Precise asymptotics for complete moment convergence in Hilbert spaces

Abstract

Let {X, X _n ; n ≥ 1} be a sequence of i.i.d. random variables taking values in a real separable Hilbert space $(\textbf{H},\|\cdot\|)$ with covariance operator Σ. Set $S_n=\sum_{i=1}^nX_i,$ n ≥ 1. We prove that for 1 < p < 2 and r > 1 + p/2,

$$\begin{array}{lll} &\lim\limits_{\varepsilon\searrow0}\varepsilon^{(2r-p-2)/(2-p)}\sum\limits_{n=1}^\infty n^{r/p-2-1/p}{\mbox{\rm{\textsf{E}}}}\{\|S_n\|-\sigma\varepsilon n^{1/p}\}_+\\&\quad\qquad\qquad\qquad=\sigma^{-(2r-2-p)/(2-p)}\frac{p(2-p)}{(r-p)(2r-p-2)}{\mbox{\rm{\textsf{E}}}}\|Y\|^{2(r-p)/(2-p)}, \end{array}$$

where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator Σ, and σ ² is the largest eigenvalue of Σ.