Self-normalized processes: exponential inequalities, moment bounds and iterated logarithm laws

Abstract

Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several exponential and moment inequalities, particularly those related to laws of the iterated logarithm, for self-normalized random variables including martingales. Tail probability bounds are also derived. For random variables B_t>0 and A_t, let Y_t(λ)=exp{λA_t−λ²B_t²/2}. We develop inequalities for the moments of A_t/B_t or sup_t≥0A_t/{B_t(log logB_t)^1/2} and variants thereof, when EY_t(λ)≤1 or when Y_t(λ) is a supermartingale, for all λ belonging to some interval. Our results are valid for a wide class of random processes including continuous martingales with A_t=M_t and $B_{t}=\sqrt {\langle M\rangle _{t}}$ , and sums of conditionally symmetric variables d_i with A_t=∑_i=1^td_i and $B_{t}=\sqrt{\sum_{i=1}^{t}d_{i}^{2}}$. A sharp maximal inequality for conditionally symmetric random variables and for continuous local martingales with values in R^m, m≥1, is also established. Another development in this paper is a bounded law of the iterated logarithm for general adapted sequences that are centered at certain truncated conditional expectations and self-normalized by the square root of the sum of squares. The key ingredient in this development is a new exponential supermartingale involving ∑_i=1^td_i and ∑_i=1^td_i². A compact law of the iterated logarithm for self-normalized martingales is also derived in this connection.