Let ${X_n : n = 1,2,cdots}$ be a sequence of pairwise independent identically distributed random vectors taking values in a separable Hilbert space H such that $E Vert X_1 Vert = infty$. Let $S_n = X_1 + X_2 + cdots + X_n$ and for any real $alpha$ with $0 < alpha < 1$ define a sequence ${gamma_n(alpha)}$ as $gamma_n(alpha) = inf {r : P(Vert S_n Vert leq r) geq alpha}$. Then $$ lim_{n o infty} sup Vert S_n Vert/gamma_n(alpha) = infty $$ holds. This is a generalization of Vvedenskaya[2].