Password: U0BP
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.
