• Title/Summary/Keyword: sub-Gaussian inequality

Search Result 2, Processing Time 0.016 seconds

THE UNIFORM CLT FOR MARTINGALE DIFFERENCE ARRAYS UNDER THE UNIFORMLY INTEGRABLE ENTROPY

  • Bae, Jong-Sig;Jun, Doo-Bae;Levental, Shlomo
    • Bulletin of the Korean Mathematical Society
    • /
    • v.47 no.1
    • /
    • pp.39-51
    • /
    • 2010
  • In this paper we consider the uniform central limit theorem for a martingale-difference array of a function-indexed stochastic process under the uniformly integrable entropy condition. We prove a maximal inequality for martingale-difference arrays of process indexed by a class of measurable functions by a method as Ziegler [19] did for triangular arrays of row wise independent process. The main tools are the Freedman inequality for the martingale-difference and a sub-Gaussian inequality based on the restricted chaining. The results of present paper generalizes those of Ziegler [19] and other results of independent problems. The results also generalizes those of Bae and Choi [3] to martingale-difference array of a function-indexed stochastic process. Finally, an application to classes of functions changing with n is given.