Self-normalized limit theorems: A survey

Abstract

Let X1,X2,..., be independent random variables with EXi = 0 and write Sn = Σ ni=1 Xi and V 2n = Σ ni=1 X 2i. This paper provides an overview of current developments on the functional central limit theorems (invariance principles), absolute and relative errors in the central limit theorems, moderate and large deviation theorems and saddle-point approximations for the self-normalized sum Sn/Vn. Other self-normalized limit theorems are also briefly discussed.