We establish strong and weak approximations for quadratic forms of weakly and strongly dependent random variables and obtain necessary and sufficient conditions for the weak convergence of weighted functions of quadratic forms. The results are applied to get the asymptotic distributions of some tests which can be used to detect possible changes in the long-memory parameter.
"Limit theorems for quadratic forms with applications to Whittle's estimate." Ann. Appl. Probab. 9 (1) 146 - 187, February 1999. https://doi.org/10.1214/aoap/1029962600