A class of strong deviation theorems and an approach of Laplace transformation (Q5955929)

Let \(\{X_n\}\) be an arbitrary sequence of nonnegative (dependent) random variables such that \((X_1,\dots,X_n)\) has a joint density \(f_n\) for each \(n\). An almost sure lower bound for \(\lim \inf_n{1\over n}(X_1+ \cdots+ X_n)\) and an upper bound for \(\lim\sup_n {1\over n}(X_1+ \cdots+ X_n)\) are found in terms of the Laplace transform of an auxiliary density \(f\) on \((0, \infty)\) and the limit of the likelihood ratio \[ r={1\over n}\lim \sup_n\log (f_n (X_1,\dots, X_n)/[f(X_1) \cdot\dots \cdot f(X_n)]). \] It is also shown how the difference between these bounds depends on the magnitude of the random variable \(r\). The proof exploits the martingale convergence theorem and Laplace transform technique.