On the asymptotics of one-sided large deviation probabilities
The one-side large deviation theorem given by \textit{S. V. Nagaev} [Theory Probab. Appl. 26, 362-366 (1982); translation from Teor. Veroyatn. Primen. 26, 369-372 (1981; Zbl 0457.60020)] in the case when a distribution function \(F\) of summands \(X_1,\dots, X_n\) of \(S_n = \sum^n_{k = 1} X_k\) satisfies \(1 - F(t) \sim t^{-\alpha} h(t)\), \(t \to \infty\), \(a > 1\), and \(h(t)\) is a slowly varying function, is extended to a larger class of distribution functions. Namely, it is proved that \(P[S_n \geq t] \sim n(1 - F(t))\) as \(n \to \infty\) and \(\limsup_{n \to \infty} {n\over t} < \infty\), when \(F\) satisfies \(EX_1 = 0\), \[ \limsup_{t \to \infty} ((1 - F(t))/(1 - F(2t))) < \infty,\qquad \lim_{t \to \infty} ((1 - F(t - y))/(1 - F(t))) = 1,\quad \forall y \in \mathbb{R}, \] the hazard rate \(q\) of \(F(t)\), \(t \in \mathbb{R}^+\), satisfies \(\limsup_{t \to \infty} tq(t) < \infty\), and moreover \[ \liminf_{t \to \infty} (-\log (1 - F(t)))/\log t > 1,\qquad \liminf_{t \to \infty} (-\log(1 - F(t)))/\log|t|> 1. \]
scientific article; zbMATH DE number 846590