Decomposition and asymptotic properties of quadratic forms in linear variables

An asymptotic theory is developed for a quadratic form Q_{n,X} in linear random variables X1,…,X_{n} which can exhibit long, short, or negative dependence and whose kernel depends on n. It offers conditions under which Q_{n,X} can be approximated in the L1 and L2 norms by a form Q_{n,Z} in i.i.d. random variables Z1,…,Z_{n}. In some cases, the rate of approximation is faster by the factor n^{-1/2} compared to existing results. The approximation, together with a new CLT for quadratic forms in i.i.d. variables Z_{k} with non-zero diagonal elements, allows us to derive the CLT for the quadratic form Q_{n,X} in the linear variables X_{k}. The assumptions are similar to the well-known classical conditions for the validity of the CLT in the i.i.d. case and require the existence of the fourth moment of the Z_{k}, and in some cases only the (2+e)-th moment where e>0 and small. The results have a number of statistical applications.