A class of limit distributions for maximum cumulative sum

Let $X_1,X_2,\dots$ be a sequence of independent identically distributed random variables with zero mathematical expectation and finite variances. $S_0=0$ and $S_n\sum^n_{i=1}X_i$. It is proved that $G_a(x)= \begin{cases} 0, & \text{\rm{ if }}x\leqslant a,\\ \dfrac{\Phi(x)-\Phi(a)}{1-\Phi(a)}, & \text{\rm{ if }}x\geqslant a. \end{cases}$ is the limit distribution function of the normalized random variable $\overline S_n=\max_{0\leqslant k\leqslant n}\{S_k+a(k,n)\}$ for some sequence of centering constants $a(k,n)$.