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Probabilistic Interpretation of Number Operator Acting on Bernoulli Functionals

Author

Listed:
  • Jing Zhang

    (School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China)

  • Lixia Zhang

    (School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China)

  • Caishi Wang

    (School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China)

Abstract

Let N be the number operator in the space H of real-valued square-integrable Bernoulli functionals. In this paper, we further pursue properties of N from a probabilistic perspective. We first construct a nuclear space G , which is also a dense linear subspace of H , and then by taking its dual G * , we obtain a real Gel’fand triple G ⊂ H ⊂ G * . Using the well-known Minlos theorem, we prove that there exists a unique Gauss measure γ N on G * such that its covariance operator coincides with N . We examine the properties of γ N , and, among others, we show that γ N can be represented as a convolution of a sequence of Borel probability measures on G * . Some other results are also obtained.

Suggested Citation

  • Jing Zhang & Lixia Zhang & Caishi Wang, 2022. "Probabilistic Interpretation of Number Operator Acting on Bernoulli Functionals," Mathematics, MDPI, vol. 10(15), pages 1-11, July.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2635-:d:873249
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    References listed on IDEAS

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    1. Zheng, Guangqu, 2017. "Normal approximation and almost sure central limit theorem for non-symmetric Rademacher functionals," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1622-1636.
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