# pr.probability – Assigning negative integer moments to random variables with Hadamard regularization

Let $$Xsim F_X$$ denote a continuous random variable that admits a density $$f_X$$ with support $$mathcal S=operatorname{supp}(X)ni 0$$ and assume $$f_X(0)>0$$. I am interested in defining a regularization $$#$$ of
$$mathsf EX^{-n}:=#int_{mathcal S} x^{-n}f_X(x)mathrm dx,quad ninBbb N,$$
which would assign values to negative integers moments of $$X$$. I am aware that uniqueness of such a regularization is unlikely and so it is important to find a regularization that suits this particular context. My thought was to use Hadamard regularization. In particular, one could define
$$M_{X^{-1}}(t)=mathcal Pint_{mathcal S} frac{f_X(x)}{x-t},mathrm dx,$$
where $$mathcal P$$ denotes the usual principal value. In this manner, $$M_{X^{-1}}$$ could serve as a sort of moment generating function so that
$$mathsf EX^{-n}:=frac{1}{(n-1)!}partial_t^{n-1}M_{X^{-1}}(t)|_{t=0}.$$

This approach does show promise however there are some undesirable aspects which I show through an example:

Suppose $$Xsimmathcal N(mu,sigma^2)$$. The principal value for $$mathsf E(X-t)^{-1}$$ can be found in the literature as
$$M_{X^{-1}}(t)=frac{sqrt 2}{sigma}mathcal Dleft(frac{mu-t}{sqrt 2sigma}right),$$
where $$mathcal{D}(z)=e^{-z^{2}}int_{0}^{z}e^{t^{2}},mathrm{d}t$$ is the Dawson function. If we derive the first and second “moments” we can then also find $$mathsf{Var}X^{-1}=mathsf EX^{-2}-(mathsf EX^{-1})^2$$. Evaluating the asymptotic expansions for $$mathsf EX^{-1}$$ and $$mathsf{Var}X^{-1}$$ as $$sigmasearrow 0$$ from their explicit expressions using the procedure above gives
$$mathsf EX^{-1}simfrac{1}{mu}+frac{sigma^2}{mu^3}+mathcal O(sigma^4)$$
and
$$mathsf{Var}X^{-1}simfrac{sigma^2}{mu^4}+8frac{sigma^4}{mu^6}+mathcal O(sigma^6).$$
Note that if we expand $$g(X)=X^{-1}$$ about $$mu$$ and compute the expected value of the first few terms w.r.t. $$mathcal N(mu,sigma^2)$$ we can compare the results to the above which show that our asymptotic expansions agree with what you obtain with the $$delta$$-method. So our regularized values for $$mathsf EX^{-n}$$ do encode information about the moments in the limiting sense where $$mathsf{Var}X=sigma^2approx 0$$ when $$muneq 0$$. However, let’s consider the special case $$mu=0$$ so that $$mathsf EX^{-n}$$ are interpreted as the negative central moments of $$X$$. If our regularization is to make any sense in this context we would hope the odd-central moments are zero while the even moments are positive. Computing the first several moments gives
$$left( begin{array}{cc} n & (mathsf EX^{-n})|_{mu=0}\ 1 & 0 \ 2 & -frac{1}{sigma ^2} \ 3 & 0 \ 4 & frac{1}{3 sigma ^4} \ 5 & 0 \ 6 & -frac{1}{15 sigma ^6} \ 7 & 0 \ 8 & frac{1}{105 sigma ^8} \ end{array} right)$$
which bear a remarkable resemblance to the positive central moments of the normal distribution with the addition of alternating negative signs. So it would seem that the method of regularization used here may not make total sense.

My question has to do with approaches one could take to alter this procedure to give positive even moments. Of course, I could simply force the even moments to be positive by introducing some sort of oscillating term into the definition of $$M_{X^{-1}}$$ that is negative when $$n=2,6,10,dots$$; however, without any rational for doing so my concern is taking such an ad hoc approach is simply putting a band aid on the problem. How else might I alter my definition of $$M_{X^{-1}}$$ above to give more appropriate results for these negative moments? Are there other regularization procedures that might be more useful for this particular application?

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