Previously, a robust and sparse Direction of Arrival (DOA) estimator was derived for array data that follows a Complex Elliptically Symmetric (CES) distribution with zero-mean and finite second-order moments which is evaluated by iterations. Here, the convergence of the iterations is investigated analytically and numerically. The analytical investigation proceeds in two steps: Firstly it is shown that the true solution is a fixed point of the iteration. Then, it is shown that the iteration update decreases the DOA estimation error. Finally, the required number of iterations until convergence is evaluated numerically for exemplary DOA estimation scenarios and discussed