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On the error of the Gaussian approximation for convolutions
V. V. Yurinskiĭ Novotcherkassk
Abstract:
Let
$F_n$ be the distribution in
$R^k$ of the sum of independent random vectors
$\xi_1,\dots,\xi_n$, and
$G_n$ be the normal distribution with the same means and ńīvariances as
$F_n$.
If
$\mathfrak L_{\Pi}$ is the Lévy–Prohorov distance between distributions in
$R^k$ defined by the Euclidean norm in
$R^k$, Theorem 1 yields the estimate
\begin{gather*}
\mathfrak L_{\Pi}(F_n,G_n)\le ck^{1/4}\mu_1^{1/4}[|\ln\mu_1|^{1/2}+(\ln k)^{1/2}],
\\
\mu_1=\mathbf E|\xi_1-\mathbf E \xi_1|^3+\dots+\mathbf E|\xi_n-\mathbf E \xi_n|^3,
\end{gather*}
with
$c$ being an absolute constant.
A similar bound holds when
$\mathfrak L_{\Pi}$ is defined using a non-Hilbert but sufficiently smooth norm in
$R^k$ (Theorem 2).
Finite-dimensional bounds are used in Section 2 to obtain coarse power convergence rate in the multidimensional invariance principle for a random walk.
Received: 16.12.1975
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