A short proof of Lévy’s continuity theorem without using tightness

Christian Döbler

Statistics & Probability Letters, 2022, vol. 185, issue C

Abstract: In this note we present a new short and direct proof of Lévy’s continuity theorem in arbitrary dimension d, which does not rely on Prohorov’s theorem, Helly’s selection theorem or the uniqueness theorem for characteristic functions. Instead, it is based on convolution with a small (scalar) Gaussian distribution as well as on basic facts about weak convergence and measure theory. Moreover, we show how, by similar means, one may prove the fact that a distribution with integrable characteristic function is absolutely continuous with respect to d-dimensional Lebesgue measure and derive the formula for its density.

Keywords: Lévy’s continuity theorem; Characteristic functions; Prohorov’s theorem; Tightness; Uniqueness theorem (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1016/j.spl.2022.109438

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