Young tableaux is also obtained. Some other tidbits discussed include some universality properties of random surfaces and the tails of Mittag-Leffler distributions.

MSC:
60C05 Combinatorial probability
05A15 Exact enumeration problems, generating functions
60F05 Central limit and other weak theorems
60K99 Special processes
Software:
gfun; TDDS; DLMF
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