Fast Triangular Factorization of Covariance Matrices of Differenced Time Series

An algorithm is presented for computing the triangular factors of the covariance matrix of a random vector whose elements are the successive differences of a stationary time series. This algorithm has applications in the modeling of time series using a difference operator rather than using a shift operator, as is done in the conventional autoregressive model. The difference-operator based models offer the benefit of better numerical conditioning when applied to series that are obtained by sampling continuous-time processes at rapid rates. The covariance matrix of diferenced data is not Toeplitz; however, it has a Toeplitz-like structure that is exploited in this paper to obtain an $O( n^2 )$ algorithm. This algorithm is amenable to parallelization; i.e., it requires only $O( n )$ time when using n processors in parallel.