$Y_{1},Y_{2},\ldots$ are i.i.d. random variables (on some ($\Omega$,F,$P$)) whose common distribution function is $F(y)=\begin{cases} 0,& y<0\\y^{\alpha}, & 0\leq y\leq 1 \\ 1,& y>1 \end{cases}$
where $\alpha \in (0,\infty )$. For any $\beta\in (0,\infty )$, determine $P( \left\{w:n^{\beta}Y_{n}(w))\to \infty \text{ as } n \to \infty \right\})$.