Abstract:

Let $\{X_{\alpha}\}$ be a family of random variables following a certain type of distributions with finite expectation $\E[X_{\alpha}]$

and finite variance $\Var(X_{\alpha})$, where $\alpha$ is a parameter. Motivated by the recent paper of Hollom and Portier (arXiv: 2306.07811v1), we study the anti-concentration function

$(0, \infty)\ni y\to \inf_{\alpha}\P\left(|X_{\alpha}-\E[X_{\alpha}]|\geq y \sqrt{\Var(X_{\alpha})}\right)$ and find its explicit expression.

We show that, for certain familiar families of distributions, including the uniform, exponential, non-degenerate Gaussian and student's $t$-distributions, the anti-concentration function is not identically zero, which means that the corresponding families of random variables

have some sort of anti-concentration property; while for some other familiar families of distributions, including the binomial, Poisson, negative binomial, hypergeometric, Gamma, Pareto, Weibull, log-normal and Beta distributions, the anti-concentration function is identically zero.

Key words: Distribution, Measure anti-concentration