The limiting distributions are obtained for two estimators of variance components: C. R. Rao's MINQUE, and an estimator produced by an iterative procedure, referred to as I-MINQUE. Limits are taken as the number of independent and identically distributed vector observations on the model assumed gets large. This approach provides the asymptotics of interest when an experiment with a large number of observations can be thought of as independent replications of a smaller experiment, a condition applying to some common experimental designs. The main result, from which the limiting distributions are obtained, is essentially an extension of a theorem due to T. W. Anderson (1973), who provides an application in time series. Both estimators considered here are consistent, and require only modest assumptions on the sampled distribution. The I-MINQUE has a limiting distribution which is functionally independent of the choice of norm; when it is further assumed that the sampled distribution is normal, the estimator is asymptotically equivalent to the m.1.e. and asymptotically efficient. The MINQUE itself is less robust in the sense that these two properties do not always apply, the conditions being dependent on the choice of design.