# karhunen-Loeve expansion of Poisson process

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.

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Wouldn't it better to ask this question on math.stackexchange.com? –  Ilya May 4 '12 at 7:58