Question about asymptotic distribution of order statistics.

Question

Let $U_i$'s be i.i.d. Uniform$(0,1)$ random variables, and let $X_n = min\{U_1, \dots , U_n \}, Y_n = max\{ U_1 , \dots , U_n \}, Z_n = Y_n - X_n $. Then find asymptotic distribution of $n(1-Z_n)$.

I understand how to find asymptotic distribution of $X_n , Y_n$. But this problem makes me confused. Also I tried to use this method($X_{(r)} \sim Beta(r,n-r+1)$), but it failed.

Please help!

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I know $Z_n=Y_n-X_n \sim Beta(n-1,2)$. So, cdf of $Z_n$ is

$P(Z_n \leq t) = n(n-1)\int_{0}^{t} x^{n-2}(1-x)dx$

$P(1-Z_n \leq t) = 1-n(n-1) \int_{0}^{1-t}x^{n-2}(1-x)dx$

$P(n(1-Z_n) \leq t) = 1-n(n-1)\int_{0}^{1-t/n}x^{n-2}(1-x)dx$

I don't know what to do after this.

Answer 1

starting from your Range density that is a $Z_n\sim Beta(n-1;2)$ transforming your

$$Y_n=n(1-Z_n)$$

with usual technics:

$$f_Y(y)=f_X(g^{-1}(y))|\frac{d}{dy}g^{-1}(y)|$$

you get

$$f_{Y_n}(y)=\frac{n-1}{n}\cdot y\cdot \left(1-\frac{y}{n}\right)^{n-2}$$

whose limit results to me

$$\lim\limits_{n \to \infty}f_{Y_n}(y)=ye^{-y}$$

which is the density of a $Gamma(2;1)$