Bunches of Random Cross-correlated Sequences

TL;DR

This paper analyzes the statistical properties and generation algorithms of random cross-correlated sequences created via convolution, exploring their correlation limits, distribution tails, and anisotropic behaviors.

Contribution

It provides a detailed examination of the properties, generation methods, and decomposition techniques for random cross-correlated sequences, including special distribution cases.

Findings

Limits of weak and strong correlations are characterized.

Algorithms for sequence generation via correlation matrix decomposition are discussed.

Heavy-tailed distributions and anisotropic properties are analyzed.

Abstract

Statistical properties of random cross-correlated sequences constructed by the convolution method (likewise referred to as the Rice's or the inverse Fourier transformation) are examined. Algorithms for their generation are discussed. They are frequently reduced to solving the problem for decomposition of the Fourier transform of the correlation matrix into a product of two mutually conjugate matrices; different decompositions of the correlation matrix are considered. The limits of weak and strong correlations for the one-point probability and pair correlation functions of the sequences are studied. Special cases of heavy-tailed distributions resulting from the convolution generation are analyzed. Anisotropic properties of statistically homogeneous random sequences related to asymmetry of a filtering function are discussed.