An example of a non-log-concave distribution where the difference has a log-concave density (Q6633343)

A function \(f: \mathbb{R}^n \to [0, \infty)\) is said to be log-concave if, \N\[\Nf (\lambda x+ (1- \lambda)y) \geq f(x)^\lambda f(y)^{1- \lambda}, \quad \forall x,y \in \mathbb{R}^n, \quad\forall \lambda \in [0,1].\N\]\NA continuous random variable is log-concave iff it has a log-concave density. Many frequently encountered probability distributions are log-concave. Examples include Gaussian, exponential and uniform random variables.\N\NAn important property of the class of log-concave functions is that it is closed under convolution (follows from the Prékopa-Leindler inequality). As a consequence, the sum of independent log-concave random variables is also log-concave. In particular, if \(X\) and \(X^\prime\) are i.i.d. random variables with log-concave density, then \(X-X^\prime\) is log-concave. This paper proves that the converse does not necessarily hold true by giving an example. More specifically:\N\N\textbf{Theorem.} Let \(X_1,X_2,X_3\) and \(X_4\) be independent standard normal distributions. Then, \(X_1X_2 - X_3X_4\) has a log-concave density, while \(X_1X_2\) has a non log-concave density.\N\NThe proof works as follows: First, the author shows that \(X_1X_2 - X_3X_4\) has Laplace (0,1) distribution (or two-sided exponential distribution symmetrical about the origin), by comparing the moment generating function. Finally, by simple calculation, it is shown that the density of the product \(X_1X_2\) is \(\frac{K_0(|x|)}{\pi}\), where \(K_0\) is the modified second class Bessel function (also known as `Hankel function'), which is log-convex as shown in the paper [\textit{Á. Baricz} et al., Expo. Math. 29, No. 4, 399--414 (2011; Zbl 1237.39023)].