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On weak convergence to Brownian motion

  • Inge S. Helland
Article

Summary

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.

Keywords

Stochastic Process Brownian Motion Probability Theory Limit Theorem Mathematical Biology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Billingsley, P.: Weak Convergence of Probability Measures. New York: Wiley 1968Google Scholar
  2. 2.
    Brown, B.M.: Martingale central limit theorems. Ann. Math. Statist. 42, 59–66 (1971)Google Scholar
  3. 3.
    Dugundji, J.: Topology. Boston: Allyn and Bacon 1966Google Scholar
  4. 4.
    Durrett, R., Resnick, S.I.: Functional limit theorems for dependent variables. Ann. Probability 6 829–846 (1978)Google Scholar
  5. 5.
    Dvoretzky, A.: Asymptotic normality for sums of dependent random variables. In: Proc. Sixth Berkeley Sympos. Math. Statist. Probab. Berkeley: Univ. of Calif. Press 1972Google Scholar
  6. 6.
    Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independent Random Variables. Menlo Park, California: Addison-Wesley 1968Google Scholar
  7. 7.
    Lindvall, T.: Weak convergence of probability measures and random functions in the function space D[0,∞). J. Appl. Probability 10, 109–121 (1973)Google Scholar
  8. 8.
    McLeish, D.L.: Dependent central limit theorems and invariance principles. Ann. Probability 2, 620–628 (1974)Google Scholar
  9. 9.
    Rootzén, H.: On the functional central limit theorem for martingales. Z. Wahrscheinlichkeitstheorie verw. Gebiete 38, 199–210 (1977). Part II. Z. Wahrscheinlichkeitstheorie verw. Gebiete 51, 79–93 (1980)Google Scholar
  10. 10.
    Rosenkrantz, W.A.: A convergent family of diffusion processes whose diffusion coefficients diverge. Bull. Amer. Math. Soc. 80, 973–976 (1974)Google Scholar
  11. 11.
    Rosenkrantz, W.A.: Limit theorems for solutions to a class of stochastic differential equations. Indiana Univ. Math. J. 24, 613–625 (1975)Google Scholar
  12. 12.
    Rudin, W.: Real and Complex Analysis. New York: McGraw-Hill 1970Google Scholar
  13. 13.
    Scott, D.J.: Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation opproach. Advances in Applied Probability 5, 119–137 (1973)Google Scholar

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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