Gaussian limit for Pfaffian point processes

We prove a central limit theorem for linear statistics of a broad class of Pfaffian point processes. As an application, we derive Gaussian limits for scaled linear statistics of step functions in the Pfaffian $\mathrm{S}\mathrm{i}\mathrm{n}{\mathrm{e}}_4$ and ${\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{e}}_1$ processes.