Let $X_t,t\geq 0$ be a Poisson process with rate parameter $\lambda$. Compute the Karhunen-Loève 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 $R(s,t)=\lambda^2st+\lambda \min(s,t)$. By definition, KL expansion should satisfy $\int^T_0 R(s,t)\phi_n(t)dt=\lambda_n \phi_n(s)$.
I've problems figuring out how to solve the integrated equation.
For Wiener process, this link and Wikipedia article on KL expansion was useful.
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