Some Studies on the Structure of Covariance Matrix of Discrete-Time fBm

This paper presents some results on the structure of autocovariance matrix of discrete-time fractional Brownian motion. Since these processes are nonstationary, the autocovariance matrix is a function of time. The eigenvalues associated with the autocovariance matrix are dependent on Hurst exponent characterizing the discrete-time fractional Brownian motion. However, it is seen that only one eigenvalue of this autocovariance matrix depends on time index $n$ in the asymptotic sense, and it increases as the time index increases. All other eigenvalues are observed to be invariant with time index in the asymptotic sense. It is observed that the eigenvectors associated with these eigenvalues have an interesting structure and these eigenvectors can be used as filters of an $M$-band orthogonal wavelet system. The eigenvector associated with the time-varying eigenvalue is a lowpass filter. The subband signal from this lowpass filtered branch imbibes the nonstationary attribute of the input process and is a nonstationary signal whereas all other subband signals are stationary in nature.