Failure of the first-order Rytov approximation to properly predict the scintillation index of a large-aperture focused beam, or an uplink collimated (or focused) beam, has been discussed in several recent publications, which cite beam wander effects as the main reason for this failure. We use computer simulations to examine several aspects of beam wander phenomena on a propagating convergent beam in the weak-fluctuation regime over a horizontal path at high altitude for which the refractive index structure parameter is on the order of C=1.39×10−16 m−2/3. Simulation results are presented at various ranges up to 10 km for (1) the beam wander centroid displacement, (2) the kurtosis excess of the irradiance profile, (3) the irradiance profile, (4) the mean-square hot spot displacement from the boresight and from the centroid, and (5) the scintillation index at the optical axis of the beam. In addition, simulation results are compared with theoretical models.