Weak convergence of linear and quadratic forms and related statements on Lp-approximability

In this paper we obtain central limit theorems for quadratic forms of non-causal short memory linear processes with independent identically distributed innovations. Nabeya and Tanaka (1988) suggested the format, which links the asymptotic distribution to integral operators. In their approach, integral operators had to have continuous symmetric kernels. Mynbaev (2001) employed the theory of approximations to get rid of the continuity requirement. Here we go one step further by lifting the kernel symmetry condition. Also, we establish Lp-approximability of the special sequences which arise in the theory of regressions with slowly varying regressors.