Local nondeterminism 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
 A: 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,
$${\rm Var}  (X\, |\, Y_1,\ldots Y_m)={\rm Var}\, (Y+Z\, |\, Y_1,\ldots Y_m) ={\rm Var} ( Z ) \,.$$
A: $\newcommand\Si\Sigma\newcommand\Var{\operatorname{\mathsf{Var}}}$More generally, if two random vectors $Y$ and $Z$ are jointly normal, then $\Si_Y-\Si_{Y|Z}$ is positive semidefinite, where $\Si_Y$ is the covariance matrix of the distribution of $Y$ and $\Si_{Y|Z}$ is the covariance matrix of the conditional distribution of $Y$ given $Z$. (One may note here that $\Si_{Y|Z}$ actually is a non-random matrix.)
In particular, if the random vector $Y$ is one dimensional (that is, a real-valued normal random variable), then
$$\Var Y\ge\Var(Y|Z).$$
Hence, your desired conclusion immediately follows.
