Let \(G\) be a probability measure on \([0,\infty)\) with unbounded support such that (i) the limit \(\lim_{x\to\infty}G*G([x,\infty))/G([x,\infty))=c\) exists; (ii) for every fixed \(y\) and some \(\gamma\geq 0\), \(G([x+y,\infty))/G([x,\infty))\to e^{-\gamma y}\) as \(x\to\infty\). In this case \(G\) is said to belong to the class of \({\mathcal S}(\gamma)\)-distributions. The classes \({\mathcal S}(\gamma)\) for \(\gamma>0\) were introduced by \textit{J. Chover}, \textit{P. Ney}, and \textit{S. Wainger} [Ann. Probab. 1, 663-673 (1973; Zbl 0387.60097) and J. Anal. Math. 26, 255-302 (1973; Zbl 0276.60018)]. The authors of the present article consider mainly the case \(\gamma>0\) and prove that the constant \(c\) in (i) is equal to \(2\int_0^\infty e^{\lambda x}G(dx)\) by necessity.