Self - Normalized Processes : Exponential Inequalities , Moment Bounds and Iterated Logarithm Laws

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 − λ B t /2}. We develop inequalities for the moments of At/Bt or supt≥0 At/{Bt(log logBt) } 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 Bt = √ 〈M〉t, and sums of conditionally symmetric variables di with