Let $X_1,X_2,...$ a sequence of independent identically distributed random variables with Frechét's distribution, i.e. $F_X(x)=e^{-\frac{1}{x}},x>0$. Let $M_n=\max{(X_1,...,X_n)}$.
- Find the CDF of $M_n$ and $\frac{M_n}{n}$.
$\rightarrow F_{M_n}(m_n)=\mathbb{P}(M_n\leq m_n)=e^{-\frac{n}{m_n}}$ and $F_{\frac{M_n}{n}}(m_n)=\mathbb{P}(M_n\leq n m_n)=e^{-\frac{1}{m_n}}$
Let $Y_0,Y_1,Y_2,...$ a sequence of independent identically distributed random variables with $F_Y(y)=e^{-\frac{1}{y(a+1)}},y>0,a \in [0,1]$. Let $V_i$ a random process such that $V_1=Y_1$ and $V_i=\max{(aY_{i-1},Y_i)}, \forall i=2,...,n$.
- Find the CDF of $V_i$ and $\frac{M_{n}^{*}}{n}$, where $M_{n}^{*}=\max{(V_1,...,V_n)}$.
I'm stuck on the second point. Can you help me? Thanks in advance.