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 Bt>0 and At, let Yt(λ)=exp{λAt−λ2Bt2/2}. We develop inequalities for the moments of At/Bt or supt≥0At/{Bt(log logBt)1/2} and variants thereof, when EYt(λ)≤1 or when Yt(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 At=Mt and $B_{t}=\sqrt {\langle M\rangle _{t}}$ , and sums of conditionally symmetric variables di with At=∑i=1tdi 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 Rm, 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=1tdi and ∑i=1tdi2. A compact law of the iterated logarithm for self-normalized martingales is also derived in this connection.