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

\begin{multline*}\lim\limits_{\varepsilon\searrow 0}\varepsilon^{(2r-p-2)/(2-p)}\sum\limits^\infty_{n=1}n^{r/p-2-1/p}E\{\|S_n\|-\sigma\varepsilon n^{1/p}\}+\\ =\sigma^{-(2r-2-p)/(2-p)}\frac{p(2-p)}{(r-p)(2r-p-2)}E\|Y\|^{2(r-p)/(2-p)},\end{multline*}

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