My problem:
Let $(U_k)_{k\geqslant 1}$ be an i.i.d. sequence random variables, and are uniform distributed over the unit interval, i. e. $U_k\sim \mathrm{Uni}([0; 1]), k = 1,2,… $. With $Z$ we denote the
Random variable, which is constantly equal to 1, and define
$Z_n:=\max \{U_1,…,U_n\} , n=1,2,…$
Prove that
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$(Z_n)_{n\in \mathbb{N}}$ converges in probability to $Z$, and
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$(Z_n)_{n \in \mathbb{N}}$ converges to $Z$ almost surely.
I have no idea how to start. can someone help?
Best Answer
For $1$ you need to show that $$ \lim_{n\to \infty} P( |Z_n - 1| > \epsilon) = 0, $$ since $Z_n \le 1$ for all $n$, thus \begin{align} \lim_{n\to \infty} P( |Z_n - 1| > \epsilon) &= \lim_{n\to \infty} P( 1 - Z_n > \epsilon) \\ & = \lim_{n\to \infty} P( Z_n < 1 - \epsilon)\\ & = \lim_{n\to \infty} F_{Z_n}(1-\epsilon)\\ & = \lim_{n\to \infty} (1-\epsilon)^n \\ & = 0, \end{align} for all $\epsilon \in (0,1)$. The last step stems from the fact that for $U_1,...,U_n$ i.i.d r.vs $$ F_{\max\{U_1,...,U_n\}}(x) = \prod_{i=1}^n F_{U_i}(x) = (F_U(x))^n, $$ where for $U\sim U[0,1]$, $F_U(x) = x$.