A Note on Weighted Invariance Principle

In this note we generalize Davydov's\ucite1 weak invariance principle for stationary processes to a weighted partial sums of long memory infinite moving average processes. This note also contains some bounds on the second moments of increments of some weighted partial sum processes of a general long memory time series, not necessarily moving average type. These bounds are useful in proving the tightness in uniform metric of these processes. As a consequence of continuous mapping theorem, the probability bounds on certain functions of random variables can be established.