Abstract
It is known that in an irreducible small Polya urn process, the composition of the urn after suitable normalization converges in distribution to a normal distribution. We show that if the urn also ...
Highlights
A Pólya urn process is defined as follows
Results on the asymptotic distribution of Xn as n → ∞ have been given by many authors under varying assumptions, using different methods. It is well-known that the asymptotic behaviour of Xn depends on the eigenvalues of R, or equivalently of its transpose A = Rt, see e.g. [8, Theorems 3.22–3.24]
We say that an eigenvalue λ is large if λ1 and strictly small if we say that the Pólya process is small if λ1 is simple and all other eigenvalues are small; a process is large whenever it is not small
Summary
A Pólya urn process is defined as follows. Consider an urn containing balls of different colours, with s possible colours which we label 1 . . . , s. If the urn is irreducible and small, Xn is asymptotically normal [8, Theorems 3.22–3.23]. Suppose that the urn is balanced, irreducible and strictly small.
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