On sequential estimation of a certain estimable function of the mean vector of a multivariate normal distribution

Consider a k-variable normal distribution Ν (μ,Σ where mgr; = (μ1,μ2, … μk)' and Σ is diagonal matrix of unknown elements >0,i = 1,2, … k. The problem of sequential estimation of = 1 αiμi is considered. The stopping rule is shown to have some interesting limiting properties when the σi's become infinite.