An exact bound for tail probabilities for a class of conditionally symmetric bounded martingales

Abstract

We consider the class, say ℳn,sym, of martingales Mn = X1 + ⋯ + Xn with conditionally symmetric bounded differences Xk such that |Xk | ≤ 1. We find explicitly a solution, say Dn(x), of the variational problem Dn(x) ≝ sup Mn ∈ℳn,sym ℙ {Mn ≥ x}. We show that this problem is equivalent to one when you want to find out the symmetric random walk with bounded length of steps which maximizes the probability to visit an interval [x;∞]. The function x \mapsto Dn(x) allows a simple description and is closely related to the binomial tail probabilities. We can interpret the result as a final and optimal upper bound ℙ{Mn ≥ x} ≤ Dn(x), x ∈ ℝ, for the tail probability ℙ {Mn ≥ x}.