I am working with a sum of variables $X_i$; they are all independent, but not identically distributed. For any $i$, I can show the bound $$\Lambda^*_{X_i}(t) := \sup_t \langle t, x \rangle - \Lambda_X(t) \ge f_i(t)$$ for some concave $f_i$, i.e. the Legendre transform of the log-moment generating function for each of the $X_i$ from below by some concave function (not sure about the convention on negative signs, but my bound is in the direction that says that tails of the $X_i$ are very light).

Can I then conclude that: $$\Lambda^*_{\sum_{i=1}^n X_i} (t) \ge n \min_{i} f_i(t/n)$$

intuitively, this seems like it should be trivial, e.g. by Gartner Ellis, but I am finding that the $^*$ operation does not respect sums nicely. (For now, I am not worried about convergence issues and such).