\end{aligned}$$ The Appell rank is similar in spirit with the Hermite rank introduced in the previous section, but it is much harder to prove limit theorems for the statistic $S_n$ for an arbitrary Appell rank $k^{\star}$. The main problem is that, in contrast to Hermite expansion, the expansion with respect to Appell polynomials typically does not exist. Furthermore, the decomposition of $S_n$ becomes more complex when $k^{\star}$ increases. For this reason only the cases $k^{\star}=1,2,3$ are usually treated in the literature in the framework of heavy tailed innovations. In particular, [@s04] investigates the cases $k^{\star}=2,3$.

At this stage we compare the introduced setting of discrete moving average with our framework of . For the sake of exposition, we will rather consider the tangent process $\widetilde{X}$ defined at driven by a symmetric $\beta$-stable Lévy motion $L$. We immediately see that our assumption on $\beta$, namely $\beta \in (0,2)$, does not satisfy the tail behaviour condition on the innovations introduced above (in particular, ${\mathbb{E}}[L_t^2]=\infty$). As for the weight coefficients, our kernel function satisfies that $$c_0 ((x+1)^{\alpha} - x^\alpha) \sim \alpha c_0 x^{\alpha -1} \quad \text{as } x\rightarrow \infty.$$ Thus, the connection to the setting of discrete moving averages is given via $k_0=\alpha 