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Zapiski Nauchnykh Seminarov POMI, 2023, Volume 525, Pages 86–95
(Mi znsl7369)
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This article is cited in 1 scientific paper (total in 1 paper)
Estimates of stability with respect to the number of summands for distributions of successive sums of i.i.d. vectors
A. Yu. Zaitsevab a St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
b Saint Petersburg State University
Abstract:
Let $X_1, X_2,\dots$ be i.i.d. random vectors in $\mathbf R^d$ with 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 $\mathbf R^d$. For any nontrivial distribution $F$ there is $c_1(F)$ such that $ \rho(F^n, F^{n+1})\leq \frac{c_1(F)}{\sqrt n} $ for any natural $n$. The distribution $F$ is called 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 for the Prokhorov distance is also formulated. For any $d$-dimensional distribution $F$ there is $c_2(F)$ such that $ (F^n)\{A\}\le (F^{n+1})\{A^{c_2(F)}\}+\frac{c_2(F)}{\sqrt{n}}$ and $(F^{n+1})\{A\}\leq (F^n)\{A^{c_2(F)}\}+\frac{c_2(F)} {\sqrt{n}} $ for all Borel set $ A $ and all natural $n$. Here $A^{\varepsilon }$ is $ \varepsilon $-neighborhood of the set $ A $.
Key words and phrases:
sums of independent random variables, proximity of successive convolutions, convex sets, Prokhorov distance, inequalities.
Received: 27.10.2023
Citation:
A. Yu. Zaitsev, “Estimates of stability with respect to the number of summands for distributions of successive sums of i.i.d. vectors”, Probability and statistics. Part 34, Zap. Nauchn. Sem. POMI, 525, POMI, St. Petersburg, 2023, 86–95
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https://www.mathnet.ru/eng/znsl7369 https://www.mathnet.ru/eng/znsl/v525/p86
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