I am just learning about order statistics and limiting distributions. I was looking through the many examples in my book covering this subject and there is one small detail I can't figure out.
Some context: suppose that $X_1,...,X_n$ is a random sample from a Pareto distribution, $X_{i} \sim PAR(1,1)$, and let $Y_{n} = nX_{1:n}$. The CDF of $X_{i}$ is $F(x)=1-(1+x)^{-1};x>0$. Then the book gives the CDF of $Y_{n}$, denoted by $G_{n}(y)$:
$$G_{n}(y) = 1 - \left(1 + \frac{y}{n}\right)^{-n}, \quad y>0.$$
I understand how the CDF of the smallest order statistics is derived. It is given as $G_{1}(y_{1}) = 1 - [1-F(y_{1})]^{n}$. This, I can derive without any problems, however I don't understand how the times $n$ in front of $X_{1:n}$ is used in determining $G_{n}(y)$. Sometimes the smallest or the biggest order statistics is multiplied by $n$ or $1/n$ and in both cases I can't do the necessary calculations. Can someone point me in the right direction?
I can give more examples if needed.