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Comparing the distributions of sums of independent random vectors. (English) Zbl 1249.60086

Summary: Let \((X_n, n\geq 1)\), \((\tilde {X}_n, n\geq 1)\) be two sequences of i.i.d. random vectors with values in \(\mathbb {R}^k\) and \(S_n=X_1+\dots +X_n\), \(\tilde {S}n=\tilde {X}_1+\dots +\tilde {X}_n\), \(n\geq 1\). Assuming that \(\operatorname{E}X_1=\operatorname{E}\tilde {X}_1\), \(\operatorname{E}| X_1| ^2<\infty \), \(\operatorname{E}| \tilde {X}_1| ^{k+2}<\infty \) and the existence of a density of \(\tilde {X}_1\) satisfying certain conditions, we prove the following inequalities \[ \mathbf {v}(S_n,\tilde {S}_n)\leq c\;\max \big \{\mathbf {v}(X_1,\tilde {X}_1),\zeta _2(X_1,\tilde {X}_1)\big \}, \quad n=1,2,\dots , \] where \(\mathbf {v}\) and \(\zeta _2\) are the total variation and Zolotarev’s metric, respectively.

MSC:

60G50 Sums of independent random variables; random walks
60F99 Limit theorems in probability theory
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References:

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