Comparing the distributions of sums of independent random vectors

Gordienko, Evgueni

Kybernetika, Tome 41 (2005) no. 4, p. [519]

Résumé

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.

Détail

Classification : 60F99, 60G50
Keywords: sum of random vectors; the total variation distance; bound of closeness; Zolotarev’s metric; characteristic function