Suppose $A_i$ is $1.4$ with 0.3 probability, $1$ with 0.4 probability, and $0.6$ with 0.3 probability. Also, $A_0, A_1, A_2, ...$ are i.i.d random variables.
A Martingale sequence of interest is $X_n$ where $X_0 = 1$ and $X_n = \prod_{=1}^{n} A_i$ for $n>0$. I am considering a stopping time $T$ such that $X_n \geq 2$ for the first time.
I want to derive the probability $P(T=n)$ for each $n=1,2,...$, the sum of those probabilities ($\sum_{i=1}^n P(T=i)$) for each $n=1,2,...$, and the limit of $\sum_{i=1}^n P(T=i)$ as $n$ goes to infinity. I have a simulation result that says the limit is around 0.4. However, I am stuck figuring out the exact likelihood and limit (I tried to apply the optional stopping theorem, but I found that the boundedness assumption does not hold here). How can I derive those probabilities?