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 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 $n\rightarrow \infty$. 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 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.