I think results from extreme value theory will be helpful here. The standard condition to put on $p(x)$ in these kinds of situations is that $p$ is regularly varying in $x$, and that there exists a constant $\gamma > 0$ such that $$ \lim_{x \rightarrow \infty} \frac{1 - F(tx)}{1 - F(x)} = t^{-1/\gamma} \text{ for all } t > 0, $$ where $F$ is the CDF of $p$. The constant $\gamma$ is called the tail index. (There also exists a more general theory with $\gamma \leq 0$ that applies to thin-tailed distributions like the Gaussian.)
Once you assume that your density is regularly varying, the quantity $f(X, \, t)$ becomes easier to analyze. For example, you can show that $f(X, \, t)$ scales with $t$.
A good reference for getting started is Chapter 4.3 of Coles (2001) "An Introduction to Statistical Modeling of Extreme Values." For a more theoretical approach, the textbook by de Haan and Ferreira (2006) is excellent.
