Given a probability space $(\Omega,\mathcal{F},(\mathcal{F}_n)_n)$ where $(\mathcal{F}_n)_n$ is a filtration of $\mathcal{F}$. Consider the sequence of random variables/vectors $(X_n)_n$ and $(\tilde{X}_n)_n$ defined on the above probability space with distribution $\mathsf{P}_{X}$ and $\mathsf{\tilde{P}_{\tilde{X}}}$, respectively. Note that there is no assumptions on the independence of $(X_n)_n$ and $(\tilde{X}_n)_n$. Define the random variable $f_n = \prod_{i=1}^n \frac{\mathsf{P}_X(dx_i)}{\tilde{\mathsf{P}}_{\tilde{X}}(d\tilde{x}_i)}$. It is well-known that $f_n$ is a likelihood function and hence a non-negative martingale with mean $1$. Obviously the actual question has not been posted here; nevertheless if I get a hint on how to proceed with this then I might be able to work out the actual problem. To that end, I have the following doubts/queries:
I know that $f_n$ will converge almost surely because it is a non-negative martingale. I am interested in the value or a reasonable estimate of the value of the limit of $f_n$.
- I know there is a strong law of large number for square-integrable martingales that can be used to determine the limit, provided $f_n$ is square-integrable.
- If I take log on both the sides, I will end up getting a supermartingle that has a Doob's decomposition. I came across an article: https://doi.org/10.1016/S0019-9958(80)90306-X along with some other articles by the same author.
I understand that the nature of the distributions $\mathsf{P}_{X}$ and $\mathsf{\tilde{P}_{\tilde{X}}}$ are important and it will drive to what value the random variable $f_n$ will converge. How do I proceed with the analysis here? Is there anything in the literature (except the above) that can be used here? Note that I am not interested in the computation of the exact value of the limit. A good estimate of the limit of $f_n$ will also work fine. Any suggestions/comments will be very helpful. Let me know if any more information is needed. Thank you
Note: I have posted this question in Mathflow but I have not received any responses. So I am posting this here.