Let $X_t$ be a stochastic process over $[a, b]$. with continuous autocovariance function $C(s, t)$. The Karhunen-Loeve expansion consists in representing $X_t$ as $$ X(t) = \sum_{i=1}^{\infty} Z_i \, e_i(t), \qquad Z_i = \int_a^b X(t) \, e_i(t) \, dt. $$ where $e_i$ is an orthonormal basis of $L^2(a, b)$ given by Mercer's theorem. My question is: without using Mercer's theorem, what can we say a priori on the type of convergence of the previous equation?
My reasoning is the following: since $\mathbb E(X_t^2)$ is bounded from above uniformly in time by assumption, then using Fubini $$ \mathbb E \int_a^b X_t^2 \, dt = \int_a^b \mathbb E(X_t^2) \, dt < \infty, $$ and so, almost surely, the paths of $X$ are in $L^2(a, b)$. It therefore seems to me that the convergence above should be in $L^2(a, b)$ almost surely, i.e. almost surely $$ \sum_{i=1}^n Z_i e_i \to X \quad \text{in $L^2(a, b)$ as $n \to \infty$}. $$ Is this correct? I am wondering because in this freely available paper, it is claimed that the convergence is a priori in $L^2(\Omega \times (a, b))$.
Convergence in $L^{2}(a,b)$ for almost all paths is correct.
Convergence in the norm of $L^{2} (\Omega \times (a,b))$ is also true. Since we already have convergence $L^{2}(a,b)$ for almost all paths it is enough to prove convergence of the series $\sum Z_i e_i(t)$ in $L^{2} (\Omega \times (a,b))$. But $(Z_ie_i)$ is an orthogonal sequence in this space so we only have to check that $\sum \|Z_ie_i\|^{2} <\infty$. This last fact follows from the fact that $E\int_a^{b} |X(t)|^{2}dt <\infty$ and the way $Z_i$'s are constructed from the process: Recall that $Z_n=\int_a^{b} X(t)e_n(t)dt$.