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February, 1979 Sign Changes of the Difference of Convex Functions and their Application to Large Deviation Rates
James Lynch
Ann. Probab. 7(1): 96-108 (February, 1979). DOI: 10.1214/aop/1176995151

Abstract

The relationship of the large deviation rate, $\psi^\ast(a)$, of the mean of independent and identically distributed random variables to their cumulant generating function, $\psi(\lambda)$, is well known. This paper studies how the behavior of the sign changes of $\psi_1^\ast(a) - \psi_2^\ast(a)$ is related to that of $\psi_1(\lambda) - \psi_2(\lambda)$ for cumulant generating functions $\psi_1$ and $\psi_2$ with rates $\psi_1^\ast$ and $\psi_2^\ast$, respectively. Use is made of the fact that the rate $\psi^\ast$ is nothing more than the conjugate convex function of $\psi$. Results concerning the relationship of the behavior of the difference of convex functions to that of the difference of their conjugates are first proven and then applied to determine the relationship of the behavior of the sign changes of $\psi_1^\ast - \psi_2^\ast$ to that of $\psi_1 - \psi_2$. Results are also given relating this behavior to that of $F_1 - F_2$ and $f_1 - f_2$, where $F_i$ and $f_i(i = 1$ and 2) are the distribution function and the density function, respectively, corresponding to $\psi_i$.

Citation

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James Lynch. "Sign Changes of the Difference of Convex Functions and their Application to Large Deviation Rates." Ann. Probab. 7 (1) 96 - 108, February, 1979. https://doi.org/10.1214/aop/1176995151

Information

Published: February, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0392.60029
MathSciNet: MR515816
Digital Object Identifier: 10.1214/aop/1176995151

Subjects:
Primary: 26A51‎
Secondary: 60R10 , 62E10 , 62E20

Keywords: conjugate convex functions , large deviations , sign changes

Rights: Copyright © 1979 Institute of Mathematical Statistics

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Vol.7 • No. 1 • February, 1979
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