Departure from normality of increasing-dimension martingales

In this paper, we consider sequences of vector martingale differences of increasing dimension. We show that the Kantorovich distance from the distribution of the k ( n ) -dimensional average of n martingale differences to the corresponding Gaussian distribution satisfies certain inequalities. As a consequence, if the growth of k ( n ) is not too fast, then the Kantorovich distance converges to zero. Two applications of this result are presented. The first is a precise proof of the asymptotic distribution of the multivariate portmanteau statistic applied to the residuals of an autoregressive model and the second is a proof of the asymptotic normality of the estimates of a finite autoregressive model when the process is an AR( ∞ ) and the order of the model grows with the length of the series.