## The Annals of Probability

- Ann. Probab.
- Volume 7, Number 1 (1979), 96-108.

### Sign Changes of the Difference of Convex Functions and their Application to Large Deviation Rates

#### 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 $\psi_i$.

#### Article information

**Source**

Ann. Probab., Volume 7, Number 1 (1979), 96-108.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995151

**Digital Object Identifier**

doi:10.1214/aop/1176995151

**Mathematical Reviews number (MathSciNet)**

MR515816

**Zentralblatt MATH identifier**

0392.60029

**JSTOR**

links.jstor.org

**Subjects**

Primary: 26A51: Convexity, generalizations

Secondary: 60R10 62E10: Characterization and structure theory 62E20: Asymptotic distribution theory

**Keywords**

Large deviations conjugate convex functions sign changes

#### Citation

Lynch, James. Sign Changes of the Difference of Convex Functions and their Application to Large Deviation Rates. Ann. Probab. 7 (1979), no. 1, 96--108. doi:10.1214/aop/1176995151. https://projecteuclid.org/euclid.aop/1176995151