Approximation Of Quadratic Forms Of Independent Random Vectors By Accompanying Laws

Let X;X1 ; X2 ; : : : be i.i.d. random vectors taking values in R d . Assume that EX = 0, E jXj 8=3 ! 1 and that X is not concentrated in a proper subspace of R d . Let Y; Y1 ; Y2 ; : : : denote i.i.d. random vectors with common distribution which is accompanying to that of X. We compare the distributions of the non-degenerate quadratic forms Q[SN ] and Q[TN ] of the normalized sums SN = N \Gamma1=2 (X1 + \Delta \Delta \Delta +XN ) and TN = N \Gamma1=2 (Y1 + \Delta \Delta \Delta + YN ) and prove that sup x fi fi fi P \Phi Q[SN \Gamma a ] ! x \Psi \Gamma P \Phi Q[TN \Gamma a ] ! x \Psi fi fi fi = O \Gamma (1 + jaj 4 ) N \Gamma1 \Delta ; a 2 R d ; provided that 9 6 d 6 1. The constant in this bound depends on E jXj 8=3 , on Q and on the covariance operator of X. We also show the optimality of the bound O \Gamma N \Gamma1 \Delta