Limiting distribution of first order statistic x

Question

Good evening everyone, I am attempting a problem on limiting distributions. The problem is as seen in the picture above. Could you please help me with part c) of this problem. The answer as per the solutions at the back is

$$ 0 \,\text{for}\, x\le 1 \\ 1-\frac{1}{y}\, \text{for} \,x\gt1$$

I dont quite understand why, I thought it would be $1 \, \text{for} \, x\gt 1$. Thank you!

Answer 1

From the specified distribution, for all $x \geqslant 1$ you have:

$$\begin{equation} \begin{aligned} F_{X_{1:n}}(x) \equiv \mathbb{P}(X_{1:n} \leqslant x) &= 1 - \mathbb{P}(X_{1:n} > x) \\[6pt] &= 1 - \mathbb{P}(\min \{ X_1,...,X_n \} > x) \\[6pt] &= 1 - \prod_{i=1}^n \mathbb{P}(X_i > x) \\[6pt] &= 1 - \prod_{i=1}^n x^{-1} \\[6pt] &= 1 - x^{-n}. \end{aligned} \end{equation}$$

Hence, for all $x \geqslant 1$ you have:

$$\begin{equation} \begin{aligned} F_{X_{1:n}^n}(x) \equiv \mathbb{P}(X_{1:n}^n \leqslant x) &= \mathbb{P}(X_{1:n} \leqslant x^{1/n}) \\[6pt] &= 1 - (x^{1/n})^{-n} \\[6pt] &= 1 - x^{-1}. \end{aligned} \end{equation}$$

This gives us back the original distribution of $X_i$. This distribution does not depend on $n$ so its limiting distribution is just the original distribution.