The Karhunen-Loeve theorem (see these notes or the wikipedia page, for example) states the following:
- Theorem: For a continuous, square-integrable, centered stochastic process $(X_t)_{t \in T}$ (with $T \subseteq \mathbb{R}^d$ compact, say), let $k(s, t) := \mathbb{E}[X_sX_t]$ be the covariance kernel and $K$ the corresponding (compact, positive, self-adjoint) integral operator on $L^2(T)$ with (continuous) eigenfunctions $(\phi_k)_k$ and (nonnegative) eigenvalues $(\lambda_k)_k$. Then, we have $$X_t= \sum_{k=1}^\infty \xi_k \phi_k(t),$$ where $\xi_k := \int_T X_t \phi_k(t) dt$ are centered, uncorrelated random variables with variance $\lambda_k$ and the convergence is in mean-square and uniform in $t$.
I would like to know if anyone knows any conditions or results related to absolute continuity of the real random variables $\xi_k$ w.r.t. Lebesgue on $\mathbb{R}$ (i.e. $\mathcal{L}(A) = 0 \implies \mathbb{P}(\xi_k \in A) = 0$ for $A \subseteq \mathbb{R}$). I am hoping for something pretty general -- for context, I have proven a result that holds for all (continuous, square-integrable, centered) stochastic processes s.t. the random vector $(\xi_1, \xi_2, \ldots)$ on $\ell^2(\mathbb{N})$ is absolutely continuous w.r.t. some nondegenerate Gaussian measure, and I would like to know where it may be applied. A related question was once asked on Math StackExchange 8 years ago with no answer.
It is known that if $(X_t)_t$ is a Gaussian process then the $\xi_k$'s are Gaussian and therefore a.c.. For Levy processes, it seems to me that Theorem 2 of this paper implies it, but I don't know enough about infinitely-divisible distributions and the Levy theory to be 200% sure. There are some criteria for absolute continuity of such distributions that I could look into, but I would like to know how general to expect the answer to be before getting into this specific case.
I also feel that I am free to choose a non-Lebesgue measure $\mu$ on $T$, diagonalize $K$ as an operator on $L^2(T, \mu)$ to get different eigenfunctions $\tilde{\phi}_k$, and then define different coefficients $\tilde{\xi}_k := \int_T X_t \tilde{\phi}_k(t) d\mu(t)$ accordingly -- Mercer's theorem and the Karhunen-Loeve proof should go through I think. I am not sure if this helps, and I have the constraint that I need to retain continuity of the eigenfunctions for my application... just an idea :)
Please let me know of any intuitions, suspicions, and guesses as well as actual results. Thank you in advance, I really appreciate MathOverflow and this is my first question, I am happy to be part of the community! Have a nice day
