Tail lightness of the iid studentized-sum density for indices below one

Determine whether the tail of the density of the studentized-sum limit ratio for an iid symmetric regularly varying sequence is lighter when the regular variation index satisfies α∈(0,1).

Background

The paper analyzes the density of the limit ratio R_{α,2}Z=ξαZ/ζ{α,2}Z arising from studentized sums of an iid symmetric regularly varying sequence. For α∈(1,2), the authors derive an explicit Gaussian-type tail expression using contour integration and the unique pole of a parabolic-cylinder-function expression.

After tabulating the pole locations for several α∈(1,2), the authors observe that the density tails become heavier as α increases, with the heaviest behavior approached as α tends to 2. They then formulate a conjecture extending the comparison to the regime α∈(0,1), where they expect the density tails to be even lighter.

References

We conjecture that the tail of the density $f_{R_{,2}Z}$ will even be lighter for $\alpha\in(0,1)$.

— Moments for self-normalized partial sums  (2609.03728 - Matsui et al., 3 Sep 2026) in Appendix, Section “The limit density of the studentized sum of an iid regularly varying sequence,” paragraph following Table 2