On transient phenomena in random walks

Let $\overline S_n=\max_{1\le k\le n}\sum_{i=1}^{k}X_{i,n}$, where for any $n=1,2,\dots$ the sequence $X_{1,n},\dots, X_{n,n}$ consists of independent and identically distributed random variables with finite positive variances. This paper studies the problem of obtaining simple and unimprovable sufficient conditions of the Lindeberg type which guarantee the convergence of the normalized variable $(\overline S_n-A_n)/B_n$ to a nondegenerate random variable when the constants $A_n$ and $B_n>0$ are chosen, respectively. The results that Prokhorov and Borovkov obtained are simplified, refined, and strengthened. In particular, an unexplored case of when $D X_{1,n}\to 0$ as $n\to\infty$ is considered in detail.