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(λA − λ²/2 B²)] ≤ 1 for λ ∈ ℝ, p(|A|/√ 2q−1/q(B² + (E[|A|^p])^2/p) ≥ x) ≤ (q/2q − 1)^q/2q−1 x^−q/2q−1 e^−x2/2 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 Lp-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.