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Distribution functions of means of a Dirichlet process. (English) Zbl 0706.62012

\(\chi\) is a random probability measure chosen by a Dirichlet process with parameter \(\alpha\) such that \(Y=\int x \chi d(x)\) is a random variable. The paper deals with the distribution function M of Y; main object is the statement of a suitable expression for this distribution. Such an expression is derived by an extension of a procedure based on generalized Stieltjes transform. The paper has four sections.
The first section deals with the basic definitions and elementary results concerning Dirichlet processes. Section 2 gives a random functional which is directly related to Y. A recurrence relation for the moments of this new functional is derived and based on this, a generalized Stieltjes transform of the probability distribution is determined.
Section 3 gives explicit expression for M through inversion formulas developed by D. B. Sumner [Bull. Am. Math. Soc. 55, 174-183 (1949; Zbl 0032.35501)] and I. I. Hirschman jun. and D. V. Widder [Duke Math. J. 17, 391-402 (1950; Zbl 0039.333)]. The last section gives three applications. The first of these relates to the case when \(\alpha\) is proportional to a Cauchy pdf. The second relates to the limit distribution of \(\sum^{n}_{k=1}x_ k/n\) as \(n\to \infty\) and the last application deals with the posterior distribution of \(\chi\).
Reviewer: G.S.Lingappaiah

MSC:

62E15 Exact distribution theory in statistics
60E05 Probability distributions: general theory
62E20 Asymptotic distribution theory in statistics
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