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1 results

  • On the Almost Sure Central Limit Theorem for Vector Martingales: Convergence of Moments and Statistical Applications

  • Part of
  • Bernard Bercu, Peggy Cénac, Guy Fayolle
  • Journal:
    Journal of Applied Probability / Volume 46 / Issue 1 / March 2009
    Published online by Cambridge University Press:
    14 July 2016, pp. 151-169
    Print publication:
    March 2009
    • Article
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  • We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of any even order converge in the almost sure central limit theorem for martingales. A conjecture about almost sure upper bounds under wider hypotheses is formulated. The theoretical results are supported by examples borrowed from statistical applications, including linear autoregressive models and branching processes with immigration, for which new asymptotic properties are established on estimation and prediction errors.