Summary
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.
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References
B. V. Gnedenko, The Theory of Probability (translated from Russian by B. D. Seckler), Chelsea Publishing Company, New York, 1962.
M. Loève, Probability Theory, (second edition), D. Van Nostrand Company, New York, 1960.
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.
E. Lukacs, “Stochastic convergence,” Heath Mathematical Monographs, D. C. Heath, and Company, Lexington, Mass., 1968.
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The work of the first author was supported by the National Science Foundation under grant NSF-GP-6175 while the second author was supported by the Air Force Office of Scientific Research grant AF-AFOSR-437-65.
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Laha, R.G., Lukacs, E. On a property of the Wiener process. Ann Inst Stat Math 20, 383–389 (1968). https://doi.org/10.1007/BF02911652
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DOI: https://doi.org/10.1007/BF02911652