  Multi-antenna relays and intelligent reflecting surfaces (IRSs) have been
utilized to construct favorable channels to improve the performance of wireless
systems. A common feature between relay systems and IRS-aided systems is the
two-hop multiple-input multiple-output (MIMO) channel. As a result, the mutual
information (MI) of two-hop MIMO channels has been widely investigated with
very engaging results. However, a rigorous investigation on the fundamental
limits of two-hop MIMO channels, i.e., the first and second-order analysis, is
not yet available in the literature, due to the difficulties caused by the
two-hop (product) channel and the noise introduced by the relay (active IRS).
In this paper, we employ large-scale random matrix theory (RMT), specifically
Gaussian tools, to derive the closed-form deterministic approximation for the
mean and variance of the MI. Additionally, we determine the convergence rate
for the mean, variance and the characteristic function of the MI, and prove the
asymptotic Gaussianity. Furthermore, we also investigate the analytical
properties of the fundamental equations that describe the closed-form
approximation and prove the existence and uniqueness of the solution. An
iterative algorithm is then proposed to obtain the solution for the fundamental
equations. Numerical results validate the accuracy of the theoretical analysis.
