A small deviation theorem for independent random variables

Let $\{X_n,\,n\ge 1\}$ be a sequence of independent, not necessarily identically distributed random variables. Put $S_k(n)=\sum_{i=1+k}^{n+k}X_i$. A small deviation theorem, i.e., the asymptotic bound of $\mathbf P(\max_{i\le n}|S_k (i)|\le x_{k,n})$ is obtained under a uniform Lindeberg's condition.