Семинары: A. Yu. Zaitsev, Estimates of the proximity of successive convolutions of the probability distributions on the convex sets and in the Prokhorov distance

СЕМИНАРЫ


Совместный общематематический семинар СПбГУ и Пекинского Университета 25 апреля 2024 г. 15:00, г. Санкт-Петербург, online

Estimates of the proximity of successive convolutions of the probability distributions on the convex sets and in the Prokhorov distance A. Yu. Zaitsev St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
Аннотация: Let $X_1,\dots, X_n,\dots$ be independent identically distributed $d$-dimensional random vectors with common distribution $F$. Then $S_n = X_1+\dots+X_n$ has distribution $F^n$ (degree is understood in the sense of convolutions). Let $$\rho(F,G) = \sup_A \|F\{A\} - G\{A\}\|,$$ where the supremum is taken over all convex subsets of $\mathbb R^d$. Basic result is as follows. For any nontrivial distribution $F$ there is $c(F)$ such that $$\rho(F^n, F^{n+1})\leq \frac{c(F)}{\sqrt n}$$ for any natural $n$. The distribution $F$ is considered trivial if it is concentrated on a hyperplane that does not contain the origin. Clearly, for such $F$ $$\rho(F^n, F^{n+1}) = 1.$$ A similar result is obtained for the Prokhorov distance between distributions normalized by the square root of $n$. Язык доклада: английский