# On measure concentration for separately Lipschitz functions in product spaces

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 ℙ{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 $$\mathbb{P}\left\{ {M_n \geqslant x} \right\} \leqslant cI(x/\sigma ), \mathbb{P}\left\{ {M_n > x} \right\} \geqslant 1 - cI( - x/\sigma )$$ 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 a m = 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Israel Journal of Mathematics Springer Journals

# On measure concentration for separately Lipschitz functions in product spaces

, Volume 158 (1) – May 17, 2007
17 pages

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# References (39)

Publisher
Springer Journals
Subject
Mathematics; Mathematics, general; Algebra; Group Theory and Generalizations; Analysis; Applications of Mathematics; Theoretical, Mathematical and Computational Physics
ISSN
0021-2172
eISSN
1565-8511
DOI
10.1007/s11856-007-0001-2
Publisher site
See Article on Publisher Site

### Abstract

Let M n = X 1 + ⋯ + X n be a martingale with bounded differences X m = M m − M m −1 such that ℙ{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 $$\mathbb{P}\left\{ {M_n \geqslant x} \right\} \leqslant cI(x/\sigma ), \mathbb{P}\left\{ {M_n > x} \right\} \geqslant 1 - cI( - x/\sigma )$$ 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 a m = 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.

### Journal

Israel Journal of MathematicsSpringer Journals

Published: May 17, 2007