On distribution tail of the maximum of a random walk

Let Sn, n ⩾ 1, be the partial sums of i.i.d. random variables with negative mean value. Many papers (see, for example, [1,2,5,6,7,9,11]) give us different theorems on the tail behavior of the distribution of sup {Sn,n ⩾ 1}. In this paper the final versions of these theorems (with necessary and sufficient conditions) are presented. The main attention is paid to the necessity part of these theorems.