An inverse problem on tail probability

Question

This is a question out of curiosity. Assume that $f(x)$ is a density function for which there is a constant $C>0$ so that $$ \int_t^\infty f(x) dx \le C f(t) $$ holds for large enough $t>0$. My question is whether this property implies that $\int_t^\infty f(x) dx$ has exponential decay? It is known that this property holds for exponential and normal distributions.

Answer 1

Let $F(t)=e^{t/C}\int_t^\infty f(x)\,dx$. The stated assumption is equivalent to $F'(t)\le 0$ for large $t$. Therefore, $\int_t^\infty f(x)\,dx=O(e^{-t/C})$.