TY - JOUR

T1 - A Bernstein-Chernoff deviation inequality, and geometric properties of random families of operators

AU - Artstein-Avidan, Shiri

N1 - Funding Information:
The first theorem in this note is a new Bernstein-type deviation inequality which we prove using Chernoff's bounds. This theorem is different from the classical Bernstein inequality in the following way: whereas the condition in the standard Bernstein inequality is on the global behavior of the random variables in question, for example a condition on the expectation of e cx2, in Theorem 1 below the condition uses only the constants appearing in the tail of the distribution, and so can reflect concentration. Sometimes one can prove very strong estimates on the tails. In the theorem below these estimates can be then used and are amplified when one averages many i.i.d, copies of the variable. This theorem in * This research was partially supported by BSF grant 2002-006. Received June 1, 2005 and in revised form August 29, 2005

PY - 2006

Y1 - 2006

N2 - In this paper we first describe a new deviation inequality for sums of independent random variables which uses the precise constants appearing in the tails of their distributions, and can reflect in full their concentration properties. In the proof we make use of Chernoff's bounds. We then apply this inequality to prove a global diameter reduction theorem for abstract families of linear operators endowed with a probability measure satisfying some condition. Next we give a local diameter reduction theorem for abstract families of linear operators. We discuss some examples and give one more global result in the reverse direction, and extensions.

AB - In this paper we first describe a new deviation inequality for sums of independent random variables which uses the precise constants appearing in the tails of their distributions, and can reflect in full their concentration properties. In the proof we make use of Chernoff's bounds. We then apply this inequality to prove a global diameter reduction theorem for abstract families of linear operators endowed with a probability measure satisfying some condition. Next we give a local diameter reduction theorem for abstract families of linear operators. We discuss some examples and give one more global result in the reverse direction, and extensions.

UR - http://www.scopus.com/inward/record.url?scp=33847736608&partnerID=8YFLogxK

U2 - 10.1007/BF02773831

DO - 10.1007/BF02773831

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AN - SCOPUS:33847736608

SN - 0021-2172

VL - 156

SP - 187

EP - 204

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

ER -