  We study the real eigenvalue statistics of products of independent real
Ginibre random matrices. These are matrices all of whose entries are real
i.i.d. standard Gaussian random variables. For such product ensembles, we
demonstrate the asymptotic normality of suitably normalised linear statistics
of the real eigenvalues and compute the limiting variance explicitly in both
global and mesoscopic regimes. A key part of our proof establishes uniform
decorrelation estimates for the related Pfaffian point process, thereby
allowing us to exploit weak dependence of the real eigenvalues to give simple
and quick proofs of the central limit theorems under quite general conditions.
We also establish the universality of these point processes. We compute the
asymptotic limit of all correlation functions of the real eigenvalues in the
bulk, origin and spectral edge regimes. By a suitable strengthening of the
convergence at the edge, we also obtain the limiting fluctuations of the
largest real eigenvalue. Near the origin we find new limiting distributions
characterising the smallest positive real eigenvalue.
