A supplement to the laws of large numbers and the large deviations

TL;DR

This paper extends classical large deviation results for sums of i.i.d. Banach space-valued variables, providing precise asymptotics for tail probabilities under certain convergence conditions, using advanced probabilistic inequalities.

Contribution

It introduces new asymptotic bounds for large deviations of sums in Banach spaces, extending and improving prior results by Hu and Nyrhinen (2004).

Findings

Established exact asymptotic limits for tail probabilities.

Extended previous large deviation results to broader conditions.

Utilized symmetrization and advanced inequalities for proofs.

Abstract

Let $0 < p < 2$. Let $\{X, X_{n}; n \geq 1\}$ be a sequence of independent and identically distributed $\mathbf{B}$-valued random variables and set $S_{n} = \sum_{i=1}^{n}X_{i},~n \geq 1$. In this paper, a supplement to the classical laws of large numbers and the classical large deviations is provided. We show that if $S_{n}/n^{1/p} \rightarrow_{\mathbb{P}} 0$, then, for all $s > 0$, \[ \limsup_{n \to \infty} \frac{1}{\log n} \log \mathbb{P}\left(\left\|S_{n} \right\| > s n^{1/p} \right) = - (\bar{\beta} - p)/p \] and \[ \liminf_{n \to \infty} \frac{1}{\log n} \log \mathbb{P}\left(\left\|S_{n} \right\| > s n^{1/p} \right) = -(\underline{\beta} - p)/p, \] where \[ \bar{\beta} = - \limsup_{t \rightarrow \infty} \frac{\log \mathbb{P}(\log \|X\| > t)}{t} ~~\mbox{and}~~\underline{\beta} = - \liminf_{t \rightarrow \infty} \frac{\log \mathbb{P}(\log \|X\| > t)}{t}. \] The main tools employed in proving this result are the symmetrization technique and three powerful inequalities established by Hoffmann-J{\o}rgensen (1974), de Acosta (1981), and Ledoux and Talagrand (1991), respectively. As a special case of this result, the main results of Hu and Nyrhinen (2004) are not only improved, but also extended.