The asymptotic behavior of one-sided large deviation probabilities. II (Q1366357)

[For part I see ibid. 36, No. 2, 125-130 (1996) resp. ibid. 36, No. 2, 155-162 (1996).] Let \(X_1\), \(X_2, \dots\) be a sequence of independent identically distributed random variables with the distribution function \(F(t)\), \(t\in\mathbb{R}\). Put \(S_n= \sum^n_{k=1} X_k\), \(R(t)= -\log (1-F(t))\), and \(q(t)\), \(t\in\mathbb{R}_+\), is a nonnegative function such that \(R(t)= R(0)+ \int^t_0 q(u)du\), \(t\in\mathbb{R}_+\). The distribution function \(F(t)\), \(t\in\mathbb{R}\), belongs to the class \({\mathcal U}\) if the following conditions are satisfied: (i) \(\limsup_{n\to\infty} tq(t) \log R(t)/R(t) <1/4\), (ii) \(tq(t)\) is nondecreasing. The following theorems are proved. Theorem 1. Assume that \(EX_1=0\), \(F\in {\mathcal U}\), and \(\limsup_{n\to\infty} R(t)/ \log t>2\). Then \[ P[S_n \geq t] \sim n\bigl(1- F(t)\bigr), \quad n\to\infty. \tag{*} \] Theorem 2. Assume that \(EX_1=0\), \(EX^2_1= +\infty\), \(F\in {\mathcal U}\), \(\liminf_{n\to\infty} R(t)/ \log t>2\). Then (*) holds true.