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

Abstract

We consider the class, say ℳ_n,sym, of martingales M_n = X₁ + ⋯ + X_n with conditionally symmetric bounded differences X_k such that |X_k | ≤ 1. We find explicitly a solution, say D_n(x), of the variational problem D_n(x) ≝ sup _{Mn ∈ℳn,sym} ℙ {M_{n ≥ 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 D_n(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{M_n ≥ x} ≤ D_n(x), x ∈ ℝ, for the tail probability ℙ {M_n ≥ x}.