On the asymptotic distribution of weighted least squares estimators

In the ELB (Empirical Linear Bayes)-approach to credibility, the unknown structural parameters are substituted by a set of parameter estimates. The weighted least squares estimators are known to be asymptotically normally distributed when the design variables are independent and identically distributed random variables. It is demonstrated that, with probability one, the conditional asymptotic distribution, given the design, is the same as the unconditional distribution. Estimation of the asymptotic covariance matrix will also be considered.