On sums of independent random variables without power moments (Q2477498)

Let \(X,X_1,X_2,\dots\) be i.i.d. random variables such that \(V(x)= P\{X\geq x\}\) is a slowly varying function as \(x\to\infty\). \textit{P. Levy} [J. Math. Pures Appl., IX. Sér. 14, 347--402 (1935; Zbl 0013.02804)] and \textit{D.A. Darling} [Trans. Am. Math. Soc. 73, 95--107 (1952; Zbl 0047.37502)] proved that \(\max\{X_1,\dots, X_n\}\) makes an overwhelming contribution to \(S= X_1+\cdots+ X_n\). These investigations are generalized in the paper. Here, we give a typical result. Denote \[ \begin{gathered} L^+(y)= [1- E(e^{yX}\mid X\geq 0)]\,P\{X\geq 0\},\\ L^-(y)= [1- E(e^{yX}\mid X< 0)]\,P\{X< 0\}.\end{gathered} \] Assume that \(L^+(y)\) is a slowly varying function as \(y\to\infty\) such that \[ \lim_{y\to\infty}\,{L^+(y)\over L^-(y)}= {p^+\over p^-} \] for some \(p\in (0,1)\), where \(p^+= p\) and \(p^-= 1- p\). Then \[ \lim_{n\to\infty}\, P\{nL^{\pm}(\pm S_n)< y;\;\pm S_n> 0\}= p^{\pm}(1- e^{-x/p^{\pm}}) \] for any \(y> 0\).