Asymptotical behavior of parameter estimation in prediction error framework

In prediction error method, it is known that the sequence of the criterion function converges uniformly in the parameter with probability one as the length of the input-output data tends to infinity. When the minimizing points of the limiting function criterion are not unique, the convergence of parameter estimation is not guaranteed in general. Two cases are distinguished. The case one is that the set of the minimizing points of the limiting function criterion is a continuum, and the case two is that the points of that set are isolated. Some interesting phenomena relating to the different cases are shown about the asymptotical behavior of parameter estimation through examples and theoretical analysis.