1$, are independent of $L$ and the properties of stable convergence. To be used in the next theorem we recall that a totally right skewed $\rho$-stable random variable $S$ with $\rho>1$, mean zero and scale parameter $\eta>0$ has characteristic function given by $${\mathbb{E}}[ e^{i \theta S} ] =\exp\Big( -\eta^\rho |\theta |^\rho \big( 1- i \mathrm{sign}(\theta)\tan(\pi \rho/2)\big)\Big),\qquad \theta\in {\mathbb{R}}.$$

For part (ii) of Theorem \[maintheorem\], which we will refer to as the ergodic case, we also show the second order asymptotic results.

\[sec-order\] Suppose that assumption (A) is satisfied and $L$ is a symmetric $\beta$-stable Lévy process with scale parameter $\sigma>0$. Let $f:[0,\infty )\mapsto {\mathbb{R}}$ be given by $f(t)=g(t)/t^{\alpha}$ for $t>0$ and $f(0)= c_0$, and assume that $f$ is $k$-times continuous right differentiable at $0$. For the below case (i) assume, in addition, that $| g'(t)|\leq K t^{\alpha-1}$ for all $t>0$.

-   If $k=1$, $\alpha <1-1/\beta$ and $p<\beta/2$, then it holds that $$n^{1-\frac{1}{(1-\alpha)\beta}}\Big(n^{-1+p(\alpha + 1/\beta)}V(p;k)_n- m_p\Big) {\stackrel{d}{\longrightarrow}}S,$$ where $S$ is a totally right skewed $(1-\alpha)\beta$-stable random variable with mean zero and scale parameter $\widetilde{\sigma}$, which is defined in Remark \[rem-const\](i).

-   \[mt-case-6\] If $k\geq 2$, $\alpha <k-2/\beta$ and $p<\beta/2$ we deduce that $$\begin{aligned}
     \label{part5}
    \sqrt{n} \Big( n^{-1+p(\alpha + 1/\beta)}V(p;k)_n - m_p 