Cram\'er's theorem is atypical

Abstract

The empirical mean of $n$ independent and identically distributed (i.i.d.) random variables $(X_1,\dots,X_n)$ can be viewed as a suitably normalized scalar projection of the $n$-dimensional random vector $X^{(n)}\doteq(X_1,\dots,X_n)$ in the direction of the unit vector $n^{-1/2}(1,1,\dots,1) \in \mathbb{S}^{n-1}$. The large deviation principle (LDP) for such projections as $n\rightarrow\infty$ is given by the classical Cram\'er's theorem. We prove an LDP for the sequence of normalized scalar projections of $X^{(n)}$ in the direction of a generic unit vector $\theta^{(n)} \in \mathbb{S}^{n-1}$, as $n\rightarrow\infty$. This LDP holds under fairly general conditions on the distribution of $X_1$, and for "almost every" sequence of directions $(\theta^{(n)})_{n\in\mathbb{N}}$. The associated rate function is "universal" in the sense that it does not depend on the particular sequence of directions. Moreover, under mild additional conditions on the law of $X_1$, we show that the universal rate function differs from the Cram\'er rate function, thus showing that the sequence of directions $n^{-1/2}(1,1,\dots,1) \in \mathbb{S}^{n-1},$ $n \in \mathbb{N}$, corresponding to Cram\'er's theorem is atypical.