A Bernstein-Chernoff deviation inequality, and geometric properties of random families of operators
نویسنده
چکیده
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 exensions. Acknowledgement: I would like to thank Prof. Vitali Milman for his support and encouragement, and mainly for his mathematical help and advice. ∗This research was partially supported by BSF grant 2002-006.
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We use the following two well known bounds in our proofs. Lemma 1 (Chernoff-Hoeffding bound). Let X1, · · · , Xn be random variables with common support [0, 1] and E[Xi] = μ. Let Sn = X1 + · · ·+Xn. Then for all t ≥ 0, Pr[Sn ≥ nμ+ t] ≤ e−2t /n and Pr[Sn ≤ nμ− t] ≤ e−2t /n Lemma 2 (Bernstein inequality). Let X1, . . . , Xn be independent zero-mean random variables. If for all 1 ≤ i ≤ n, |Xi| ≤ k...
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