On the anti-concentration functions of some familiar families of distributions

Abstract

Let $\{X_{\alpha }\}$ be a family of random variables following a certain type of distributions with finite expectation $\mathbf{E}[X_{\alpha }]$ and finite variance $\mathrm{Var}(X_{\alpha })$, where $\alpha$ is a parameter. Motivated by the recent paper of Hollom and Portier (arXiv: 2306.07811v1), we study the anti-concentration function $(0,\infty )\ni y\to {\inf }_{\alpha }\mathbf{P}\left(|X_{\alpha }-\mathbf{E}[X_{\alpha }]|\ge y\sqrt{\mathrm{Var}(X_{\alpha })}\right)$ and find its explicit expression. We show that, for certain familiar families of distributions, including uniform distributions, exponential distributions, non-degenerate Gaussian distributions and student's $t$-distribution, the anti-concentration function is not identically zero, while for some other familiar families of distributions, including binomial, Poisson, negative binomial, hypergeometric, Gamma, Pareto, Weibull, log-normal and Beta distributions, the anti-concentration function is identically zero.

15 pages