  We investigate the linear statistics of random matrices with purely imaginary
Bernoulli entries of the form $H_{pq} = \overline{H}_{qp} = \pm i$, that are
either independently distributed or exhibit global correlations imposed by the
condition $\sum_{q} H_{pq} = 0$. These are related to ensembles of so-called
random tournaments and random regular tournaments respectively. Specifically,
we construct a random walk within the space of matrices and show that the
induced motion of the first $k$ traces in a Chebyshev basis converges to a
suitable Ornstein-Uhlenbeck process. Coupling this with Stein's method allows
us to compute the rate of convergence to a Gaussian distribution in the limit
of large matrix dimension.
