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.