I am trying to find the mean and the variance of a martingale defined as the maximized likelihood ratios over some finite parameter space. The way I want to do this is through Azuma's inequality (or any other concentration inequality).
The other information I have is that the martingale as defined above, according to Doom's convergence theorem, converge to some random variable a.s. which is something that can be used to define the concentration inequalities but I am not sure about that.
Can anyone help me, using the information given above, how to estimate the mean and the variance of the sequence of maximized likelihood ratios?