Recurrence relation involving the $\max$ function (reference request): $p_{n + 1} = p_{n} + \max\{r_{p_{n} - 1} - 1, r_{p_{n}}\}$

Question

Consider two sequences $(p_{n})_{n\in\mathbb{N}}$ and $(r_{n})_{n\in\mathbb{N}}$ such that they satisfy the following recurrence relation: \begin{align*} p_{n + 1} = p_{n} + \max\{r_{p_{n} - 1} - 1, r_{p_{n}}\} \end{align*}

where $p_{0} := r_{0}$ and $r_{k} = 0$ for every $k < 0$, and $r_{n}\in\{1,2,\ldots,N\}$ are IID random variables supposed to be known whose distribution is not specified.

I would like to know if it is possible to determine the expression for $p_{n}$ in terms of $(r_{n})_{n\in\mathbb{N}}$.

I do not need a full explanation (although a starting point would be appreciated). I would like to know if anyone could provide me any reference where I could find the corresponding theory related to the solution to this problem.

The same question has been asked here.

Any suggestion (book, article, etc) is quite appreciated.