Sums of independent random variables $S_n = X_1 + cdots + X_n$ are considered, where the $X_n$ are chosen according to a stationary process of distributions. For $c > 0$, let $t_c$ be the smallest positive integer n such that $$mid$S_n$mid$ > cn^{frac{1}{2}}$</TEX>. In this set up we are concerned with finiteness of expectation of $t_c$ and we have some results of sign-invariant process as applications.