Analysis Of A Long Memory Circular Convolution Model
Robert Kimberk
TL;DR
The paper develops a long-memory time-series model based on the product of a symmetric circulant matrix and a Gaussian vector, yielding a power-law PSD $|f|^{-eta}$. By tying the eigenstructure of the circulant matrix to the spectral distribution, it connects the intrinsic dimension $d$ of the unit-sphere cosine statistics to a Beta$(\alpha,\alpha)$ distribution with $\alpha=(d-1)/2$, and provides practical estimators for $d$, $\alpha$, variance, and condition number from eigenvalues. The work presents explicit R-code implementations for generating realizations, histograms, and eigenvalue-based estimates, and discusses the geometric interpretation of long memory via unit-sphere geometry and direction statistics. Together, these results offer a concrete, computable framework for simulating and analyzing long-memory processes with a clear link between spectral structure, distribution shape, and matrix conditioning, with implications for spectral estimation and dimensionality assessment in related directional-statistics contexts.
Abstract
A stochastic model, the product of a circulant matrix and a random normal vector, is shown to produce an evolutive long memory time series with a power law spectral density. The distribution of the time series, a beta location scale family of distributions, provides a connection to the unit centered spherical distribution and directional statistics. The eigenanalysis of the deterministic circulant matrix is shown to provide estimates of the discrete Fourier spectral trend, the intrinsic dimension, the probability density shape parameter of the resulting time series, the condition number of the matrix and a principle component analysis. Examples of the R code, used as the constructive exploratory element of the work are given as constructive elements of the paper. The R code may be copied, pastedinto a R editor, and explored.