Summary.
We present a simple proof, based on modified logarithmic Sobolev inequalities, of Talagrand’s concentration inequality for the exponential distribution. We actually observe that every measure satisfying a Poincaré inequality shares the same concentration phenomenon. We also discuss exponential integrability under Poincaré inequalities and its consequence to sharp diameter upper bounds on spectral gaps.
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Received: 10 June 1996 / In revised form: 9 August 1996
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Bobkov, S., Ledoux, M. Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution. Probab Theory Relat Fields 107, 383–400 (1997). https://doi.org/10.1007/s004400050090
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DOI: https://doi.org/10.1007/s004400050090