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Haar-Based Multiresolution Stochastic Processes

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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.

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Authors and Affiliations

  1. Department of Mathematics, 651 PGH, University of Houston, Houston, TX, 77204-3008, USA

    Wei Zhang

  2. Department of Mathematics, Tufts University, 503 Boston Avenue, Medford, MA, 02155, USA

    Marjorie G. Hahn

Authors
  1. Wei Zhang
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  2. Marjorie G. Hahn
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Correspondence to Wei Zhang.

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Zhang, W., Hahn, M.G. Haar-Based Multiresolution Stochastic Processes. J Theor Probab 25, 890–909 (2012). https://doi.org/10.1007/s10959-010-0333-4

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  • DOI: https://doi.org/10.1007/s10959-010-0333-4

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