Question

Let Xi....,Xn,..., ~iid Exp(1) and let Yn) be the sample maximum of the first n observations. Show that the limiting distribution of Zn-(Y(n)-log n) has CDF F(z) exp{-e-*), z є R.

Answer 1

$CDF~of~Z_n~is\\ F_n(z)=P(Z_n\leq z)=P((Y_{(n)}-\log n)\leq z)=P(Y_{(n)}\leq z+\log n)\\ =P(X_1\leq z+\log n,~X_2\leq z+\log n,\ldots, X_n\leq z+\log n)\\ =\prod_{i=1}^nP(X_i\leq z+\log z),~since~X_i's~are~independent\\ =\prod_{i=1}^n\int_0^{z+\log n}e^{-x}dx=\left(1-e^{-z-\log n} \right )^n\\ =\left(1-\frac{e^{-z}}{n} \right )^n\rightarrow \exp(-e^{-z})~as~n\rightarrow \infty~where~z\in \mathbb{R}$