Unimodality of the probability distribution of the extensive functional of samples of a random sequence

Yu. P. Virchenko ; A. M. Tevolde

Contemporary Mathematics. Fundamental Directions, Proceedings of the Voronezh Winter Mathematical Krein School — 2024, Tome 70 (2024) no. 4, pp. 542-560

Résumé

We establish a criterion for the unimodality of the probability distribution of a functional that is represented by the sum of a set of independent identically distributed random nonnegative variables ${\tilde x}_k$ with a random number of terms distributed according to Poisson. The general distribution of terms ${\tilde x}_k$ is concentrated on the interval $[0, 1]$ and is such that $\mathrm{Pr}\,\{{\tilde x}_k = 0\} \ne 0.$ Its absolutely continuous part is asymptotically close to a uniform distribution. We introduce the concept of smoothing functions and establish an explicit form of the distribution of any fixed number of terms uniformly distributed on $[0, 1].$

Détail

Keywords: sum of independent identically distributed random variables, unimodality of probability distribution, smoothing function, single-peak function.

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