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 B-t > 0 and A(t), let Y-t(lambda) = exp{lambdaA(t) - lambda(2)B(2)/2}. We develop inequalities for the moments of A(t)/B-t or sup(tgreater than or equal to0) A(t)/{B-t(log log B-t)(1/2)} and variants thereof, when EYt(lambda) less than or equal to 1 or when Y-t(lambda) is a supermartingale, for all, 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 = root(M)(t), and sums of conditionally symmetric variables d(i) with A(t) = Sigma(i=1)(t) d(i) and B-t = rootSigma(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 greater than or equal to 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 Sigma(i=1)(t) d(i) and Sigma(i=1)(t) d(i)(2). A compact law of the iterated logarithm for self-normalized martingales is also derived in this connection.