  This work focuses on the convergence analysis of adaptive distributed
beamforming schemes that can be reformulated as local random search algorithms
via a random search framework. Once reformulated as local random search
algorithms, it is proved that under two sufficient conditions: a) the objective
function of the algorithm is continuous and all its local maxima are global
maxima, and b) the origin is an interior point within the range of the
considered transformation of the random perturbation, the corresponding
adaptive distributed beamforming schemes converge both in probability and in
mean. This proof of convergence is general since it can be applied to analyze
randomized adaptive distributed beamforming schemes with any type of objective
functions and probability measures as long as both the sufficient conditions
are satisfied. Further, this framework can be generalized to analyze an
asynchronous scheme where distributed transmitters can only update their
beamforming coefficients asynchronously. Simulation results are also provided
to validate our analyses.
