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An algebraic approach to Pólya processes. (English) Zbl 1185.60029
The paper deals with Pólya processes which are generalizations of Pólya-Eggenberger urn models with constant balance (i.e. for which the total number of added “balls” at any time is constant). The difference is that, instead counting a number of “balls”, positive real quantities corresponding to each “colour” are considered and entries of the replacement matrix $$R$$ are not integers but reals. Then the process is normalized to have balance 1. A Pólya process is called small if 1 is a simple eigenvalue of the replacement matrix $$R$$ and every other eigenvalue of $$R$$ has real part not greater than $$1/2$$. Otherwise, it is called large. The main results of the paper give an almost sure (and in $$L_p$$, $$p\geq 1$$) asymptotic representation of a large Pólya process up to the order $$o(n^\sigma)$$, where $$\sigma$$ is the maximal real part of eigenvalues $$\lambda_2,\dots,\lambda_s$$ of $$R$$ except $$\lambda_1$$ always being equal to 1. This asymptotic representation is described by finitely many random variables that appear as limits of martingales. The proofs of main results are based on estimates of moments of a Pólya process which have been obtained by an application of the spectral decomposition of a suitable finite difference transition operator on polynomial functions.

##### MSC:
 60F15 Strong limit theorems 60F17 Functional limit theorems; invariance principles 60F25 $$L^p$$-limit theorems 60G05 Foundations of stochastic processes 60G42 Martingales with discrete parameter 60J05 Discrete-time Markov processes on general state spaces 68W40 Analysis of algorithms
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