Sums of Independent Random Vectors: Proximity Estimating

Let S n  = X 1 + ··· + X n , , n ≥ 1, where {X n } n≥1 and be sequences of i.i.d. random vectors in ℝ k . We assume the equality of means and covariance matrices of X 1 and , and also the existence of “smooth enough” densities f X and of X and respectively. Under these hypotheses we show that for some finite constant c, where V is the total variation metric.