Journal of Theoretical Probability

, Volume 25, Issue 3, pp 890–909 | Cite as

Haar-Based Multiresolution Stochastic Processes

  • Wei Zhang
  • Marjorie G. Hahn


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 Open image in new window , 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 Open image in new window , 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.


Sample path properties Generalized Brownian-type construction Fractional dimensions Hölder continuity 

Mathematics Subject Classification (2000)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Boggess, A., Narcowich, F.J.: A First Course in Wavelets with Fourier Analysis. Prentice Hall, New York (2001) MATHGoogle Scholar
  2. 2.
    Bryc, W.: Normal Distribution Characterizations with Applications. Lecture Notes in Statistics, vol. 100, Springer, Berlin (2005) Google Scholar
  3. 3.
    Dobric, V., Ojeda, F.: Natural wavelet expansions for Gaussian Markov processes. Manuscript (2005) Google Scholar
  4. 4.
    Falconer, K.: Fractal Geometry, Mathematical Foundations and Applications, 2nd edn. Wiley, West Sussex (2003) MATHCrossRefGoogle Scholar
  5. 5.
    Hamadouche, D., Suquet, C.: Weak holder convergence of processes with application to the perturbed empirical process. Appl. Math., 26(1), 63–83 (1999) MathSciNetMATHGoogle Scholar
  6. 6.
    Lévy, P.: Processus Stochastiques et Mouvement Brownien, 1st edn. Gautier-Villars, Paris (1948). 2nd ed. 1965 MATHGoogle Scholar
  7. 7.
    Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes and Martingales, 2nd edn. Itô Calculus, vol. 2. Cambridge University Press, Cambridge (2000) Google Scholar
  8. 8.
    Steele, J.M.: Stochastic Calculus and Financial Applications. Springer, Berlin (2000) Google Scholar
  9. 9.
    Zhang, W.: Haar-based multi-resolution stochastic processes. Dissertation, Tufts University, Medford, MA, USA (2008) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics, 651 PGHUniversity of HoustonHoustonUSA
  2. 2.Department of MathematicsTufts UniversityMedfordUSA

Personalised recommendations