Characterization of Wiener process by symmetry

Prakasa Rao , B. L. S. (1972) Characterization of Wiener process by symmetry Sankhya - Series A, 34 (3). pp. 227-234. ISSN 0581-572X

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Recently Heyde (1970) has proved that if the conditional distribution of a linear function of n independent random variables Xi, 1 ≤ i ≤ n on another linear function of the same set of random variables is symmetric, then each of the Xi, 1 ≤ i ≤ n is normally distributed or degenerate under some conditions. We prove a similar result for Wiener process by means of stochastic integrals defined in probability.

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