# Power spectrum of the difference of two Poisson processes with equal rates

I am studying the asymptotic properties of a dynamic a model involving the difference of two balanced Poisson processes (i.e., $\lambda_1 = \lambda_2$). I recently discovered the Skellam Distribution for the difference of two Poisson distributions. This distribution tells me the distribution of my process at different points in time, but it does not give any insight into the dynamics of this process.

In particular, I am interested in references or pointers/formulae for the power spectrum of this process as the rate parameter approaches infinity.

My conjecture at this point based on the asymptotic normality of the Skellam Distribution is:

$$\frac{X_1(t;\lambda)-X_2(t;\lambda)}{\sqrt{2}\lambda} \xrightarrow{d} W_t\;\textrm{as}\;\;\lambda \to \infty$$

Where $X_i(t;\lambda)$ is a Poisson counting process with rate $\lambda$ and $X_i(0;\lambda)=0$

Hence, the power spectrum will approach that of a Wiener process.

• You will have an easier time finding the limiting behaviour of $X_i - \lambda t$ and putting them together using independence. I would expect billingsley to discuss it, but I don't know for sure that he does. – user83457 Jun 28 '16 at 10:48
• @michael could you explain your proposal in a little more detail? – Bey Jun 29 '16 at 11:53
• As you say , it has to be normalized. $\frac {X_i - \lambda t} {\sqrt { \lambda}}$ should be converging to a wiener process. As I said, I would think you could find this, maybe in Billingsley's book, but it is intuitive and not hard. If you believe that, then you have the difference of independent processes each of which is converging to a wiener process. – user83457 Jun 29 '16 at 14:04
• @michael ah, got it. Thanks for the suggestion. – Bey Jun 29 '16 at 14:44