Estimates of the proximity of successive convolutions of the probability distributions on the convex sets and in the Prokhorov distance

• 主讲人：Zaitsev Andrei Yurievich (St.-Petersburg Department of Steklov Mathematical Institute)
• 举办方： Beijing-Saint Petersburg Mathematics Colloquium
• 时间： 2024-04-25 20:00 - 2024-04-25 21:00
• 地点： online

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Abstract: 

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 $n$.

 

Bio: Zaitsev Andrei works at the St.-Petersburg Department of Steklov Mathematical Institute (1978–now);

Area of interest now: Probability Theory, Probability Limit Theorems, Invariance Principles, Infinitely Divisible Distributions, Strong Approximation, Kernel density Estimators, Concentration Functions.