Let $X:=[X_1,\dots, X_n]$ be a random vector with $X_i \in (0,2)$ and having a joint distribution $F_X$. Take a constant vector $a:=[a_1,\dots, a_n]$ with $a_i \in [0,1]$ and $\sum_{i=1}^n a_i = 1$. Consider the variance $$ v(a):={\rm var}( \log a^\top X) $$ Is it possible to tell if the function $v(a)$ is convex or concave in $a$?
My attempts: We know that $a^\top X$ is linear in $a$ and $\log ()$ is concave, hence $\log a^\top X$ is concave in $a$. However, when taking the variance, it involves a difference between the second moments and the squared mean, which complicates the computation.
I tried to go with the second-order derivative test for Hessian, but the computation again became very complicated. So, I came back and tested some simple cases such as the Bernoulli case. With various simulations, I find it seems to be a concave function; however, only for the testing cases. Any suggestion is appreciated.
As I understood problem well it isn't possible without knowlegde about covariances pairs $(X_{1},X_{2})$since ${\rm var}( \log a^\top X)= {\rm var} (\sum^{n}_{i=1} ( \log a_{i})X_{i}) = \sum^{n}_{i=1} {\rm var} ( \log a_{i}X_{i} ) + 2 \cdot\sum_{i \neq j \leq n } {\rm cov}((\log a_{i})X_{i};(\log a_{j})X_{j})$