Large Deviations For Randomly Weighted Sums of Random Measures

2021/06/23 by Shui Feng, Feng, Shui
Computer Science · Mathematics · #60G57(Primary) 62F15(Secondary) #Bayesian Methods and Mixture Models #FOS: Mathematics #Mathematical Approximation and Integration #Probability (math.PR) #Statistical Methods and Inference

paper · pdf · doi:10.48550/arxiv.2106.12493

openalex publication_date 2021/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \\bf Z_n:n≥ 1\ be a sequence of i.i.d. random probability measures. Independently, for each n≥ 1, let (X_n1,…, X_nn) be a random vector of positive random variables that add up to one. This paper studies the large deviation principles for the randomly weighted sum ∑_i=1ⁿ X_ni Z_i. In the case of finite Dirichlet weighted sum of Dirac measures, we obtain an explicit form for the rate function. It provides a new measurement of divergence between probabilities. As applications, we obtain the large deviation principles for a class of randomly weighted means including the Dirichlet mean and the corresponding posterior mean. We also identify the minima of relative entropy with mean constraint in both forward and reverse directions.

Citations

• Random weighted averages, partition structures and generalized arcsine laws

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