Haar-Based Multiresolution Stochastic Processes

Wei Zhang () and Marjorie G. Hahn ()
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Wei Zhang: University of Houston
Marjorie G. Hahn: Tufts University

Journal of Theoretical Probability, 2012, vol. 25, issue 3, 890-909

Abstract: Abstract Modifying a Haar wavelet representation of Brownian motion yields a class of Haar-based multiresolution stochastic processes in the form of an infinite series $$X_t = \sum_{n=0}^\infty\lambda_n\varDelta _n(t)\epsilon_n,$$ where λ n Δ n (t) is the integral of the nth Haar wavelet from 0 to t, and ε n are i.i.d. random variables with mean 0 and variance 1. Two sufficient conditions are provided for X t to converge uniformly with probability one. Each stochastic process , the collection of all almost sure uniform limits, retains the second-moment properties and the same roughness of sample paths as Brownian motion, yet lacks some of the features of Brownian motion, e.g., does not have independent and/or stationary increments, is not Gaussian, is not self-similar, or is not a martingale. Two important tools are developed to analyze elements of , the nth-level self-similarity of the associated bridges and the tree structure of dyadic increments. These tools are essential in establishing sample path results such as Hölder continuity and fractional dimensions of graphs of the processes.

Keywords: Sample path properties; Generalized Brownian-type construction; Fractional dimensions; Hölder continuity; 60G17 (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/s10959-010-0333-4

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