Let $(a_n)_n$ be a sequence of positive numbers, $(X_n)_n$ - sequence of independent random variables, with density $f_{X_n}(x)=a_nx^{a_n-1}1_{[0,1]}(x)$.
Show than for any $(a_n)_n$: $Y_n=max\{X_1,...,X_n\}$ converges in distribution and find that distribution.
I was doing it this way:
$\int_0^ta_nx^{a_n-1}dx=t^{a_n}$
$\sum {a_n}>1$:
$P(Y_n<t)=\prod_{i=1}^n t^{a_i} = t^{\sum_{i=1}^n{a_i}} \to \begin{cases} 0, &t<1 \\ 1,&t \ge1\end{cases} $
$0<\sum {a_n} \le 1$:
$P(Y_n<t)=\prod_{i=1}^n t^{a_i} = t^{\sum_{i=1}^n{a_i}} \to \begin{cases} 0, &t<0 \\ 1,&t \ge0\end{cases} $
So in both cases there is a convergence in distribution - first to a variable $X$ such that $P(X=1)=1$ and second, $P(X=0)=0$.
Is it okay?
$$ F_{Y_n}(t)=F_{X_1}(t)\cdot\ldots\cdot F_{X_n}(t)=\begin{cases}0, & t\leq 0, \cr t^{a_1+\ldots+a_n}, & 0<t<1, \cr 1, & t\ge 1\end{cases} $$ This converges to degenerate at $1$ distribution if $\sum_{i=1}^n a_i \to\infty$ as $n\to\infty$, and to distribution with CDF $$ F_{Y}(t)=\begin{cases}0, & t\leq 0\cr t^{a}, & 0<t<1, \cr 1, & t\ge 1\end{cases} $$ if $\sum_{i=1}^n a_i \to a$ as $n\to\infty$.