Cram\'er's theorem is atypical

TL;DR

This paper extends Cramér's theorem to large deviations of scalar projections of i.i.d. vectors in arbitrary directions, revealing that the classical case is atypical among all directions.

Contribution

It proves a universal large deviation principle for projections in generic directions, showing the classical Cramér's theorem case is atypical.

Findings

Universal rate function independent of directions

Cramér's theorem is atypical among directions

Large deviations hold under general conditions

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.