Suppose a random variable X has cdf $F$ has rapidly varying tail $\overline{F} =1-F$, such that: $$ \lim_{x \to \infty} \frac{\overline{F}(x\lambda)}{\overline{F}(x)}= 0 $$ if $\lambda >1$, and $\infty$ if $\lambda \in (0,1)$. Then I want to show that X is relatively stable, that is, there exists a sequence $a_n > 0$, $a_n \to \infty$, such that:
$$ \frac{M_n}{a_n} \overset{p}{\to} 1 $$ Where $M_n$ is the maximum observation.
attempt: Need to show that $$ \lim_{n\to\infty}P\left (\big|\frac{M_n}{a_n}-1 \big |\ge \epsilon \right) =0, $$ for $\epsilon >0$. \begin{align*} &\lim_{n\to\infty}P\left (\big|\frac{M_n}{a_n}-1 \big |\ge \epsilon \right) = \lim_{n \to \infty}[P(\frac{M_n}{a_n} \le 1-\epsilon) + P(\frac{M_n}{a_n} \ge 1+\epsilon) ]\\ &= \lim_{n \to \infty} F^n((1-\epsilon) a_n) + 1 - \lim_{n \to \infty} F^n ((1+\epsilon)a_n)\\ &= \lim_{n \to \infty} \left ( 1 - \frac{n \overline{F}((1-\epsilon)a_n)}{n} \right)^n +1 -\lim_{n \to \infty} \left ( 1 - \frac{n \overline{F}((1+\epsilon)a_n)}{n} \right)^n\\ &=\exp (-\lim_{n \to \infty} n \overline{F}( (1-\epsilon) a_n) ) +1 -\exp (-\lim_{n \to \infty} n \overline{F}( (1+\epsilon) a_n) ) \end{align*} Now, the last line is equal to zero if and only if: $$ \lim_{n \to \infty} n \overline{F}( (1-\epsilon) a_n)=\lim_{n \to \infty} n \overline{F}( (1+\epsilon) a_n) =0 $$ I have no clue if this is going in the right direction though, any hints?
