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 likelihoods.
One way I tried is by taking the log of $X_n$. Then, $Y_n = log(X_n) = \sum_{i=1}^{n} log(A_i)$, and $Y_n$ seems like an asymmetric random walk. Is there any theorem to apply to this example to get the probability of $Y_n$ reaching $log(2)$ for the first time at each period $n$? Or is there any other way to calculate the aforementioned probabilities instead of taking the log?