Random weighted averages, partition structures and generalized arcsine laws

Abstract

This article offers a simplified approach to the distribution theory of randomly weighted averages or $P$-means $M_P(X):={\sum }_j{X}_j{P}_j$, for a sequence of i.i.d.random variables $X,X_1,X_2,\dots$, and independent random weights $P:=(P_j)$ with $P_j\ge 0$ and ${\sum }_j{P}_j=1$. The collection of distributions of $M_P(X)$, indexed by distributions of $X$, is shown to encode Kingman's partition structure derived from $P$. For instance, if $X_p$ has Bernoulli$(p)$ distribution on $\{0,1\}$, the $n$th moment of $M_P(X_p)$ is a polynomial function of $p$ which equals the probability generating function of the number $K_n$ of distinct values in a sample of size $n$ from $P$: $E(M_P(X_p){)}^n=E{p}^{K_n}$. This elementary identity illustrates a general moment formula for $P$-means in terms of the partition structure associated with random samples from $P$, first developed by Diaconis and Kemperman (1996) and Kerov (1998) in terms of random permutations. As shown by Tsilevich (1997) if the partition probabilities factorize in a way characteristic of the generalized Ewens sampling formula with two parameters $(\alpha ,\theta )$, found by Pitman (1992), then the moment formula yields the Cauchy-Stieltjes transform of an $(\alpha ,\theta )$ mean. The analysis of these random means includes the characterization of $(0,\theta )$-means, known as Dirichlet means, due to Von Neumann (1941), Watson (1956) and Cifarelli and Regazzini (1990) and generalizations of Lévy's arcsine law for the time spent positive by a Brownian motion, due to Darling (1949) Lamperti (1958) and Barlow, Pitman and Yor (1989).

arXiv comment76 pages, 2 figures

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• Robust Mean Estimation with the Bayesian Median of Means
• Arcsine laws for random walks generated from random permutations with applications to genomics
• Large Deviations For Randomly Weighted Sums of Random Measures