Summary
Sharp lower bounds are found for the concentration of a probability distribution as a function of the expectation of any given convex symmetric function ϕ. In the case ϕ(x)=(x-c)2, wherec is the expected value of the distribution, these bounds yield the classical concentration-variance inequality of Lévy. An analogous sharp inequality is obtained in a similar linear search setting, where a sharp lower bound for the concentration is found as a function of the maximum probability swept out from a fixed starting point by a path of given length.
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Research partially supported by NSF Grant SES-88-21999
Research partially supported by NSF Grants DMS-87-01691 and DMS-89-01267 and a Fulbright Research Grant
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Foley, R.D., Hill, T.P. & Spruill, M.C. A generalization of Lévy's concentration-variance inequality. Probab. Th. Rel. Fields 86, 53–62 (1990). https://doi.org/10.1007/BF01207513
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DOI: https://doi.org/10.1007/BF01207513
Keywords
- Probability Distribution
- Lower Bound
- Stochastic Process
- Probability Theory
- Mathematical Biology