Asymptotic Distribution of Order Statistic for a uniform random variable

Question

I am trying to find the asymptotic distribution of an order statistic $X_{(n)}$ for iid RVs $X_1, ..., X_n \sim \mathrm{Unif}(0,a)$, where $a>0$.

The distribution for $X_{(n)}$

$$f(X_{(n)}=x) = \frac{nx^{n-1}}{a^n}I(0<x<a)$$

My gut reaction was to find the limit as $n\rightarrow \infty$ of the CDF, and see if this resembled another distribution. This gave me

$$\frac{x^n}{a^n}$$

My guess is there is a trick or a property I am forgetting. Note: the actual problem was more involved, I simplified it to just include the component I unsure about.

Answer 1

Let $Z_n = n(X_{(n)}/a-1).$ Then $$ P(Z_n\le z) = P(X_{(n)}/a\le 1+z/n) =(1+z/n)^n\sim e^z.$$ So the distribution is asymoptitically reverse-exponential.