Let $Y_1,Y_2,...,$ be independent $C(0,1)$ random variables, determine the limit distribution of :
$Z_n = \dfrac{1}{n} \cdot max\{Y_1, Y_2,..Y_n \} $ as $n \rightarrow \infty $,
Here is my approach:
$F_{Z_n} (x) = \mathbb{P}(Z_n \leq x) = \mathbb{P}(\dfrac{1}{n} max\{ Y_1,Y_2,...Y_n \} \leq x ) = \mathbb{P}( Y_1 \leq x\cdot n , Y_2 \leq x\cdot n, ....,Y_n \leq x\cdot n ) = \Big(F_{Y}(x\cdot n)\Big)^n $ (where $Y \in C(0,1)$ )
I have used that they are independent and identically distributed.
The next step is to find $F_Y(x\cdot n)$ which can be found by integration of the density of $C(0,1)$ , that is $F_Y(x\cdot n) = \dfrac{1}{\pi} \int_{-\infty}^{x\cdot n} \dfrac{1}{1+x^2} =\dfrac{1}{\pi} \big( arctan(x \cdot n) + \dfrac{\pi}{2} \big)$ .
Here is where Iam stuck , however, i know that $\dfrac{\pi}{2} = arctan(u) + arctan(1/u) $ and that $arctan(z) \approx z - \dfrac{z^3}{3} + \dfrac{z^5}{5} + ...$
I tried to use it somehow to get some "nice" expression for: $\dfrac{1}{\pi} \big( arctan(x \cdot n) + \dfrac{\pi}{2} \big)$, but failed
With "nice" i refer to the fact that $lim_{n \rightarrow \infty} \Big(F_{Y}(x\cdot n)\Big)^n $ could be recognized "easily"
Can someone give me a helping hand ?
Your approach is correct. You obtain that $F_Y(nx) = 1 - \arctan (1/nx) / \pi = 1 - 1/(\pi n x) + \ldots$, so you deduce an approximation of $\log F_Y(nx)$ since $\log(1+x)$ is equivalent to $x$, and you get the result.
For a more general approach to the problem of finding the distribution of the maximum of random variables, see the sub-area of probability and statistics called Extreme Value Theory.