On weak convergence to Brownian motion

  • Inge S. Helland


We consider a minimal form of the usual conditions for the dependent central limit theorem and invariance principle for “near martingales”. We show that these conditions imply convergence to Brownian motion in a way that is slightly stronger than weak convergence in D[0,∞). On the other hand, if a sequence of processes with paths in D[0,∞) converges to Brownian motion in this way, then we can always find a sequence of partitions of the time axis that is such that these conditions hold for the corresponding array of increments.


Stochastic Process Brownian Motion Probability Theory Limit Theorem Mathematical Biology 
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Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Inge S. Helland
    • 1
  1. 1.Department of Mathematics and StatisticsAgricultural University of NorwayAas-NLHNorway

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