Does a Gaussian Process with Diagonal Covariance Matrix Exist

Question

From reading Showing that there do not exist uncountably many independent, non-constant random variables on $ ([0,1],\mathcal{B},\lambda) $. and the answer to What is meant by a continuous-time white noise process?, I believe we cannot construct a stochastic process with covariance given by the dirac delta $\delta$ without resorting to generalized functions.

My understanding is that this implies that we cannot construct a stochastic process $X(t)$ where for any collection of unique $t_1,t_2,…$ the random variables $X(t_1), X(t_2), …$ are mutually orthogonal.

Does this imply that is it is impossible to construct a Gaussian Process that has a diagonal covariance matrix? If not, why?

Answer 1

This is not true. Limiting to $\Big([0,1], \mathcal{B}, \lambda \Big)$ is crucial in cited problem.

On the other hand, from Kolmogorov existence theorem it is straight forward to check that:

If $\ (\mu_{t})_{t \in T} \ $ is any family of distributions on $\ \mathbb{R}, \ $ then there exists an independent family $\ (X_{t})_{t\in T} \ $ of such random variables, that $\ X_{t} \ $ is distributed according to $\ \mu_{t} \ $ for all $\ t\in T. \ $

In the above statement $\ T \ $ is an interval contained in $\ \mathbb{R}_{+}. \ $

By taking $\ \mu_{t}=\mathcal{N}(0,1) \ $ for all $ \ t\in \mathbb{R}_{+} \ $ we see that:

• $(X_{t})_{t\in \mathbb{R}_{+}}$ is gaussian.
• $\mathrm{Cov}(X_{t},X_{s})=0 \ $ for all $\ s\not= t.$

This approach has nothing to do with generalized functions.