Limit Theorems for Self-Normalized Large Deviation

Self-normalized limit theorems: A survey

Let X1,X2, . . . , be independent random variables with EXi = 0 and write Sn = ∑ n i=1 Xi and V 2 n = ∑ n i=1 X i . 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 S...

Cramér Type Moderate Deviation Theorems for Self-Normalized Processes

Cramér type moderate deviation theorems quantify the accuracy of the relative error of the normal approximation and provide theoretical justifications for many commonly used methods in statistics. In this paper, we develop a new randomized concentration inequality and establish a Cramér type moderate deviation theorem for general self-normalized processes which include many well-known Studentiz...

Self-normalized Weak Limit Theorems for a Φ-mixing Sequence

Let {Xj , j ≥ 1} be a strictly stationary φ-mixing sequence of non-degenerate random variables with EX1 = 0. In this paper, we establish a self-normalized weak invariance principle and a central limit theorem for the sequence {Xj} under the condition that L(x) := EX2 1 I{|X1| ≤ x} is a slowly varying function at ∞, without any higher moment conditions.

Limit Theorems For Large Tandem Queueing

Limit Theorems for Self-Similar Tilings

We study deviation of ergodic averages for dynamical systems given by self-similar tilings on the plane and in higher dimensions. The main object of our paper is a special family of finitely-additive measures for our systems. An asymptotic formula is given for ergodic integrals in terms of these finitely-additive measures, and, as a corollary, limit theorems are obtained for dynamical systems g...