Precise rates in the law of the logarithm in the Hilbert space
Let \(\{X,X_n\); \(n\geq 1\}\) be a sequence of i.i.d. random variables taking values in a real separable Hilbert space (\(\mathbb H,\|\cdot\|)\) with covariance operator \(\Sigma\), and set \(S_n=X_1+\cdots+X_n\), \(n\geq 1\). Let \(a_n=o(\sqrt{n/\log n})\). We prove that, for any \(1< r < 3/2\) and \(a>-d/2\), \[ \begin{multlined} \lim_{\varepsilon\searrow\sqrt{r-1})}[\varepsilon^2-(r-1)]^{a+d/2}\sum^\infty_{n=1}n^{r-2}(\log n)^a{\mathbf P}\{\|S_n\|\geq \sigma\phi(n)\varepsilon + a_n\}\\ =\Gamma^{-1}(d/2)K(\Sigma)(r-1)^{(d-2)/2}\Gamma(a+d/2)\end{multlined} \] holds if \[ {\mathbf E} X=0,\quad {\mathbf E}[\|X\|^{2r}(\log\| X\|)^{a-r}]<\infty. \]
Tail probabilities of sums of i.i.d. random variables
scientific article; zbMATH DE number 2157713
scientific article