I'm trying to understand something in Adler and Taylor's book, Random Fields and Geometry.
Let $T \subset \mathbb{R}^N$ be a compact parameter set (for simplicity, suppose it is a closed hypercube) and let $f : T \to \mathbb{R}^N$ be a random field that is a.s. in $C^1(T)$.
We define by $\nabla f$ the $N\times N$ matrix of first-order partial derivatives $$\nabla f(t) = \left(\frac {\partial f_i}{\partial t_j}(t)\right)_{i,j}$$
In Theorem 11.2.1 (page 267), several assumptions are made, including the following "assumption (d)":
(d) The conditional densities $p_t(z|f(t)=x)$ of $\det \nabla f(t)$ given $f(t)=x$, are continuous for $z$ and $x$ in neighborhoods of $0$ and $u$, respectively, uniformly in $t \in T$.
On the beginning of the next page (page 268), the following paragraph appears:
[When the random field $f$ is Gaussian,] the marginal and conditional densities appearing in the conditions of Theorem 11.2.1 are also Gaussian, and so their boundedness and continuity are immediate, as long as all the associated covariance matrices are nondegenerate, which is what we'll need to assume in this case.
I don't understand why this is true. (Specifically, I'm interested only in condition (d) which I quoted.)
Regardless of the above, ultimately, I want to prove the following: For any $t$ and small enough $A$, we have $\mathbb P\left\{|\det \nabla f(t)| \le A\right\} \le MA$ for some constant $M$. ($f$ is Gaussian, and maybe satisfies certain nondegeneracy conditions.) It doesn't seem like $\det \nabla f(t)$ should be Gaussian, since it is a sum of products of Gaussians... But I don't need it to be Gaussian, just the above bound.
Thank you for any help!