Example of sum of log-concave is not always log-concave I know that sum of log-concave is not always log-concave. Could anyone provides me with an example to prove this? Like probability distribution fn (pdf) of normal distribution is log-concave; on what condition, the sum of two pdf will not be log-concave? Thanks!! :)

Here is an example. Consider the density of two exponential random variables as \begin{align*} f_1(x) &=\lambda_1 \exp(-\lambda_1x) \\\ f_2(x) &=\lambda_2 \exp(-\lambda_2x), \end{align*} where $\lambda_1 \
eq \lambda_2$.

Consider the sum of these two log concave functions \begin{align*} g(x)=f_1(x)+f_2(x), \end{align*} where the first and second derivatives of $g(x)$ are given as: \begin{align*} g^{'}(x) &=- \lambda_1 ^2 \exp(-\lambda_1x) - \lambda_2^2 \exp(-\lambda_2x) \\\ g^{''}(x) &= \lambda_1 ^3 \exp(-\lambda_1x) + \lambda_2^3 \exp(-\lambda_2x) \end{align*}

One can show that $g^{'}(x)g^{'}(x)\le g^{''}(x)g(x)$, hence $g(x)$ is not log-concave. In fact, it is log-convex.