In two different journal articles:

The First Negative Moment of Skew-t and Generalized Student's t-Distributions in the Principal Value Sense


The first negative moment in the sense of the Cauchy principal value

the concept of the principal valued first moment is presented in the case where the first moment of a r.v. does not exist. The principal valued first moment (a.k.a. principal valued first negative moment FNM) for a continuous r.v. is defined by \begin{equation} PV\!E(Y^{-1}) = \lim_{\epsilon\to0^+}\left(\int_{-\infty}^{-\epsilon}+\int_{\epsilon}^\infty\right)\frac{1}{y}f_Y(y)\,\mathrm dy. \end{equation}

The author makes the following assertion in both articles:

Hence, through different viewpoints of the integral of FNM, the concept of the (Cauchy) principal value, widely used in the probability theory such as the weak law of large numbers [11], can be used to avoid the nonexistence of the FNM in the usual sense.

The same citation is used both times which is Feller's An Introduction to Probability Theory and its Applications vol. 2.

My question is simple: How is the FNM used in the weak law of large numbers? Is there a version of the WLLN for r.v's without moments that makes use of the FNM?

I found the following generalization of the WLLN in Feller's book (VII.7) for sequences of i.i.d. r.v's

\begin{equation} \frac{S_n-n\mathrm EXI\{|X|\leq n\}}{n}\overset{p}{\to}0\ \ \text{as}\ \ n\to\infty\iff x\,\mathsf P(|X|>x)\to 0\ \ \text{as}\ \ x\to\infty. \end{equation}

I checked this condition for the reciprocal normal, i.e. $X=Y^{-1}$ for $Y\sim\mathcal N(\mu,\sigma^2)$ and found that \begin{equation} \lim_{x\to\infty}x\,\mathsf P(|X|>x)=2f_Y(0)\neq 0. \end{equation} So it would seem that Feller's WLLN does not apply. Can someone help me understand the claim made in the papers? Where is the FNM used in the WLLN?


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