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On some characterization of probability distributions in Hilbert spaces

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Summary

The aim of this paper is to prove the following theorem about characterization of probability distributions in Hilbert spaces:Theorem. — Let x1, x2, …, xn be n (n≥3) independent random variables in the Hilbert spaceH, having their characteristic functionals fk(t) = E[ei(t,x k)], (k=1, 2, …, n): let y1=x1 + xn, y2=x2 + xn, …, yn−1=xn−1 + xn.

If the characteristic functional f(t1, t2, …, tn−1) of the random variables (y1, y2, …, yn−1) does not vanish, then the joint distribution of (y1, y2, …, yn−1) determines all the distributions of x1, x2, …, xn up to change of location.

References

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    U. Grenander,Probabilities on algebraic stuctures, Stockholm, 1963.

  2. [2]

    F. Riesz, B. Sz. Nagy,Leçons sur d'analyse fonctionelle, Budapest, 1952.

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Kotlarski, I. On some characterization of probability distributions in Hilbert spaces. Annali di Matematica 74, 129–134 (1966). https://doi.org/10.1007/BF02416453

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Keywords

  • Probability Distribution
  • Hilbert Space
  • Joint Distribution
  • Characteristic Functional
  • Independent Random Variable