Local Limit Theorems for Poisson's Binomial in the Case of Infinite Expectation

2018/03/12 by Italo Simonelli, Simonelli, Italo, Lucia D. Simonelli +1
Mathematics · #FOS: Mathematics #Mathematical Dynamics and Fractals #Point processes and geometric inequalities #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1803.04153

openalex publication_date 2018/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let V_n = X_1,n + X_2,n + ⋯ + X_n,n where X_i,n are Bernoulli random variables which take the value 1 with probability b(i;n). Let λ_n = ∑_i=1ⁿ b(i;n) , λ= lim_{n → ∞} λ_n, and m_n = max_{1 ≤ i ≤ n} b(i;n). We derive asymptotic results for P(V_n=k) that hold without assuming that λ< +∞ or m_n → 0. Also, we do not assume k to be fixed, but instead, our results hold uniformly for all k which satisfy particular growth conditions with respect to n. These results extend known Poisson local limit theorems to the case when λ= +∞. While our results apply to triangular arrays, without the assumption that \(m_n → 0\) they continue to hold for sums of Bernoulli random variables. In this setting, our growth conditions cover a range of values for k not centered at λ_n, thus complementing known local limit theorems based on approximation by the normal distribution. In addition, we show that our local limit theorems apply to a scheme of dependent random variables introduced in the work of Sevast'yanov.

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