Feeds:
Posts
The setting is a measure space with self-transformation, $(X,\mathcal{B},\mu,T)$ with measure $\mu$ and transformation $T:X\rightarrow X$.  The von Neumann mean ergodic theorem says that if $f\in L^2$, then the average
$Avg(f,N) = \frac{1}{N}\sum_{k=0}^{N-1} f \circ T^k$
converges in $L^2$ to a limit $\bar{f}$ that is invariant even when $T$ is not ergodic.  When it is, then $\bar{f} = \int f d\mu$.  The Birkhoff pointwise ergodic theorem says the convergence of the average above is almost everywhere when $f\in L^1$ with the constant above for limit.
These theorems would be interesting to volatility dynamics if $v_{k+1} = T v_k$ for example for some invariant transformation with noise such as multiplication by a matrix.  This is actually the ARMA type setting of time series analysis.