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On measure concentration for separately Lipschitz functions in product spaces

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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 21 + ⋯ + σ 2n . 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.

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Authors and Affiliations

  1. Vilnius institute of mathematics and informatics, Akademijos 4, LT-08663, Vilnius, Lithuania

    Vidmantas Bentkus

  2. Vilnius Pedagogical university, Studentu 39, LT-08106, Vilnius, Lithuania

    Vidmantas Bentkus

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  1. Vidmantas Bentkus
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Research supported by Max Planck Institute for Mathematics, Bonn

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Bentkus, V. On measure concentration for separately Lipschitz functions in product spaces. Isr. J. Math. 158, 1–17 (2007). https://doi.org/10.1007/s11856-007-0001-2

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  • DOI: https://doi.org/10.1007/s11856-007-0001-2

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