  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 scheme-independent. The power moment is usually a
positive-integer-power moment; in this paper, we introduce
nonpositive-integer-power 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 cut-off scheme, the
characteristic function scheme, the Mellin transformation scheme, and the
power-logarithmic moment scheme. The probability distributions considered are
the Cauchy distribution, the Levy distribution, the q-exponential distribution,
the q-Gaussian distribution, the normal distribution, the Student's
t-distribution, and the Laplace distribution.
