Error bounds for the normal approximation to the length of a Ewens partition

04/03/2019

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by   Koji Tsukuda, et al.

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Let K(=K_n,θ) be a positive integer-valued random variable whose distribution is given by P(K = x) = s̅(n,x) θ^x/(θ)_n (x=1,...,n) , where θ is a positive number, n is a positive integer, (θ)_n=θ(θ+1)...(θ+n-1) and s̅(n,x) is the coefficient of θ^x in (θ)_n for x=1,...,n. This formula describes the distribution of the length of a Ewens partition, which is a standard model of random partitions. As n tends to infinity, K asymptotically follows a normal distribution. Moreover, as n and θ simultaneously tend to infinity, if n^2/θ→∞, K also asymptotically follows a normal distribution. In this paper, error bounds for the normal approximation are provided. The result shows that the decay rate of the error changes due to asymptotic regimes.