  Reconfigurable intelligent surface (RIS) has attracted extensive attention in
recent years. However, most research focuses on the scenario of the narrowband
and/or instantaneous channel state information (CSI), while wide bandwidth with
the use of millimeter-wave (mmWave) (including sub-Terahertz) spectrum is a
major trend in next-generation wireless communications, and statistical CSI is
more practical to obtain in realistic systems. Thus, we {consider} the ergodic
achievable rate of RIS-assisted mmWave multiple-input multiple-output
orthogonal frequency division multiplexing communication systems. The widely
used Saleh-Valenzuela channel model is adopted to characterize the mmWave
channels and only the statistical CSI is available. We first derive the
approximations of the ergodic achievable rate by means of the majorization
theory and Jensen's inequality. Then, an alternating optimization based
algorithm is proposed to maximize the ergodic achievable rate by jointly
designing the transmit covariance matrix at the base station and the reflection
coefficients at the RIS. Specifically, the design of the transmit covariance
matrix is transformed into a power allocation problem and solved by
spatial-frequency water-filling. The reflection coefficients are optimized by
the Riemannian conjugate gradient algorithm. Simulation results corroborate the
effectiveness of the proposed algorithms.
