EXPONENTIAL INEQUALITIES FOR SELF-NORMALIZED PROCESSES WITH APPLICATIONS

We prove the following exponential inequality for a pair of random variables (A, B) with B > 0 satisfying the canonical assumption, E[exp(lambda A - lambda(2)/2B(2))] <= 1 for lambda is an element of R, [GRAPHICS] for x > 0, where 1/p + 1/q = 1 and p >= 1. Applying this inequality, we obtain exponential bounds for the tail probabilities for self-normalized martingale difference sequences. We propose a method of hypothesis testing for the L-p-norm (p >= 1) of A ( in particular, martingales) and some stopping times. We apply this inequality to the stochastic TSP in [0, 1](d) (d >= 2), connected to the CLT.