Let $(M_n)_{n\in \mathbb N}, M_n\geq 0$ be a positive martingale and define $T:=\inf \{n\in \mathbb N \vert M_n=0\}$ as the first hitting time of $0$, knowing that $\mathbb P(T < \infty)=1 $. Let $N$ be a random variable on $\mathbb N_0$, such that $\mathbb P(N=k)=:p_k$ and $(T_n)_{n\in \mathbb N}$ be an iid sequence distributed as $T$.
What can I say about the distribution of
$$\max\{T_1,...,T_N\}$$
Suppose I know the characteristic function of $T$, can I infer information about the characteristic function of $\max\{T_1,...,T_N\}$ ? (After all, I only want to relate the distributions of $T$ and $\max\{T_1,...,T_N\}$)