Let $X_t, t\geq 0$ be a Poisson process with rate parameter $\lambda$. Compute the karhunen-loeve expansion of $X$ in interval [0, T]. How about the KL expansion of the centered process $X_t-\lambda t$?
The auto-correlation function of poisson process is $R(s,t)=\lambda^2 st + \lambda \min(s,t)$. By definition, KL expansion should satisfy $\int_0^T R(s,t) \phi_n(t) dt = \lambda_n \phi_n(s)$.
I've problems figuring out how to solve the integrated equation.
For wiener process, this link (Karhunen–Loève approximation of Brownian motion and diffusions) and wikipedia article on KL expansion was useful.