On confidence sequences for the mean vector of a multivariate normal distribution

Let X1, X2,... be idd random vectors with a multivariate normal distribution N([mu], [Sigma]). A sequence of subsets {Rn(a1, a2,..., an), n >= m} of the space of [mu] is said to be a (1 - [alpha])-level sequence of confidence sets for [mu] if P([mu] [set membership, variant] Rn(X1, X2,..., Xn) for every n >= m) >= 1 - [alpha]. In this note we use the ideas of Robbins Ann. Math. Statist. 41 (1970) to construct confidence sequences for the mean vector [mu] when [Sigma] is either known or unknown. The constructed sequence Rn(X1, X2, ..., Xn) depends on Mahalanobis' or Hotelling's according as [Sigma] is known or unknown. Confidence sequences for the vector-valued parameter in the general linear model are also given.