2021/11/01 by Xiao Fang, Fang, Xiao, Songhao Liu +3
Mathematics · #60F05 #60F10 #62E17 #Advanced Algebra and Geometry #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2111.00679
openalex publication_date 2021/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X1,…, Xn be independent and identically distributed random vectors in ℝd. Suppose 𝔼 X1=0, Cov(X1)=Id, where Id is the d× d identity matrix. Suppose further that there exist positive constants t0 and c0 such that 𝔼 et0|X1|≤ c0x)ℙ(|Q1/2Z|>x)-1 |≤ C ( \frac1+x5det(Q1/2)n+(x6)/(n)) for d≥ 5 \endequation* and | \fracℙ(|Q1/2W|gt;x)ℙ(|Q1/2Z|gt;x)-1 |≤ C ( \frac1+x3det(Q1/2)n(d)/(d+1)+(x6)/(n)) for 1≤ d≤ 4, where ε and C are positive constants depending only on d, t0, and c0. This is a first extension of Cramér-type moderate deviation to the multivariate setting with a faster convergence rate than 1/√(n). The range of x=o(n1/6) for the relative error to vanish and the dimension requirement d≥ 5 for the 1/n rate are both optimal. We prove our result using a new change of measure, a two-term Edgeworth expansion for the changed measure, and cancellation by symmetry for terms of the order 1/√(n).