Find the limiting distribution of the following random variable

Question

Let $X_1,X_2,...$ Be independent random variables with common density:

$$f_X(x)=\alpha x^{-(\alpha+1)}. x>1$$

Where $\alpha>0$. Define a new sequence of random variables:

$$Y_n=(1/n^{1/\alpha})X_{(n)}$$

Where $X_{(n)}$ is the highest observation of n I.I.d. r.v. $X_1,…,X_n$. Show that $Y_n$ converges in distribution as $n\to \infty$ and find the limiting distribution.

Answer 1

Hint: $$ \Pr(Y_n\leq y)=\Pr(\max X_n\leq yn^{\frac 1\alpha})=\left(\Pr( X_1\leq yn^{\frac 1\alpha})\right)^n\\ =\left(\int_1^{yn^{\frac 1\alpha}} \alpha x^{-\alpha-1}dx\right)^n=\left(\frac{-1}{x^\alpha}\Bigg|_1^{yn^{\frac 1\alpha}}\right)^n=\left(1-\frac{1}{ny^\alpha}\right)^n. $$