Rather than measuring the rate  of convergence of a sequence, it is more widely accepted that we measure the rate of convergence of a stochastic process that converges to a stationary measure.For a general stochastic process that generates the data $x_1,x_2,\dots x_n$  there is not a uniform rate available since we can always construct a (normal) ergodic process that breaks down the rate. The common measure of convergence rate mentioned in ergodic theory is the so-called *rate function $r_k:\mathbb{R}^+\times\mathbb{Z}^+\mapsto\mathbb{R}^+$ of frequencies for a process with joint meausre $\mu$*
if $$\mu_n(\{ x_1^n: |\mu_k-p_k(\bullet\mid x_1^n)| \})\leq r_k(\epsilon,n)\text{ for fixed k,}\epsilon>0 \text{ as }n\rightarrow\infty$$
where $\mu_k$ is the finite dimensional distribution (of a sample of size $k$) and $p_k$ is the empirical measure defined by a sample of size $k$. Note that we also require $r_k(\epsilon,n)\rightarrow 0$ as $n\rightarrow \infty$. If the log-rate function $$\frac{1}{n}logr_k(\epsilon,n)>0$$ is bounded from zero then we usually want to choose an optimal value for $\frac{1}{n}logr_k(\epsilon,n)$ which is studies by large deviation theory.

But if you want to measure the convergence rate between two filtrations then a more appropriate (yet not equivalent) notion is the *[Rosenblatt's mixing coefficient][1]*. This notion measures the similarity between two $\sigma$-algebras directly, when two algebras are independent their mixing coefficient is zero. 

This is more like an entrophy costraint on convergence rate as following.
The measure of rate in terms of entropy we be specified similarly as 
$$\mu_n(\{ x_1^n: 2^{-n(h+\epsilon)}\le\mu_n(x_1^n)\le2^{-n(h-\epsilon)}\})\geq 1-r_k(\epsilon,n)\text{ for fixed }\epsilon>0 \text{ as }n\rightarrow\infty$$
where $h$ is the entropy of the joint measure $\mu$.

Since a martingale adapted to a filtration sequence can as well be regarded as a stochastic process, its convergence rate in terms of filtration can also be measured by the rate function defined above.

[Shiedls]Shields, Paul C. "The ergodic theory of discrete sample paths." Graduate Studies in Mathematics, American Mathematics Society (1996).


  [1]: http://www.stat.cmu.edu/~cshalizi/754/2006/notes/lecture-27.pdf