A counterexample of proof of convergence using asymptotic averaging

A counterexample of convergence of a parameter estimator, the recursive prediction error method (RPEM), is presented. Here, a case where the RPEM fails to reach a minimum of the negative log likelihood function is examined. The cause of the failure of asymptotic averaging is disclosed and shown to be generic for any proof of convergence using asymptotic averaging techniques. The assumptions made in the asymptotic averaging theory proof of convergence are strengthened to assure convergence wp1 to a minimum of the asymptotic negative log likelihood function. This is illustrated by means of a simple example