2015/08/18 by Nina Gantert, Gantert, Nina, Steven Soojin Kim +3
Decision Sciences · Mathematics · #60D05 (Secondary) #60F10 (Primary) #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #math.PR #msc:60D05 #msc:60F10
paper · pdf · doi:10.48550/arxiv.1508.04402
16 pages, simplified proof of Theorem 2.4, result slightly strengthened, added references, corrected typos, clarified some language
openalex publication_date 2015/08/18 · arxiv created 2015/10/06 · arxiv updated 2015/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The empirical mean of n independent and identically distributed (i.i.d.) random variables (X1,…,Xn) can be viewed as a suitably normalized scalar projection of the n-dimensional random vector X(n)\doteq(X1,…,Xn) in the direction of the unit vector n-1/2(1,1,…,1) ∈ \mathbbSn-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) ∈ \mathbbSn-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) ∈ \mathbbSn-1, n ∈ ℕ, corresponding to Cramér's theorem is atypical.