Comparing the distributions of sums of independent random vectors

Evgueni I. Gordienko

Kybernetika (2005)

  • Volume: 41, Issue: 4, page [519]-529
  • ISSN: 0023-5954

Abstract

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Let ( X n , n ≥ 1 ) , ( X ˜ n , n ≥ 1 ) be two sequences of i.i.d. random vectors with values in ℝ k and S n = X 1 + ⋯ + X n , S ˜ n = X ˜ 1 + ⋯ + X ˜ n , n ≥ 1 . Assuming that E X 1 = E X ˜ 1 , E | X 1 | 2 < ∞ , E | X ˜ 1 | k + 2 < ∞ and the existence of a density of X ˜ 1 satisfying the certain conditions we prove the following inequalities: v ( S n , S ˜ n ) ≤ c max { v ( X 1 , X ˜ 1 ) , ζ 2 ( X 1 , X ˜ 1 ) } , n = 1 , 2 , ⋯ , where v and ζ 2 are the total variation and Zolotarev’s metrics, respectively.

How to cite

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Gordienko, Evgueni I.. "Comparing the distributions of sums of independent random vectors." Kybernetika 41.4 (2005): [519]-529. <http://eudml.org/doc/33769>.

@article{Gordienko2005,
abstract = {Let $(X_n, n\ge 1), (\tilde\{X\}_n, n\ge 1)$ be two sequences of i.i.d. random vectors with values in $\{\mathbb \{R\}\}^k$ and $S_n=X_1+\cdots +X_n$, $\tilde\{S\}_n=\tilde\{X\}_1+\cdots +\tilde\{X\}_n$, $n\ge 1$. Assuming that $EX_1=E\tilde\{X\}_1$, $E|X_1|^2<\infty $, $E|\tilde\{X\}_1|^\{k+2\}<\infty $ and the existence of a density of $\tilde\{X\}_1$ satisfying the certain conditions we prove the following inequalities: \[v(S\_n,\tilde\{S\}\_n)\le c\;\max \big \lbrace v(X\_1,\tilde\{X\}\_1), \zeta \_2(X\_1,\tilde\{X\}\_1)\big \rbrace , \quad n=1,2,\dots ,\] where $v$ and $\zeta _2$ are the total variation and Zolotarev’s metrics, respectively.},
author = {Gordienko, Evgueni I.},
journal = {Kybernetika},
keywords = {sum of random vectors; the total variation distance; bound of closeness; Zolotarev’s metric; characteristic function; sum of random vectors; the total variation distance; bound of closeness; Zolotarev's metric; characteristic function},
language = {eng},
number = {4},
pages = {[519]-529},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Comparing the distributions of sums of independent random vectors},
url = {http://eudml.org/doc/33769},
volume = {41},
year = {2005},
}

TY - JOUR
AU - Gordienko, Evgueni I.
TI - Comparing the distributions of sums of independent random vectors
JO - Kybernetika
PY - 2005
PB - Institute of Information Theory and Automation AS CR
VL - 41
IS - 4
SP - [519]
EP - 529
AB - Let $(X_n, n\ge 1), (\tilde{X}_n, n\ge 1)$ be two sequences of i.i.d. random vectors with values in ${\mathbb {R}}^k$ and $S_n=X_1+\cdots +X_n$, $\tilde{S}_n=\tilde{X}_1+\cdots +\tilde{X}_n$, $n\ge 1$. Assuming that $EX_1=E\tilde{X}_1$, $E|X_1|^2<\infty $, $E|\tilde{X}_1|^{k+2}<\infty $ and the existence of a density of $\tilde{X}_1$ satisfying the certain conditions we prove the following inequalities: \[v(S_n,\tilde{S}_n)\le c\;\max \big \lbrace v(X_1,\tilde{X}_1), \zeta _2(X_1,\tilde{X}_1)\big \rbrace , \quad n=1,2,\dots ,\] where $v$ and $\zeta _2$ are the total variation and Zolotarev’s metrics, respectively.
LA - eng
KW - sum of random vectors; the total variation distance; bound of closeness; Zolotarev’s metric; characteristic function; sum of random vectors; the total variation distance; bound of closeness; Zolotarev's metric; characteristic function
UR - http://eudml.org/doc/33769
ER -

References

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