Consider a random process $\{X_k\}^\infty_{k=1}$ of independent and identically distributed random variables, with mean value $\mathbb{E}[X_k] = \mu $ and $0<\mu <\infty$. Consider a random process $W_n$ given by the recursion$W_n = \max\{W_{n-1}+X_n,0\}$, with $W_0 =0$. We define $\lambda_n$ as the last time $W_n$ becomes equal to zero (we call this event a 'renewal'), i.e.
\begin{equation} \lambda_n = \sup\{ \ell \leq n : W_\ell =0\}. \end{equation}
I am trying to verify whether $n - \lambda_n \rightarrow \infty$ in some sense (e.g. almost surely, or in probability) as $n\rightarrow \infty$. Intuitively, I think that should hold because of the positive mean value assumption. I would really appreciate if someone could provide any hints or references for this kind problem. Thanks!