Cram's theorem is atypical

The empirical mean of n independent and identically distributed (i.i.d.) random variables (X₁, ⋯, X_n) can be viewed as a suitably normalized scalar projection of the n-dimensional random vector X(n)=·(X1,⋯,Xn) in the direction of the unit vector n^{- 1 / 2}(1, 1, ⋯, 1 ) ∈ S^n-1. The large deviation principle (LDP) for such projections as n→ ∞ 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 θ⁽ⁿ⁾∈ S^n-1, as n→ ∞. This LDP holds under fairly general conditions on the distribution of X₁, and for “almost every” sequence of directions (θ(n))n∈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 X₁, 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, ⋯, 1 ) ∈ S^n-1, n∈ N, corresponding to Cramér’s theorem is atypical.