Maximum of exponentially-tailed variables

Question

Suppose that $X$ is a random variable on supported $\mathbb{R}$ and denote $f_X$ its density. Assume also that $f_X$ has "exponential tailed", i.e.:

$$f_X(x)\sim_{x\to+\infty} Ze^{-ax}$$

where $Z,a>0$. If we consider $n$ independent variables with the density $f_X$, can we say something on the behaviour of $Y_n:=\max_{1\leq i \leq n} X_i$ (in probability for example) ?

My intuition is that we have $Y_n\sim\log n$ in probability (up to some multiplicative constant) but I was not able to derive something until now.

Do you have some references or solution for this topic ?

Thank you very much!

Answer 1

We do have $Y_n\sim x_n:=\frac1a\,\ln n$ in probability as $n\to\infty$.

Indeed, $G(x):=P(X>x)=(c+o(1))e^{-ax}$ as $x\to\infty$, where $c:=Z/a$. So, for any real $p>0$, $$P(Y_n>px_n)=1-(1-G(px_n))^n=1-(1-(c+o(1))n^{-p})^n \to \begin{cases} 0&\text{ if }p>1 \\ 1&\text{ if }p<1 \end{cases}$$ as $n\to\infty$. So, $Y_n/x_n\to1$ in distribution and hence in probability. $\quad\Box$