Let $X_1,X_2,...$ a sequence of random variables identically distributed following a Pareto distribution of parameter $\alpha>0$ such that $P(X_1>x)=x^{-\alpha}$ for $x\geqslant 1$. Show that for a sequence of constants $c_1,c_2,...$ we have:

$\limsup\frac{\log(X_n)}{c_n}=\frac{1}{\alpha}$ almost surely.

I thought of the following possible resolution however I am stuck:

I need to prove $P(\limsup\frac{\log(X_n)}{c_n}\neq \frac{1}{\alpha})=0$. By Borel Cantelli if I prove $\sum_{n=1}^{\infty}P(\frac{\log(X_n)}{c_n}\neq \frac{1}{\alpha}))<\infty$ then it implies $P(\limsup\frac{\log(X_n)}{c_n}\neq \frac{1}{\alpha})=0$

However I do not know how to make the computations.


Is there something wrong with my reasoning? How should I solve this question?

Thanks in advance!


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