Limit theorems for aggregated linear processes (Q2837758)

The author studies aggregated linear processes, i.e., the component-wise sum of \(N\) independent linear processes with random coefficients, where the number \(N\) of linear processes might go to infinity as the number \(n\) of observations grows. Linear processes show a short range dependent behavior if the coefficients are absolutely summable, if they are not summable, the process exhibits long range dependence. The author distinguishes between short and long memory by imposing a similar condition on the \(L_2\)-norms of the random coefficients. As expected, the limit of the partial sum of the aggregated linear process is a Brownian motion in the short range dependent case. Due to the randomness of the coefficients, the Brownian motion might have a random variance parameter if the number \(N\) of linear processes is bounded. More general, the author considers the partial sum of a \(d\)-variate function of d consecutive observations from the aggregated linear processes. In the long range dependent case, the function \(f\) is assumed to be dominated by a power of the difference of two random variables times a polynomial. Differences of long range dependent linear processes have a reduced dependence. So with the help of this assumption, the author is able to establish a Brownian motion as limit distribution also in the long range dependent case.