On a property of the Wiener process

  • R. G. Laha
  • E. Lukacs


In an earlier paper [3] the authors gave a characterization of the Wiener process by the property that a stochastic integral\(\int_A^B {a\left( t \right)dX\left( t \right)} \) (taken in the sense of convergence in the quadratic mean) has the same distribution as α[X(t+1)−X(t)]. The assumption that the integral was defined in the sense of convergence in the quadratic mean necessitated a number of hypotheses; for instance one had to suppose that the process was of the second order and that its mean value function and its covariance function were of bounded variation.

In the present paper these assumptions are avoided by assuming that the stochastic integral is defined in the sense of convergence in probability.


Characteristic Function Wiener Process Stochastic Integral Homogeneous Process Independent Increment 


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  1. [1]
    B. V. Gnedenko, The Theory of Probability (translated from Russian by B. D. Seckler), Chelsea Publishing Company, New York, 1962.MATHGoogle Scholar
  2. [2]
    M. Loève, Probability Theory, (second edition), D. Van Nostrand Company, New York, 1960.MATHGoogle Scholar
  3. [3]
    R. G. Laha and E. Lukacs, “On linear forms and stochastic integrals,” Proceedings of the 35th Session of the International Statistical Institute, Beograd, II (1965), 828–840.MathSciNetGoogle Scholar
  4. [4]
    E. Lukacs, “Stochastic convergence,” Heath Mathematical Monographs, D. C. Heath, and Company, Lexington, Mass., 1968.MATHGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics 1968

Authors and Affiliations

  • R. G. Laha
    • 1
  • E. Lukacs
    • 1
  1. 1.The Catholic University of AmericaUSA

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