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Pairwise Distance-Diffusion Analysis (PDDA): A Geometric Framework for Estimating Hurst Exponents in Multivariate Long-Memory Processes

Year: 2026

Type: preprint

Abstract: We introduce Pairwise Distance-Diffusion Analysis (PDDA), a geometric framework that connects the Hurst exponent to the scaling of pairwise distances in long-memory stochastic processes. From a single distance-plot representation, PDDA yields three complementary routes to persistence: R/S-PDDA, based on the growth of geometric extrema; MSD-PDDA, based on the scaling of the second moment of lagged distances; and recurrence-volume scaling, in which the decay of close returns with increasing temporal separation reflects the expansion of the lagged displacement cloud. The framework extends naturally to multivariate isotropic and anisotropic processes, where the local spatial dimension specifies the available degrees of freedom while the Hurst exponents govern their temporal expansion. These results establish the distance plot as a common geometric representation of persistence, providing a unified distance-based foundation for Hurst analysis.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2605.21530

Authors: Soriano, Diogo C., Vanheusden, Frederique, Nasuto, Slawomir J.

Institutions: Universidade Federal do ABC, Nottingham Trent University, University of Reading

Topic: Complex Systems and Time Series Analysis

Subfield: Economics and Econometrics

Field: Economics, Econometrics and Finance

Domain: Social Sciences

Open Access status: green

Funders: Fundação de Amparo à Pesquisa do Estado de São Paulo, Conselho Nacional de Desenvolvimento Científico e Tecnológico

Awards: G295657101, G4124199609, G8166476623

OpenAlex ID: W7162116014

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