On measure concentration for separately Lipschitz functions in product spaces

Let M n = X 1 + ⋯ + X n be a martingale with bounded differences X m = M m - M m -1 such that double-struck p sign{a m - σ m ≤ X m ≤ a m + σ m } = 1 with nonrandom nonnegative σ m and σ(X 1, ⋯, X m -1)-measurable random variables a m . Write σ 2 = σ 1 2 + ⋯ + σ n 2 . Let I(x) = 1 - Φ(x), where Φ is the standard normal distribution function. We prove the inequalities double-struck p sign{Mn ≥ x} ≤ cI(x/σ ), double-struck p sign{ {M n > x} ≥1 - cI(- x/σ) with a constant c such that 3.74 ⋯ ≤ c ≤ 7.83 ⋯. The result yields sharp bounds in some models related to the measure concentration. In the case where all am = 0 (or a m ≤ 0), the bounds for constants improve to 3.17 ⋯ ≤ c ≤ 4.003 ⋯. The inequalities are new even for independent X 1, ⋯, X n, as well as for linear combinations of independent Rademacher random variables. © 2007 The Hebrew University of Jerusalem.