Self-normalized large deviations

Let ${X, X_n, n \geq 1}$ be a sequence of independent and identically distributed random variables. The classical Cramér-Chernoff large deviation states that $\lim_{n\to\infty} n^{-1} \ln P((\sum_{i=1}^n X_i)/n \geq x) = \ln \rho (x)$ if and only if the moment generating function of $X$ is finite in... View More