Precise Asymptotics in Strong Limit Theorems for Self-normalized Sums of Multidimensionally Indexed Random Variables

Abstract

AbstractThe present paper discusses the precise asymptotic behaviours for the deviation probabilities of self-normalized sums of multidimensionally indexed random variables. The precise asymptotics for the general deviation probabilities are derived. Many known theorems for self-normalized sums of random variables can follow from the given results, and thus the precise asymptotics in the complete moment convergence, law of iterated logarithm and large deviation for self-normalized sums are generalized from one-dimensionally indexed random variables to multidimensionally indexed random variables.KeywordsSelf-normalized SumsPrecise AsymptoticsStrong Limit TheoremsComplete Moment ConvergenceDifference Propagation ProbabilityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.