Suppose that I have a random Gaussian variable (a vector), $x$, that has zero mean and covariance matrix as $\Sigma$. Suppose that its CDF is denoted by $F(x)$. We know that $F(x)$ is log concave. But I wonder if the following function is also log-concave:
$G(z) = F(z) - F(z-1)$
I know that generally speaking, adding up (or the difference) between two log concave functions is not necessarily concave. But I wonder if $G(z)$ is log concave given that $F(x)$ has this specific structure, i.e., CDF of a multivariate Gaussian distribution?
I tried to convert $-F(z-1)$ by the survival function: $-F(z-1) = S(z-1)-1$, which means that $-F(z-1)$ is indeed log concave. But I have no clue as to how to prove/or disagree that $G(z)$ is a log concave function?
Thanks!