Approximation of Quadratic Forms of Independent Random Vectors by Accompanying Laws

Let X, X 1 , X 2 ,... be independent and identically distributed random vectors taking values in R d . Assume that EX =0, E|X| 8/3 <∞$ and that X is not concentrated in a proper subspace of R d . Let Y, Y 1 , Y 2 ,...denote i.i.d. random vectors with common distribution which is accompanying to that of X. We compare the distributions of the nondegenerate quadratic forms Q[SN ] and Q[TN ] of the normalized sums SN=N-1/2\, (X₁+⋯ +XN) and TN=N-1/2\, (Y₁+⋯ +YN) and prove that %& ₓ | \ Q[SN-a ]< x \ - \ Q[TN-a ]< x \| %& = O ((1+|a|⁴)\,N⁻¹ ), a∈ᵈ, % provided that 9 d∞. The constant in this bound depends on E|X| 8/3 , Q, and the covariance operator of X. We also show the optimality of the bound O(N -1 ).