Cramér'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 $n\to \infty$ is given by the classical Cramér'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 $n\to \infty$. 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 Cramér 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 Cramér's theorem is atypical.

16 pages, simplified proof of Theorem 2.4, result slightly strengthened, added references, corrected typos, clarified some language