  This paper concerns with the convergence analysis of a fourth order singular
perturbation of the Dirichlet Monge-Amp\`ere problem in the $n$-dimensional
radial symmetric case. A detailed study of the fourth order problem is
presented. In particular, various {\em a priori} estimates with explicit
dependence on the perturbation parameter $\vepsi$ are derived, and a crucial
convexity property is also proved for the solution of the fourth order problem.
Using these estimates and the convexity property, we prove that the solution of
the perturbed problem converges uniformly and compactly to the unique convex
viscosity solution of the Dirichlet Monge-Amp\`ere problem. Rates of
convergence in the $H^k$-norm for $k=0,1,2$ are established, and illustrating
numerical experiment results are also presented in the paper.
