I'm trying to understand Berman's classic paper on the subject ("Local Nondeterminism and Local Times of Gaussian Processes"). In order to define local nondeterminism, he considers the ratio


$$
\frac{Var(X\_{t\_m}-X\_{t\_{m-1}}|X\_{t\_{m-1}}, X\_{t\_{m-2}}, \ldots X\_{t\_1})}{Var(X\_{t\_m}-X\_{t\_{m-1}})}.
$$


Then he says that this is always between 0 and 1. Apparently this is an almost sure statement, because the top is a r.v., however I think I heard somewhere that with a Gaussian process (which $X$ is here) a conditional variance is actually a deterministic function. Am I right about this? I can't find a reference for it, though. I think that the conditional variance is not bounded a.s. by the variance in general, so it must be something to do with being Gaussian. Can someone help with this, please?


Thanks,
Greg



Local nondeterminism
If $X, Y\_1,\ldots Y\_m$ are jointly Gaussian variables over a probability space $\Omega$, think of them as vectors in the Hilbert space $L^2(\Omega)$. WLOG all these variables have mean zero, otherwise subtract the means without affecting variances. We can decompose $$X=Y+Z$$
where $Y\!\!\in $span$(Y\_1,\ldots Y\_m)$ and $Z$ is orthogonal to this span, so
$${\rm Var}(X)={\rm Var}(Y)+{\rm Var}(Z)\,.$$


Since uncorrelated Gaussian variables are independent,
$${\rm Var} (X\, |\, Y\_1,\ldots Y\_m)={\rm Var}\, (Y+Z\, |\, Y\_1,\ldots Y\_m) ={\rm Var} ( Z ) \,.$$


