Two-dimensional limit theorem for the particle number and energy in the grand canonical ensemble

It is shown that in the grand canonical ensemble for the case of a lattice system the fluctuations in energy and particle number have an asymptotic (for large volume) normal distribution. A similar statement is valid for fluctuations in energy and particle number for a sufficiently large region of the lattice even in the case of a limiting Gibbs distribution.
Both theorems are proved provided the activity $z$ is small enough or the temperature $T$ is sufficiently high.