Extension of the general-start Dunkl-process central limit theorem

Determine whether the central limit theorem for Dunkl processes with drift vector \(\lambda\) and starting point \(tx\), namely convergence of \((X_t^{\lambda,tx}-t(\lambda+x))/\sqrt{t}\) to \(N(0,I_N)\), remains valid under assumptions on the pair \((x,\lambda)\) more general than regularity.

Background

The paper proves a central limit theorem for Dunkl processes whose drift vector and scaled starting-point direction form a regular pair: both vectors lie in the interior of the same Weyl chamber and satisfy a quantitative distance condition. The authors note that the fixed-start theorem is formally recovered when the starting point is zero, although that case is not covered by their definition of regular pairs. They therefore conjecture that the general-start result should hold under weaker or otherwise more general assumptions, while also exhibiting a one-dimensional obstruction when the starting point and drift lie in different Weyl chambers.

References

Clearly, for $x=0$, the assertions of the CLTs \ref{clt-dunkl} and \ref{clt-dunkl-general} are equal where this case formally is not covered by Theorem \ref{clt-dunkl-general}. For this reason we conjecture that Theorem \ref{clt-dunkl-general} still holds under slightly more general assumptions.

— A central limit theorem for Bessel and Dunkl processes with drift  (2609.10040 - Voit, 9 Sep 2026) in Section 1, Introduction, immediately after Theorem 1.2 (Theorem \ref{clt-dunkl-general})

We finally notice that in the CLT \ref{clt-bessel} the regularity of the drift $\lambda$ is necessary for $k>0$, as for drift vectors $\lambda$ on the boundary of $C$ the supports of the distributions of $X_t\lambda-t\lambda)/\sqrt t$ are contained in some half space, i.e., the limits cannot be normal. We conjecture that here the limit distributions are multivariate $\chi2$-distributions (after taking squares in all coordinates).

— A central limit theorem for Bessel and Dunkl processes with drift  (2609.10040 - Voit, 9 Sep 2026) in Section 1, Introduction, paragraph following Corollary 1.4