Extension of the special SRD and LRD mixed-case limit theorem

Let $\mathbf{V}_N(t)=\left(\mathbf{S}_N(t),\mathbf{L}_N(t)\right)$ be the decomposition into short-range-dependent and long-range-dependent components described above, with $G_{j,S}$ and $G_{j,L}$ having Hermite ranks $k_{j,S}$ and $k_{j,L}$, respectively. Assume the normalizations and mixed-range condition

$$\frac{1}{2}\left(1-\frac{1}{k_{j_L,L}}\right)<d<\frac{1}{2}\left(1-\frac{1}{k_{j_S,S}}\right)$$

for all relevant $j_S$ and $j_L$.

Extension conjecture. The special SRD and LRD mixed-case limit theorem holds without the restriction that $k_{j,L}$ be $1$ or $2$; in particular,

$$(\mathbf{S}_N(t),\mathbf{L}_N(t))\overset{f.d.d.}{\longrightarrow}(\mathbf{B}(t),\mathbf{Z}_d^{(\mathbf{k}_L)}(t)),$$

with $\mathbf{B}(t)$ and $\mathbf{Z}_d^{(\mathbf{k}_L)}(t)$ independent.

The theorem is established in the paper only when the long-range-dependent components have Hermite ranks $1$ or $2$. The conjecture asks whether the same mixed Gaussian–Hermite-process limit and independence remain valid for arbitrary long-range-dependent Hermite ranks.