Firstly, this is the first question I've posted, so sorry my formatting isn't quite there yet! I've come across a paper on rumour spreading processes which uses the Optional Stopping Theorem (OST) on a martingale which doesn't appear to have an upper bound, violating the OST condition that the martingale must be bounded.
The martingale is $M(k) = s_k(\frac{n}{n-1})^k$, where $n$ is the total number of nodes in the network, $s_k$ is the number of nodes susceptible to hearing a rumour, and $k$ is the timestep.
The stopping time (T) being used is the time at which no infective nodes are left in the rumour spreading process: $i_T = 0$.
It's clear to see $M(k)$ has a lower bound of $s_0$, but as $M(k)$ is raised to the power of $k$, it has no upper bound?
Nonetheless, the OST is still applied but I don't understand why. Any ideas would be greatly appreciated. Thank you!