Assume $X_1$, ..., $X_n$ are random variables (might be independent or not) with the same marginal density $f$ having finite moments to any desired order. I want to reach a tight tail distribution of the quantity: $$ \max_{1\leq i \leq n}\frac{1}{f(X_i)} $$
- Is the order of magnitude of $E\left[\max_{1\leq i \leq n}\frac{1}{f(X_i)}\right]$ known in the bibliography? I didn't find any meaningful result.
- Are there any results regarding the tail distribution of $\max_{1\leq i \leq n}\frac{1}{f(X_i)}?$
What I can prove for the moment is that for large a, $P\left(\max_{1\leq i \leq n}\frac{1}{f(X_i)} > n^{a}\right)$ converges to 0 at a rate $n^{-\beta}$ for $\beta > 0$.
I seek a faster tail decay.