I have a question related to convergence rates of martingales:
Assume that there is a sequence of maximized likelihood ratios:
$ \frac{f_{\hat{\theta}_{n}} \left ( Y_{1},Y_{2},\dots,Y_{n} \right ) }{f_{0} \left (Y_{1},Y_{2},...Y_{n} \right )} $
where $O$ denotes the true parameter and $\hat{\theta}_{n} \in \Theta - \{0\}$ is the maximum likelihood estimate of the unknown parameter at time $n$. Moreover the observation sequence $Y_{1},Y_{2},...Y_{n}$ can in general be dependent. An important information is that the parameter set $\Theta$ is finite. I.e. without loss of generality say $\Theta = \{0,1,2,...,K\} \Leftrightarrow |\Theta|=K+1$.
The sequence of such ratios is known that forms a submartingale that converges to $0$ almost surely.
I am trying to estimate the convergence rate (of the general dependent case) but I can't. Can anybody suggest something on that?