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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2310.20283 |
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| _version_ | 1866917642188619776 |
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| author | Zaitsev, Andrei Yu. |
| author_facet | Zaitsev, Andrei Yu. |
| contents | Let $X_1,\dots, X_n,\dots$ be i.i.d.\ $d$-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 convolution). Let $$ ρ_{\mathcal{C}_d}(F,G) = \sup_A |F\{A\} - G\{A\}|, $$ 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 $c_1(F)$ such that $$ ρ_{\mathcal{C}_d}(F^n, F^{n+1})\leq \frac{c_1(F)}{\sqrt n} $$ 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 $F$ $$ ρ_{\mathcal{C}_d}(F^n, F^{n+1}) = 1. $$ A similar result for the Prokhorov distance is also obtained. For any $d$-dimensional distribution~$F$ there is a $c_2(F)>0$ that depends only on $F$ and such that \begin{multline}\nonumber (F^n)\{A\}\le (F^{n+1})\{A^{c_2(F)}\}+\frac{c_2(F)}{\sqrt{n}} \text{and}\quad (F^{n+1})\{A\}\leq (F^n)\{A^{c_2(F)}\}+\frac{c_2(F)} {\sqrt{n}} \end{multline} for any Borel set $ A $ for all positive integers $n$. Here $A^{\varepsilon }$ is $ \varepsilon $-neighborhood of the set $ A $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20283 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Estimates of stability with respect to the number of summands for distributions of successive sums of independent identically distributed vectors Zaitsev, Andrei Yu. Probability 60G50 Let $X_1,\dots, X_n,\dots$ be i.i.d.\ $d$-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 convolution). Let $$ ρ_{\mathcal{C}_d}(F,G) = \sup_A |F\{A\} - G\{A\}|, $$ 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 $c_1(F)$ such that $$ ρ_{\mathcal{C}_d}(F^n, F^{n+1})\leq \frac{c_1(F)}{\sqrt n} $$ 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 $F$ $$ ρ_{\mathcal{C}_d}(F^n, F^{n+1}) = 1. $$ A similar result for the Prokhorov distance is also obtained. For any $d$-dimensional distribution~$F$ there is a $c_2(F)>0$ that depends only on $F$ and such that \begin{multline}\nonumber (F^n)\{A\}\le (F^{n+1})\{A^{c_2(F)}\}+\frac{c_2(F)}{\sqrt{n}} \text{and}\quad (F^{n+1})\{A\}\leq (F^n)\{A^{c_2(F)}\}+\frac{c_2(F)} {\sqrt{n}} \end{multline} for any Borel set $ A $ for all positive integers $n$. Here $A^{\varepsilon }$ is $ \varepsilon $-neighborhood of the set $ A $. |
| title | Estimates of stability with respect to the number of summands for distributions of successive sums of independent identically distributed vectors |
| topic | Probability 60G50 |
| url | https://arxiv.org/abs/2310.20283 |