Uniform Central Limit Theorem for martingales
  We study some sufficient conditions imposed on the sequence of martingale differences (m.d.) in the separable Banach spaces of continuous functions defined on the metric compact set for the Central Limit Theorem in this space.   We taking into account the classical entropy terms, and use the theory of the so-called Grand Lebesgue Spaces of random variables having power and exponential decreasing tail of distribution. 
L.Sirota
math.PR