Approximations and limit theory for quadratic forms of linear processes

AbstractThe paper develops a limit theory for the quadratic form Qn,X in linear random variables X1,…,Xn which can be used to derive the asymptotic normality of various semiparametric, kernel, window and other estimators converging at a rate which is not necessarily n1/2. The theory covers practically all forms of linear serial dependence including long, short and negative memory, and provides conditions which can be readily verified thus eliminating the need to develop technical arguments for special cases. This is accomplished by establishing a general CLT for Qn,X with normalization (Var[Qn,X])1/2 assuming only 2+δ finite moments. Previous results for forms in dependent variables allowed only normalization with n1/2 and required at least four finite moments. Our technique uses approximations of Qn,X by a form Qn,Z in i.i.d. errors Z1,…,Zn. We develop sharp bounds for these approximations which in some cases are faster by the factor n1/2 compared to the existing results

AR and MA representation of partial autocorrelation functions, with applications

We prove a representation of the partial autocorrelation function (PACF), or the Verblunsky coefficients, of a stationary process in terms of the AR and MA coefficients. We apply it to show the asymptotic behaviour of the PACF. We also propose a new definition of short and long memory in terms of the PACF.Comment: Published in Probability Theory and Related Field

Convergence of quadratic forms with nonvanishing diagonal

Motivated by applications to time series analysis, we establish the asymptotic normality of a quadratic form in i.i.d. random variables which has a nonvanishing diagonal. Our theory covers the case of both the finite and the infinite fourth moment, and leads to new results also in the case of a vanishing diagonal.Asymptotic normality Quadratic form

Approximations and limit theory for quadratic forms of linear processes

The paper develops a limit theory for the quadratic form Qn,X in linear random variables X1,...,Xn which can be used to derive the asymptotic normality of various semiparametric, kernel, window and other estimators converging at a rate which is not necessarily n1/2. The theory covers practically all forms of linear serial dependence including long, short and negative memory, and provides conditions which can be readily verified thus eliminating the need to develop technical arguments for special cases. This is accomplished by establishing a general CLT for Qn,X with normalization assuming only 2+[delta] finite moments. Previous results for forms in dependent variables allowed only normalization with n1/2 and required at least four finite moments. Our technique uses approximations of Qn,X by a form Qn,Z in i.i.d. errors Z1,...,Zn. We develop sharp bounds for these approximations which in some cases are faster by the factor n1/2 compared to the existing results.Asymptotic normality Integrated periodogram Linear process Quadratic form Semiparametric and kernel estimation