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Higher variations for free Lévy processes

Volume 252 / 2020

Michael Anshelevich, Zhichao Wang Studia Mathematica 252 (2020), 49-81 MSC: Primary 46L54; Secondary 60F05, 60G51. DOI: 10.4064/sm181102-20-2 Published online: 2 December 2019

Abstract

For a general free Lévy process, we prove the existence of its higher variation processes as limits in distribution, and identify the limits in terms of the Lévy–Itô representation of the original process. For a general free compound Poisson process, this convergence holds almost uniformly. This implies joint convergence in distribution to a $k$-tuple of higher variation processes, and so the existence of $k$-fold stochastic integrals as almost uniform limits. If the existence of moments of all orders is assumed, the result holds for free additive (not necessarily stationary) processes and more general approximants. In the appendix we note relevant properties of symmetric polynomials in non-commuting variables.

Authors

  • Michael AnshelevichDepartment of Mathematics
    Texas A&M University
    College Station, TX 77843-3368, U.S.A.
    e-mail
  • Zhichao WangDepartment of Mathematics
    Texas A&M University
    College Station, TX 77843-3368, U.S.A.
    and
    Department of Mathematics
    University of California, San Diego
    La Jolla, CA 92093-0112, U.S.A.
    e-mail
    e-mail

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