Abstract
Let Y be a nonnegative random variable with mean μ and finite positive variance σ 2, and let Y s, defined on the same space as Y, have the Y size-biased distribution, characterized by
Under a variety of conditions on Y and the coupling of Y and Y s, including combinations of boundedness and monotonicity, one sided concentration of measure inequalities such as
hold for some explicit A and B. The theorem is applied to the number of bulbs switched on at the terminal time in the so called lightbulb process of Rao et al. (Sankhyā 69:137–161, 2007).
Article PDF
Similar content being viewed by others
References
Barbour, A.D., Chen, L.H.Y.: An introduction to Stein’s method. In: Chen, L.H.Y, Barbour, A.D. (eds.) Lecture Notes Series No. 4, Institute for Mathematical Sciences, National University of Singapore, Singapore University Press and World Scientific, pp. 1–59 (2005)
Chatterjee S.: Stein’s method for concentration inequalities. Probab. Theory Relat. Fields 138, 305–321 (2007)
Goldstein L., Rinott Y.: Multivariate normal approximations by Stein’s method and size bias couplings. J. Appl. Probab. 33, 1–17 (1996)
Goldstein, L., Zhang, H.: A Berry Esseen theorem for the lightbulb process. Preprint (2009)
Ledoux M.: The Concentration of Measure Phenomenon. American Mathematial Society, Providence (2001)
Raič M.: CLT related large deviation bounds based on Stein’s method. Adv. Appl. Probab. 39, 731–752 (2007)
Rao C.R., Rao B.M., Zhang H.: One bulb? Two bulbs? How many bulbs light up? A discrete probability problem involving dermal patches. Sankhyā 69, 137–161 (2007)
Stein, C.: A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In: Proceedings of the Sixth Berkeley Symp. Math. Statist. Probab., vol. 2, pp. 583–602, University of California Press, Berkeley (1972)
Stein C.: Approximate Computation of Expectations. Institute of Mathematical Statistics, Hayward (1986)
Zhou H., Lange K.: Composition Markov chains of multinomial type. Adv. Appl. Probab. 41, 270–291 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ghosh, S., Goldstein, L. Concentration of measures via size-biased couplings. Probab. Theory Relat. Fields 149, 271–278 (2011). https://doi.org/10.1007/s00440-009-0253-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-009-0253-3