Suppose $F$ is a strictly convex and increasing function, $U$ a random variable with support $[0,1]$ and density $$ f_U(u)= \frac{e^{-\frac{1}{T}F(u)}}{\int_{0}^{1} e^{-\frac{1}{T} F(x)} dx}.$$
Do we have a known bound for $\Pr\{U>y\}$ for any $y>0$ in terms of $T$ which is tight in asymptotic?
In asymptotic, $T \to 0$, one knows that $U$ converges to $0$ in probability.
Any help appreciated!