Renormalization of divergent moment in probability theory
Abstract
Some probability distributions have moments, and some do not. For example, the normal distribution has power moments of arbitrary order, but the Cauchy distribution does not have power moments. In this paper, by analogy with the renormalization method in quantum field theory, we suggest a renormalization scheme to remove the divergence in divergent moments. We establish more than one renormalization procedure to renormalize the same moment to prove that the renormalized moment is schemeindependent. The power moment is usually a positiveintegerpower moment; in this paper, we introduce nonpositiveintegerpower moments by a similar treatment of renormalization. An approach to calculating logarithmic moment from power moment is proposed, which can serve as a verification of the validity of the renormalization procedure. The renormalization schemes proposed are the zeta function scheme, the subtraction scheme, the weighted moment scheme, the cutoff scheme, the characteristic function scheme, the Mellin transformation scheme, and the powerlogarithmic moment scheme. The probability distributions considered are the Cauchy distribution, the Levy distribution, the qexponential distribution, the qGaussian distribution, the normal distribution, the Student's tdistribution, and the Laplace distribution.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 arXiv:
 arXiv:2205.09119
 Bibcode:
 2022arXiv220509119Z
 Keywords:

 Mathematics  Probability