Isotropic Random Current
PRINCETON UNIVERSITY PRINCETON United States
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The theory of isotropic random vector fields was originated by H. P. Robertson 1in his theory on isotropic turbulence. He defined the covariance bilinear form of randomvector fields which corresponds to Khinchins covariance function in the theory of stationarystochastic processes. Although in the latter theory the essential point was madeclear in connection with the theory of Hilbert space and that of Fourier analysis, we haveno corresponding theory on isotropic random vector fields.Robertson obtained a condition necessary for a bilinear form to be the covariance bilinearform of an isotropic random vector field. Unfortunately his condition is not sufficientin fact, he took into account only the invariant property of the covariance bilinearform but not its positive definite property. A necessary and sufficient condition was obtainedby S. It6 2. Although his statement is complicated, he grasped the crucial point.His result corresponds to Khinchins spectral representation of the covariance functionof stationary stochastic processes.
- Statistics and Probability