If we have $V_i$ a random variable such that $E[V_i] = 0$ and $E[V_i^2] = 1$ What distribution of $V_i$ would minimise $E[V_i^4]$?
My intuition is that it would be $V_i = \pm 1$ with probability $1/2$. I am however stuck on how to rigorously demonstrate this.
$min_{(V_i , p(V_i))}\sum_{i = 1}^n V_i^4 p(V_i)$ such that : $\sum_{i = 1}^n V_i^2 p(V_i)$.
My above intuition indeed matches the constraint. Any hints would be very much appreciated!
One strategy that comes to mind:
Remove a middle value, and assign its probability mass to the most extreme values in such a way that the mean remains zero. Now scale the values such that the second moment is 1 again. If you can show that this always results in a fourth moment that is smaller, then your desired conclusion should follow.