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 \(X^{(n)}\displaystyle \mathop {=}^{\cdot }\,(X_1,\dots ,X_n)\) 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 Cramer’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 Cramer 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 Cramer’s theorem is atypical.