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arXiv · 2605.21530

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

Diogo C. Soriano·Frederique Vanheusden·Slawomir J. Nasuto
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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.

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Diogo C. Soriano, Frederique Vanheusden, Slawomir J. Nasuto. 2026-05-19. Pairwise Distance-Diffusion Analysis (PDDA): A Geometric Framework for Estimating Hurst Exponents in Multivariate Long-Memory Processes. https://arxiv.org/abs/2605.21530

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