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In a paper that appeared in 2010, C. Tone proved a multivariate central limittheorem for some strictly stationary random fields of random vectors satisfyingcertain mixing conditions. The "normalization" of a given "partial sum" (or"block sum") involved matrix multiplication by a "standard -1/2 power" of itscovariance matrix (a symmetric, positive definite matrix), and the limitingmultivariate normal distribution had the identity matrix as its covariancematrix. The mixing assumptions in Tone's result implicitly imposed an upperbound on the ratios of the largest to the smallest eigenvalues in thecovariance matrices of the partial sums. The purpose of this note is to showthat in Tone's result, for the entire collection of the covariance matrices ofthe partial sums, there is essentially no other restriction on the relativemagnitudes of the eigenvalues or on the (orthogonal) directions of thecorresponding eigenvectors. For simplicity, the example given in this note willinvolve just random sequences, not the broader context of random fields.

On the behavior of the covariance matrices in a multivariate central limit theorem under some mixing conditions