Higher Order Fractional Stable Motion: Hyperdiffusion with Heavy Tails

Abstract

We introduce the class of higher order fractional stable motions that can exhibit hyperdiffusive spreading with heavy tails. We define the class as a generalization of higher order fractional Brownian motion as well as a generalization of linear fractional stable motions. Higher order fractional stable motions are self-similar with Hurst index larger than one and non-Gaussian stable marginals with infinite variance and have stationary higher order increments. We investigate their sample path properties and asymptotic dependence structure on the basis of codifference. In particular, by incrementing or decrementing sample paths once under suitable conditions, the diffusion rate can be accelerated or decelerated by one order. With a view towards simulation study, we provide a ready-for-use sample path simulation recipe at discrete times along with error analysis. The proposed simulation scheme requires only elementary numerical operations and is robust to high frequency sampling, irregular spacing and super-sampling.

Appendix: Proofs

The derivation of results entails rather lengthy proofs of somewhat routine nature. To avoid overloading the paper, we omit nonessential details in some instances. We start with two general notation. As usual, for $f\in L^{\alpha }(\mathbb {R})$, we write $\Vert f(\cdot )\Vert _{\alpha }:=(\int _{\mathbb {R}}|f(s)|^{\alpha }ds)^{1/\alpha }$. As discussed in brief prior to Proposition 6.3, the kernel $f_n(t,s;H,\alpha )$ is continuously differentiable in t for $t\in (s,+\infty )$. Recall that $H-1/\alpha$ is not an integer. Hence, the mean value theorem can be applied as follows:

$$\begin{aligned} f_n\left( t_2,s;H,\alpha \right) -f_n\left( t_1,s;H,\alpha \right) = \left( t_2-t_1\right) f_{n-1}\left( \Theta _s(t_1,t_2),s;H-1,\alpha \right) ,\quad s< t_1\le t_2,\nonumber \\ \end{aligned}$$

(8.1)

with the aid of the recurrence formula (3.5) and where $\Theta _s(t_1,t_2)$ is a real number in the interval $(t_1,t_2)$, depending on s. Some results are standard, for which we give proof for the sake of completeness.

Proof of Theorem 4.1

The case $n=1$ yields a linear fractional stable motion, which is known to satisfy all the results. Hence, we assume $n\ge 2$.

• (i) Since $\{L_{\alpha }(t):\,t\in \mathbb {R}\}$ is symmetric, even when $\alpha =1$, it suffices [19, Property 3.2.2] to show that for each $t>0$, $|(t-s)_+^{H-1/\alpha }|^{\alpha }$ and $|f_n(t,\cdot ;H,\alpha )|^{\alpha }$ are integrable respectively on (0, t) and $(-\infty ,0)$, with $H\in (n-1,n)\setminus \{1/\alpha \}$. The first component is straightforward; $\int _0^t (t-s)_+^{\alpha H-1}ds=t^{\alpha H}/(\alpha H),$ due to $\alpha H>0$. For the second component, it suffices to show that $F_{H,\alpha ,n}$ is well defined since then $\int _{-\infty }^0|f_n(t,s;H,\alpha )|^{\alpha }ds=t^{\alpha H}F_{H,\alpha ,n}$. As $s\downarrow -\infty$, $|f_n(1,s;H,\alpha )|^{\alpha }=O((-s)^{\alpha H-1-\alpha n})$, which is integrable at infinity due to $\alpha H-1-\alpha n\in (-1-\alpha ,-1).$ As $s\uparrow 0$, $|f_n(1,s;H,\alpha )|^{\alpha }=O((-s)^{\alpha H-1-\alpha (n-1)}),$ if $H-1/\alpha <n-1$, or O(1) if $H-1/\alpha \ge n-1$, both of which are integrable near the origin due to $\alpha H-1-\alpha (n-1)\in (-1,-1+\alpha )$.

• (ii) For any positive constant c, positive integer d and real numbers $\{\theta _k\}_{k=1,\ldots ,d}$ and $\{t_k\}_{k=1,\ldots ,d}$, it holds by elementary change of variables that $c^{-H}\Vert \sum _{k=1}^d\theta _k f_n(ct_k,\cdot ;H,\alpha )\Vert _{\alpha }=\Vert \sum _{k=1}^d\theta _k f_n(t_k,\cdot ;H,\alpha )\Vert _{\alpha }$, which is independent of $c>0$. This indicates self-similarity.

• (iii) Let $T^*$ be an arbitrary countable dense subset of $[t_1,t_2]$ with $0\le t_1<t_2<+\infty$. By [19, Corollary 10.2.4], the process is sample unbounded if $\int _{t_1}^{t_2}\sup _{t\in T^*}|f_n(t,s;H,\alpha )|^{\alpha }ds=+\infty$. This is indeed the case when $H\in (n-1,\min (n,1/\alpha ))$ since for every $s\in [t_1,t_2]$, $\sup _{t\in T^*}|f_n(t,s;H,\alpha )|=+\infty$. If $\alpha >1/(n-1)$, then the interval $(n-1,\min (n,1/\alpha ))$ of H is empty. Hence, we can restrict $\alpha <1/(n-1)$.

• (iv) Again, let $T^*$ be an arbitrary countable dense subset of $[t_1,t_2]$ with $0\le t_1<t_2<+\infty$. Notice that for each $s\le t_2$, $\sup _{t\in T^*}|f_n(t,s;H,\alpha )|=|f_n(t_2,s;H,\alpha )|\in L^{\alpha }(\mathbb {R})$, whenever $H>1/\alpha$, due to (3.7) and (3.8). If $\alpha <1/n$, then the interval $(\max (n-1,1/\alpha ),n)$ of H is empty. Hence, we can restrict $\alpha >1/n$.

• (v)–(vi) First, $\int _0^T (\int _{\mathbb {R}}|f_n(t,s;H,\alpha )|^{\alpha }ds)^{1/\alpha }dt=T^{H+1} C_{H,\alpha ,n}/(H+1)<+\infty ,$ which is equivalent to $\int _0^T|L^n_{H,\alpha }(t)|dt<+\infty$, a.s. by [19, Theorem 11.3.2]. If $\alpha \in (1,2)$, then this justifies, by Theorem 11.4.1 and Theorem 11.7.4 [19], the interchange of derivative and integral in $(d/dt)\int _{\mathbb {R}}f_{n+1}(t,s;H+1,\alpha )dL_{\alpha }(s)= \int _{\mathbb {R}}(d/dt)f_{n+1}(t,s;H+1,\alpha )dL_{\alpha }(s)$, a.s. The recurrence formula (3.5) yields the result. It remains to show the case of $\alpha =1$. Observe that for each $t>0$,

  $$\begin{aligned}&A_n(t,s;H):=\frac{|f_n(t,s;H,1)|\int _0^t\int _{\mathbb {R}}|f_n(v,u;H,1)|du\,dv}{\int _{\mathbb {R}}|f_n(t,u;H,1)|du\int _0^t|f_n(v,s;H,1)|dv}\\&\quad =\frac{t}{H+1}\frac{|f_n(t,s;H,1)|}{|f_{n+1}(t,s;H+1,1)|}\sim {\left\{ \begin{array}{ll} \frac{H}{H+1}\frac{t}{t-s},&{}\text {if }s\uparrow t,\\ \frac{H}{H+1}(<1),&{}\text {if }s\downarrow 0,\\ \frac{n}{H+1}(<1),&{}\text {if }s\uparrow 0,\\ \frac{n+1}{H+1}(\in (1,1+1/n)),&{}\text {if }s\downarrow -\infty , \end{array}\right. } \end{aligned}$$

  with the aid of the recurrence formula (3.4) and the asymptotic behaviors (3.6). Clearly, $|f_n(t,\cdot ;H,1)|\in L^1$. Moreover, it holds that $|f_n(t,\cdot ;H,1)|\ln _+A_n(t,s;H)\sim (t-s)^{H-1}|\ln (t-s)|$ as $s\uparrow t$, and $\int _0^t (t-s)^{H-1}|\ln (t-s)|ds=O(t^H|\ln (t)|)$ near the origin. Hence, we get $\int _0^T\int _{\mathbb {R}}|f_n(t,s;H,1)|\ln _+A_n(t,s;H) ds\,dt<+\infty$. This proves the case $\alpha =1$.

Proof of Theorem 5.1

It is sufficient to show that the nth order increments are stationary and that the $(n-1)$st increments are not stationary. Let k be a non-negative integer. The difference operator changes the first argument of the kernel (3.2), in the sense that

$$\begin{aligned} D_{\Delta }^kf_n\left( t,s;H,\alpha \right) =\sum _{j=0}^k(-1)^{k-j} \left( {\begin{array}{c}k\\ j\end{array}}\right) f_n\left( t+j\Delta ,s;H,\alpha \right) . \end{aligned}$$

The kernel has mainly two components $(t-s)_+^{H-1/\alpha }$ and $t^k$’s in the summation. Since the $t^k$-components are simply polynomial in t, we have

$$\begin{aligned} D^n_{\Delta }t^k=\sum _{j=0}^n (-1)^{n-j}\left( {\begin{array}{c}n\\ j\end{array}}\right) (t+j\Delta +h)^k= 0. \end{aligned}$$

(8.2)

Since the power k is at most $n-1$ in the summation term of the kernel (3.2), it holds that for $\Delta >0$ and $n\in \mathbb {N}$, the differenced kernel reduces to

$$\begin{aligned} D_{\Delta }^n f_n(t,s;H,\alpha )=\frac{D_{\Delta }^n(t-s)_+^{H-1/\alpha }}{\Gamma (H-1+1/\alpha )}. \end{aligned}$$

It then holds by a straightforward algebraic work and the change of variables that for any positive integer d and real $\{\theta _k\}_{k=1,\ldots ,d}$ and positive $\{t_k\}_{k=1,\ldots ,d}$,

$$\begin{aligned} \left\| \sum _{k=1}^d \theta _k c^{-H}D_{\Delta }^n L_{H,\alpha }^n(ct_k+h)\right\| _{\alpha }^{\alpha }&= \int _{\mathbb {R}}\left| \sum _{k=1}^d\theta _k\frac{c^{-H}D_{\Delta }^n(ct_k+h-s) _+^{H-1/\alpha }}{\Gamma (H+1-1/\alpha )}\right| ^{\alpha }ds\nonumber \\&=\int _{\mathbb {R}}\left| \sum _{k=1}^d\theta _k\frac{D_{\Delta }^n(t_k-s)_+^{H-1/\alpha }}{\Gamma (H+1-1/\alpha )}\right| ^{\alpha }ds\nonumber \\&=\left\| \sum _{k=1}^d \theta _k D_{\Delta }^n L_{H,\alpha }^n(t_k)\right\| _{\alpha }^{\alpha }, \end{aligned}$$

(8.3)

which is independent of c and h. This gives the first claim. By setting $d=1$, $\theta _1=1$ and $t_1=t$ in (8.3), we get

$$\begin{aligned} \left\| D_{\Delta }^n L_{H,\alpha }^n(t)\right\| _{\alpha }^{\alpha }= & {} \int _{\mathbb {R}}\left| \frac{D_{\Delta }^n(t-s)_+^{H-1/\alpha }}{\Gamma (H+1-1/\alpha )}\right| ^{\alpha }ds=\int _{\mathbb {R}}\left| \frac{D_{\Delta }^n(-s)_+^{H-1/\alpha }}{\Gamma (H+1-1/\alpha )}\right| ^{\alpha }ds\\= & {} \Delta ^{\alpha H}\int _{\mathbb {R}}\left| \frac{D_1^n(-s)_+^{H-1/\alpha }}{\Gamma (H+1-1/\alpha )}\right| ^{\alpha }ds, \end{aligned}$$

by changes of variables. $\square$

Proof of Theorem 5.2

It suffices to focus on the case $c>n$ since our interest is in the asymptotics as $c\uparrow +\infty$. Throughout the proof, the capital K’s indicate suitable positive constants which changes their values line to line. As previously, we use $D_{\Delta }^nf(c):=(D_{\Delta }^nf(t+c))|_{t=0}$, with a slight abuse of notation. Observe that

$$\begin{aligned}&\left\| D_{\Delta }^n L_{H,\alpha }^n\left( c\Delta \right) -D_{\Delta }^n L_{H,\alpha }^n\left( 0\right) \right\| _{\alpha }^{\alpha }-\left\| D_{\Delta } ^nL_{H,\alpha }^n\left( c\Delta \right) \right\| _{\alpha }^{\alpha }- \left\| D_{\Delta }^nL_{H,\alpha }^n\left( 0\right) \right\| _{\alpha }^{\alpha }\\&\quad =\frac{\Delta ^{\alpha H}}{\Gamma (H+1-1/\alpha )^{\alpha }}\int _{-\infty }^{c+n} \left( \left| D_1^n(c-s)_+^{H-1/\alpha }-D_1^n(-s)_+^{H-1/\alpha }\right| ^{\alpha }\right. \\&\qquad \left. -\left| D_1^n(c-s)_+^{H-1/\alpha }\right| ^{\alpha }-\left| D_1^n(-s)_+^{H-1/\alpha }\right| ^{\alpha }\right) ds\\&\quad =:\frac{\Delta ^{\alpha H}}{\Gamma (H+1-1/\alpha )^{\alpha }}\left( I_0(c)+I_1(c) +\cdots +I_n(c)+I_{n+1}(c)\right) , \end{aligned}$$

where the domain of integration is decomposed into $(n+2)$ disjoint intervals $(-\infty ,0)$, (0, 1), (1, 2), ..., $(n-1,n)$ and $(n,c+n)$;

$$\begin{aligned} I_0(c)&:=\int _{-\infty }^0\left( \left| D_1^n(c-s)^{H-1/\alpha }-D_1^n(-s)^{H-1/\alpha }\right| ^{\alpha }-\left| D_1^n(c-s)^{H-1/\alpha }\right| ^{\alpha }\right. \\&\quad \left. -\left| D_1^n(-s)^{H-1/\alpha }\right| ^{\alpha }\right) ds,\\ I_m(c)&:=\int _{m-1}^m\left( \left| D_1^n(c-s)^{H-1/\alpha }-v_m(s)\right| ^{\alpha }-\left| D_1^n(c-s)^{H-1/\alpha }\right| ^{\alpha }-\left| v_m(s)\right| ^{\alpha }\right) ds,\\&\qquad m=1,\ldots ,n,\\ I_{n+1}(c)&:=\int _n^c\left( \left| D_1^n(c-s)_+^{H-1/\alpha }-D_1^n(-s)_+^{H-1/\alpha }\right| ^{\alpha }-\left| D_1^n(c-s)_+^{H-1/\alpha }\right| ^{\alpha }\right. \\&\qquad \left. -\,\left| D_1^n(-s)_+^{H-1/\alpha }\right| ^{\alpha }\right) ds,\\ v_m(s)&:=\sum _{k=m}^n(-1)^{n-k}\left( {\begin{array}{c}n\\ k\end{array}}\right) (k-s)^{H-1/\alpha },\quad s\in (m-1,m). \end{aligned}$$

Since $I_{n+1}(c)= 0$, it suffices to investigate $I_0(c)$ and $I_m(c)$ for $m=1,\ldots ,n$. Hereafter, we will use the following results [2]

$$\begin{aligned}&\displaystyle \left| \left| \theta _1+\theta _2\right| ^{\alpha }-\left| \theta _1\right| ^{\alpha }- \left| \theta _2\right| ^{\alpha }\right| \le {\left\{ \begin{array}{ll} 2|\theta _1|^{\alpha },&{}\text {if }\alpha \in (0,1],\\ \alpha |\theta _1||\theta _2|^{\alpha -1}+(\alpha +1)|\theta _1|^{\alpha },&{}\text {if }\alpha \in (1,2), \end{array}\right. }\end{aligned}$$

(8.4)

$$\begin{aligned}&\displaystyle (1+\theta )^{\alpha }-1-\alpha \theta \le \alpha (\alpha -1)(1-\theta )^{\alpha -2}\theta ^2,\quad |\theta |<1,\quad \alpha \in (0,2),\end{aligned}$$

(8.5)

$$\begin{aligned}&\displaystyle \left| \theta _1+\theta _2\right| ^{\alpha }-\left| \theta _1\right| ^{\alpha }=\left| \theta _1\right| ^{\alpha }\left( \left| 1+\frac{\theta _2}{\theta _1}\right| ^{\alpha }-1-\alpha \frac{\theta _2}{\theta _1}\right) +\alpha \theta _2 \left| \theta _1\right| ^{\alpha -1}\mathrm{sgn}(\theta _1),\quad \alpha \in (1,2),\end{aligned}$$

(8.6)

$$\begin{aligned}&\displaystyle \left| \left| \theta _1+\theta _2\right| ^{\alpha }-\left| \theta _1\right| ^{\alpha }\right| \le K_1\left| \theta _1\right| \left| \theta _2\right| ^{\alpha -1}+K_2\left| \theta _1\right| ^{\alpha }+K_3\left| \theta _2\right| ^{\alpha },\quad \alpha \in (1,2), \end{aligned}$$

(8.7)

as well as, for $c>n$ and $m=1,\ldots ,n$,

$$\begin{aligned} D_1^n(\theta -s)^{H-1/\alpha }=n!\left( {\begin{array}{c}H-1/\alpha \\ n\end{array}}\right) \int _{(0,1)^n}\left( \theta +\left( v_1+\cdots +v_n\right) -s\right) ^{H-1/\alpha -n}dv_1\cdots dv_n,\quad s< \theta ,\nonumber \\ \end{aligned}$$

(8.8)

$$\begin{aligned}&\displaystyle \left| D_1^n(c-s)^{H-1/\alpha }\right| \le \left| n!\left( {\begin{array}{c}H-1/\alpha \\ n\end{array}}\right) \right| \left( c-\theta _2\right) ^{H-1/\alpha -n},\quad 0\le \theta _1\le s\le \theta _2,\end{aligned}$$

(8.9)

$$\begin{aligned}&\displaystyle \left| D_1^n(c-s)^{H-1/\alpha }\right| \le \left| n!\left( {\begin{array}{c}H-1/\alpha \\ n\end{array}}\right) \right| \left( c-\theta _1\right) ^{H-1/\alpha -n},\quad \theta _1\le s\le \theta _2\le 0. \end{aligned}$$

(8.10)

We investigate $I_0(c)$ and $I_m(c)$ as follows;

• (A1) The term $I_0(c)$ is of order $c^{\alpha (H-n)}$ when “$\alpha \in (0,1]$ and $H\in (n-1,n)\setminus \{1/\alpha \}$”, or “$\alpha \in (1,2)$ and $H\in (\max (n-1,n-\frac{1}{\alpha (\alpha -1)}),n)\setminus \{1/\alpha \}$”.

• (A2) The term $I_0(c)$ is of order $c^{H-1/\alpha -n}$ when $\alpha \in (1,2)$ and $H\in (n-1,n-\frac{1}{\alpha (\alpha -1)})\setminus \{1/\alpha \}$.

• (B1) The terms $I_m(c)$ is at most of order $c^{\alpha (H-n)-1}$ when $\alpha \in (0,1]$ and $H\in (n-1,n)\setminus \{1/\alpha \}.$

• (B2) The terms $I_m(c)$ is of order $c^{H-1/\alpha -n}$ when $\alpha \in (1,2)$ and $H\in (n-1,n)\setminus \{1/\alpha \}.$

As described earlier, the case (A2) is irrelevant if $\alpha \in (1,(1+\sqrt{5})/2)$.

• (A1) We rewrite $I_0(c)$ as

  $$\begin{aligned} I_0(c)&=c^{\alpha H}\int _{-\infty }^0\left( \left| D_{1/c}^n(1-s)^{H-1/\alpha } -D_{1/c}^n(-s)^{H-1/\alpha }\right| ^{\alpha }\right. \\&\quad \left. -\,\left| D_{1/c}^n(1-s) ^{H-1/\alpha }\right| ^{\alpha }-\left| D_{1/c}^n(-s)^{H-1/\alpha }\right| ^{\alpha }\right) ds, \end{aligned}$$

  By applying the bounds (8.4), we get

  $$\begin{aligned}&c^{-\alpha (H-n)}\left| I_0(c)\right| \nonumber \\&\quad \le {\left\{ \begin{array}{ll} K_1\int _{-\infty }^0(1-s)^{\alpha (H-1/\alpha -n)}ds,&{}\text {if }\alpha \in (0,1],\\ K_2\int _{-\infty }^0(1-s)^{H-1/\alpha -n}(-s)^{(\alpha -1)(H-1/\alpha -n)}ds+K_3\int _{-\infty }^0(1-s)^{\alpha (H-1/\alpha -n)}ds,&{}\text {if }\alpha \in (1,2), \end{array}\right. } \end{aligned}$$

  where all the integrals are well defined for $(\alpha ,H)$ under consideration. This ensures passage to the limit by the dominated convergence theorem, and thus yields $\lim _{c\uparrow +\infty }c^{-\alpha (H-n)}I_0(c)=B_n.$

• (A2) Next, we investigate $I_0(c)$ when $\alpha \in (1,2)$ and $H\in (n-1,n-\frac{1}{\alpha (\alpha -1)})$. We can show [2, Lemma 5.3] that for small positive $\epsilon (\ll 1)$, there exists $\gamma _{\epsilon }>0$ and $c_{\epsilon }>n$ such that for $c>c_{\epsilon }$ $\sup _{s\in [-\gamma _{\epsilon } c,0)}|D_1(c-s)^{H-1/\alpha }/D_1^n(-s)^{H-1/\alpha }|\le \epsilon .$ By applying the identity (8.6) with $\theta _1\leftarrow -D_1^n(-s)^{H-1/\alpha }$ and $\theta _2\leftarrow D_1^n(c-s)^{H-1/\alpha }$, the integral $I_0(c)$ can be rewritten as

  $$\begin{aligned}&I_0(c)=-\underbrace{\int _{-\infty }^0 \left| D_1^n(c-s)^{H-1/\alpha }\right| ^{\alpha }ds}_{I_{01}(c)(\sim K_1 c^{\alpha (H-n)})}\\&+\underbrace{\int _{-\infty }^{-\gamma _{\epsilon }c} \left( \left| D_1^n(c-s)^{H-1/\alpha }-D_1^n(-s)^{H-1/\alpha }\right| ^{\alpha } -\left| D_1^n(-s)^{H-1/\alpha }\right| ^{\alpha }\right) ds}_{I_{02}(c)(= O(c^{\alpha (H-n)}))}\\&+\underbrace{\int _{-\gamma _{\epsilon }c}^0\left| D_1^n(-s)^{H-1/\alpha }\right| ^{\alpha } \left( \left| 1-\frac{D_1^n(c-s)^{H-1/\alpha }}{D_1^n(-s)^{H-1/\alpha }}\right| ^{\alpha } -1+\frac{D_1^n(c-s)^{H-1/\alpha }}{D_1^n(-s)^{H-1/\alpha }}\right) ds}_{I_{03}(c)(= O(c^{\alpha (H-n)}))}\\&-\alpha \underbrace{\int _{-\gamma _{\epsilon }c}^0 D_1^n(c-s)^{H-1/\alpha }\left| D_1^n(-s)^{H-1/\alpha }\right| ^{\alpha -1}\mathrm{sgn}\left( D_1^n(-s)^{H-1/\alpha }\right) ds}_{I_{04}(c)(\sim K_4 c^{H-1/\alpha -n})}. \end{aligned}$$

  Note that $\alpha (H-n)<H-1/\alpha -n(<0)$ if and only if $H<n-\frac{1}{\alpha (\alpha -1)}$. First, with the aid of the mean value representation (8.8), we get $c^{-\alpha (H-n)}I_{01}(c)\rightarrow |\frac{n!}{\alpha (H-n)}\left( {\begin{array}{c}H-1/\alpha \\ n\end{array}}\right) |^{\alpha }$. For the second term $I_{02}(c)$, by setting $\theta _1\leftarrow D_1^n(-s)^{H-1/\alpha }$ and $\theta _2\leftarrow D_1^n(c-s)^{H-1/\alpha }$ in the inequality (8.7), we get

  $$\begin{aligned} \left| I_{02}(c)\right|&\le K_1\int _{-\infty }^{\gamma _{\epsilon }c}\left| D_1^n(-s)^{H-1/\alpha }\right| \left| D_1^n(c-s)^{H-1/\alpha }\right| ^{\alpha -1}ds\\&+K_2\int _{-\infty }^{\gamma _{\epsilon }c} \left| D_1^n(-s)^{H-1/\alpha }\right| ^{\alpha }ds+K_3\int _{-\infty }^{\gamma _{\epsilon }c} \left| D_1^n(c-s)^{H-1/\alpha }\right| ^{\alpha }ds\\&\sim c^{\alpha (H-n)}\left( K_1 \int _{-\infty }^{\gamma _{\epsilon }}(-s)^{H-1/\alpha -n} (1-s)^{(H-1/\alpha -n)(\alpha -1)}ds\right. \\&\left. +K_2\int _{-\infty }^{\gamma _{\epsilon }} (-s)^{\alpha (H-n)-1}ds+K_3\int _{-\infty }^{\gamma _{\epsilon }}(1-s)^{\alpha (H-n)-1}ds\right) , \end{aligned}$$

  where the three integrals are clearly well defined. For the third term $I_{03}(c)$, the bound (8.5) yields

  $$\begin{aligned}&\left| I_{03}(c)\right| \le C\int _{-\gamma _{\epsilon }c}^0\left| D_1^n(-s)^{H-1/\alpha }\right| ^{\alpha -2} \left| D_1^n(c-s)^{H-1/\alpha }\right| ^2ds\le O(c^{2(H-1/\alpha -n)})\\&\int _{-\gamma _{\epsilon }c}^0 \left| D_1^n(-s)^{H-1/\alpha }\right| ^{\alpha -2}ds, \end{aligned}$$

  for sufficiently large c, with the aid of the bound (8.10). The last integral is well defined because the summands with indexes $k\ge 1$ in the difference $D_1^n(-s)(=\sum _{k=0}^n(-1)^{n-k}\left( {\begin{array}{c}n\\ k\end{array}}\right) (k-s)^{H-1/\alpha })$ are bounded above and below away from zero, so the integrability depends only on the summand with index $k=0$. Clearly, this leading term $(-s)^{H-1/\alpha }$ is in $L^{\alpha -1}(-\gamma _{\epsilon }c,0)$ whenever $H>0$. For the fourth term $I_{04}(c)$, $\limsup _c c^{-(H-1/\alpha -n)}|D_1^n(c-s)^{H-1/\alpha }||D_1^n(-s)^{H-1/\alpha }|^{\alpha -1}\le K|D_1^n(-s)^{H-1/\alpha }|^{\alpha -1}$, where is integrable on $(-\gamma _{\epsilon },0)$, for the same reason as for the second term. This justifies passage to the limit by the dominated convergence theorem, which yields $\lim _c c^{-(H-1/\alpha -n)}I_{04}(c)=n!\left( {\begin{array}{c}H-1/\alpha \\ n\end{array}}\right) \int _{-\infty }^0|D_1^n(-s)^{H-1/\alpha }|^{\alpha -1}\mathrm{sgn}(D_1^n(-s)^{H-1/\alpha })ds$. Hence, the leading term in $I_0(c)$ is $I_{04}(c)$ alone, of order $c^{H-1/\alpha -n}$.

• (B1) We turn to the terms $I_m(c)$ when $\alpha \in (0,1]$ and $H\in (n-1,n)$. Using the bound (8.9) with $\theta _1\leftarrow D_1^n(c-s)^{H-1/\alpha }$ and $\theta _2\leftarrow v_m(s)$ in the inequality (8.4) for $\alpha \in (0,1]$ yields $|I_m(c)|\le \int _{m-1}^m |n!\left( {\begin{array}{c}H-1/\alpha \\ n\end{array}}\right) (c-m)^{H-1/\alpha -n}|^{\alpha }ds=|n!\left( {\begin{array}{c}H-1/\alpha \\ n\end{array}}\right) |^{\alpha }(c-m)^{\alpha (H-n)-1}$. This, along with (A1), ensures the result for $\alpha \in (0,1]$.

• (B2) Finally, the terms $I_m(c)$ when $\alpha \in (1,2)$ and $H\in (n-1,n)$ can be dealt with in a similar manner to (A2). By applying the identity (8.6) with $\theta _1\leftarrow -v_m(s)$ and $\theta _2\leftarrow D_1^n(c-s)^{H-1/\alpha }$, the integral $I_m(c)$ can rewritten as

  $$\begin{aligned}&I_m(c)=-\underbrace{\int _{m-1}^m \left| D_1^n(c-s)^{H-1/\alpha }\right| ^{\alpha }ds}_{I_{m1}(c)(=O(c^{\alpha (H-n)-1}))}\\&+\underbrace{\int _{m-1}^m\left| v_m(s)\right| ^{\alpha } \left( \left| 1-\frac{D_1^n(c-s)^{H-1/\alpha }}{v_m(s)}\right| ^{\alpha }-1 +\frac{D_1^n(c-s)^{H-1/\alpha }}{v_m(s)}\right) ds}_{I_{m2}(c)(=O(c^{2(H-1/\alpha -n}))}\\&-\,\alpha \underbrace{\int _{m-1}^m D_1^n(c-s)^{H-1/\alpha }\left| v_m(s)\right| ^{\alpha -1}\mathrm{sgn}\left( v_m(s)\right) ds}_{I_{m3}(c)(\sim Kc^{H-1/\alpha -n})}. \end{aligned}$$

  Note that $\alpha (H-n)-1<H-1/\alpha -n(<0)$ if and only if $H<n+1/\alpha$, which thus holds true always with $H\in (n-1,n)$. The first term $I_{m1}(c)$ is at most $O(c^{\alpha (H-n)-1})$ due to the bound (8.9). For the second term $I_{m2}(c)$, we can show [2, Lemma 5.3] that for small positive $\epsilon (\ll 1)$, there exists $c_{\epsilon }$ such that for $c>c_{\epsilon }$ $\sup _{s\in [m-1,m]}|D_1(c-s)^{H-1/\alpha }/v_m(s)|\le \epsilon .$ Hence, the bound (8.5) yields $|I_{m2}(c)|\lesssim K\int _{m-1}^m|v_m(s)|^{\alpha -2}|D_1^n(c-s)^{H-1/\alpha }|^2ds\le O(c^{2(H-1/\alpha -n)})\int _{m-1}^m|v_m(s)|^{\alpha -2}ds$, with the aid of the bound (8.9). The last integral is well defined as in (A2). For the third term $I_{m3}(c)$, $\limsup _c c^{-(H-1/\alpha -n)}|D_1^n(c-s)^{H-1/\alpha }||v_m(s)|^{\alpha -1}\le K\left| v_m(s)\right| ^{\alpha -1}$, for some positive constant C, where is integrable on $(m-1,m)$, for the same reason as for the second term. This justifies passage to the limit by the dominated convergence theorem, which yields $c^{-(H-1/\alpha -n)}I_{m3}(c)\rightarrow n!\left( {\begin{array}{c}H-1/\alpha \\ n\end{array}}\right) \int _{m-1}^m|v_m(s)|^{\alpha -1}\mathrm{sgn}\left( v_m(s)\right) ds$, as $c\uparrow +\infty$. Hence, the leading term in $I_m(c)$ is $I_{m3}(c)$ alone, of order $c^{H-1/\alpha -n}$. $\square$

Proof of Lemma 6.1

It is well known [19] that the stable Lévy process (2.2) on the compact time interval $[-\kappa ,T]$ can be written in the form of infinite sum;

$$\begin{aligned}&\left\{ L_{\alpha }(t):\,t\in [-\kappa ,T]\right\} \\&\quad \mathop {=}\limits ^{\mathcal {D}} \left\{ \sum _{k=1}^{+\infty }r_k\left( \frac{\Gamma _k}{(T+\kappa )c_{\alpha }}\right) ^{-1/\alpha } \left( \mathbbm {1}\left( U_k \in (-\kappa ,t]\right) -\mathbbm {1}\left( U_k\in (-\kappa ,0]\right) \right) :\,t\in [-\kappa ,T]\right\} , \end{aligned}$$

where the infinite sum converges almost surely uniformly on the compact interval $[-\kappa ,T]$. Here, $\{\Gamma _k\}_{k\in \mathbb {N}}$ is a sequence of the standard Poisson arrival times. By the inverse Lévy measure series representation [9], the summands corresponding to the outer part of the Lévy measure $\nu (dz)\mathbbm {1}(|z|>\eta (m))$ are the ones indexed with $\{k\in \mathbb {N}:\Gamma _k\le (T+\kappa )m\}$. Since $\{\Gamma _k\}_{k\in \mathbb {N}}$ is a (increasing) sequence of the standard Poisson process, it is clear that $\{k\in \mathbb {N}:\Gamma _k\le (T+\kappa )m\}\mathop {=}\limits ^{\mathcal {L}}\{1,\ldots ,Z_{\kappa ,m}\}$, where $Z_{\kappa ,m}$ is a Poisson random variable with mean $(T+\kappa )m$, and the Poisson arrival times $\{\Gamma _k\}_{k\in \mathbb {N}}$ on $[0,(T+\kappa )m]$ are identical in law to the ascending order statistic of $Z_{\kappa ,m}$ iid uniform random variables on $(0,(T+\kappa )m)$. The representation (6.5) now follows from the definition (6.3). $\square$

Proof of Proposition 6.3

(i) Fix $d\in \mathbb {N}$, a sequence $\{\theta _k\}_{k=1,\ldots ,d}$ of real numbers and $0\le t_1<t_2<\ldots <t_d\le T$, and set the notation $\psi (s):=\sum _{k=1}^d \theta _k (t_k-s)_+^{H-1/\alpha }/\Gamma (H+1-1/\alpha )$. In light of (2.2), the linear combination can be written in the Lévy-Ito representation

$$\begin{aligned} \sum _{k=1}^d \theta _k \frac{X(t_k;m)}{\sigma (m)}=\int _0^T \int _{|z|\le \eta (m)}\frac{\psi (s)}{\sigma (m)}z \left( \mu -\nu \right) (dz,ds), \end{aligned}$$

where $\mu$ is the Poisson random measure and its compensator $\nu (dz)=\alpha c_{\alpha }|z|^{-\alpha -1}dz$, satisfying $\int _0^t\int _{|z|>\eta (m)}z(\mu -\nu )(dz,ds)=L_{\alpha }(t;m)$. Hence, with the aid of (6.1), the convergence of the finite dimensional distributions holds by

$$\begin{aligned}&\mathbb {E}\left[ \exp \left[ iy\sum _{k=1}^d \theta _k \frac{X(t_k;m)}{\sigma (m)}\right] \right] \nonumber \\&\quad =\exp \left[ \alpha c_{\alpha }\int _0^T \int _{|z|\le \eta (m)} \left( \cos \left( y\frac{\psi (s)}{\sigma (m)}z\right) -1\right) \frac{dz}{|z|^{\alpha +1}}ds\right] \nonumber \\&\quad =\exp \left[ \int _0^T \int _{|z|\le 1}\frac{\alpha c_{\alpha }}{\eta (m)^{\alpha }} \left( \cos \left( y\frac{\psi (s)\eta (m)}{\sigma (m)}z\right) -1\right) \frac{dz}{|z|^{\alpha +1}}ds\right] \nonumber \\&\quad \rightarrow \exp \left[ -\frac{y^2}{2}\int _0^T\psi (s)^2ds\right] ,\quad m\uparrow +\infty , \end{aligned}$$

(8.11)

where the passage to the limit is justified by the dominated convergence theorem with

$$\begin{aligned} \frac{\alpha c_{\alpha }}{\eta (m)^{\alpha }}\left| \cos \left( y\frac{\psi (s)\eta (m)}{\sigma (m)}z\right) -1 \right| \le \frac{1}{2}\frac{\alpha c_{\alpha }}{\eta (m)^{\alpha }}\left| y\frac{\psi (s)\eta (m)}{\sigma (m)}z\right| ^2=\frac{y^2}{2}\psi (s)^2\frac{2-\alpha }{2}z^2,\nonumber \\ \end{aligned}$$

(8.12)

which is integrable with respect to the measure $|z|^{-\alpha -1}dz\,ds$ on $(-1,+1)\times (0,T)$ iff $H>1/\alpha -1/2$. (This condition is required for $\int _0^T \psi (s)^2ds<+\infty$.)

Next, recall that the integer and fractional parts of $H-1/\alpha$ are denoted, respectively, by $\zeta :=\lfloor H-1/\alpha \rfloor$ and $\theta :=H-1/\alpha -\zeta$. Note that $\theta \in (0,1)$, since $H-1/\alpha$ is assumed to be non-integer. Also, noting that $\eta (m)\le z_0$, the stochastic process $\{\int _0^t\int _{|z|\le \eta (m)}z(\mu -\nu )(dz,ds):\,t\in [0,T]\}$ is a locally square-integrable martingale. Observe that for $0\le t_1\le t_2\le T$ and $k=0,1,\ldots ,\zeta$,

$$\begin{aligned}&\int _0^T\left( \frac{(t_2-s)_+^{H-1/\alpha -k}-(t_1-s)_+^{H-1/\alpha -k}}{\Gamma (H-1/\alpha -k+1)}\right) ^2ds \nonumber \\&\quad \le \int _{\mathbb {R}} \left( \frac{(t_2-s)_+^{H-1/\alpha -k}-(t_1-s)_+^{H-1/\alpha -k}}{\Gamma (H-1/\alpha -k+1)} \right) ^2ds\nonumber \\&\quad =\left( t_2-t_1\right) ^{2(H-1/\alpha -k+1/2)}\int _{\mathbb {R}}\left( \frac{(1-s)_+^{H-1/\alpha -k} -(-s)_+^{H-1/\alpha -k}}{\Gamma (H-1/\alpha -k+1)}\right) ^2ds, \end{aligned}$$

(8.13)

uniformly in m, where the first equality holds by the Wiener-Ito isometry. Hence, with $k=0$ in (8.13), it holds by Lemma 16.2 and Theorem 16.3 of [11] that the weak convergence holds in the space $\mathcal {C}([0,T];\mathbb {R})$. Noting that $H-1/\alpha -\zeta >0$, the Kolmogorov- C̆ hentsov theorem ensures that for each $k=0,1,\ldots ,\zeta$, the stochastic process

$$\begin{aligned} \left\{ \frac{1}{\sigma (m)}\int _0^t\int _{|z|\le \eta (m)}\frac{\partial ^k}{\partial t^k}\frac{(t-s)_+^{H-1/\alpha }}{\Gamma (H-1/\alpha +1)}z \left( \mu -\nu \right) (dz,ds):\,t\in [0,T]\right\} \end{aligned}$$

is square-integrable and continuous as well as is tight in m (as $m\uparrow +\infty$). Moreover, it holds by [7, Theorem 2.2] that for each $k=0,1,\ldots ,\zeta$,

$$\begin{aligned}&\frac{\partial }{\partial t}\int _0^t\int _{|z|\le \eta (m)}\frac{\partial ^{k-1}}{\partial t^{k-1}}\frac{(t-s)_+^{H-1/\alpha }}{\Gamma (H-1/\alpha +1)} z\left( \mu -\nu \right) (dz,ds)\\&\quad =\int _0^t\int _{|z|\le \eta (m)}\frac{\partial ^k}{\partial t^k}\frac{(t-s)_+^{H-1/\alpha }}{\Gamma (H-1/\alpha +1)} z\left( \mu -\nu \right) (dz,ds), \end{aligned}$$

almost surely. This justifies the $\zeta$-times differentiability of sample paths of $\{X(t;m)/\sigma (m):\,t\in [0,T]\}$. With $k=\zeta$ in (8.13), the local Hölder continuity of the $\zeta$th derivative of sample paths of $\{X(t;m)/\sigma (m):\,t\in [0,T]\}$ holds true by [11, Corollary 16.9].

(ii)–(iii) Those can be proved in a similar manner to (i). In particular, for (iii), we set the kernel $\psi (s):=\sum _{k=1}^d \theta _k f_n(t_k,s;H,\alpha )$ with $0\le t_1<t_2<\cdots <t_d\le T$, and the domain $(-\infty ,0)$ of integration with respect to s. The only difference is the square integrability of the kernel $\psi (s)$ on $(-\infty ,0)$ in (8.12), which holds true when $H>n-1+1/\alpha -1/2$. We can exclude $\alpha \in (0,2/3)$, since the range of H would be empty otherwise.

(iv) Next, observe that for $0\le t_1\le t_2\le T$,

$$\begin{aligned} \mathbb {E}\left[ \left| \frac{Y(t_2;m)}{\sigma (m)}-\frac{Y(t_1;m)}{\sigma (m)}\right| ^2\right]&=\int _{-\infty }^0 \left| f_n\left( t_2,s;H,\alpha \right) -f_n\left( t_1,s;H,\alpha \right) \right| ^2ds\\&=\left( t_2-t_1\right) ^2\int _{-\infty }^0 \left| f_{n-1}\left( \Theta _s(t_1,t_2),s;H-1,\alpha \right) \right| ^2ds\\&=\left( t_2-t_1\right) ^2 \left( \Theta _s(t_1,t_2)\right) ^{2(H-1)}C^2_{H-1+1/\alpha -1/2,2,n-1}\\&\le \left( t_2-t_1\right) ^2 T^{2(H-1)}C^2_{H-1+1/\alpha -1/2,2,n-1}, \end{aligned}$$

uniformly in m, where the first equality holds by the Wiener-Ito isometry, where the second equality holds by the mean value theorem (8.1), and where the last inequality holds by $H>1$ since $n\ge 2$. The Kolmogorov- C̆ hentsov theorem ensures that the sequence $\{\sigma (m)^{-1}Y(t;m):\,t\in [0,T]\}$ is tight in m (as $m\uparrow +\infty$). Hence, together with (ii), the weak convergence holds in the space $\mathcal {C}([0,T];\mathbb {R})$ by Lemma 16.2 and Theorem 16.3 of [11]. $\square$

Proof of Theorem 6.4

• (i) The tail of a linear combination of $L^{\alpha }$-functions vanishes, that is, for $d\in \mathbb {N}$, real numbers $\{\theta _k\}_{k=1,\ldots ,d}$ and $0\le t_1\le \ldots \le t_d\le T$, we have $\int _{-\infty }^{-\kappa }|\sum _{k=1}^d \theta _k f_n(t_k,s;H,\alpha )|^{\alpha }ds\rightarrow 0$, as $\kappa \uparrow +\infty$. This shows the convergence of the finite dimensional distributions.

• (ii) Observe that for $\theta \in (0,\alpha )$ and $0\le t_1\le t_2\le T$,

  $$\begin{aligned}&\mathbb {E}\left[ \left| R\left( t_2;\kappa \right) -R\left( t_1;\kappa \right) \right| ^{\theta }\right] =(t_2-t_1)^{\theta }\mathbb {E}\left[ \left| \int _{-\infty }^{-\kappa }f_{n-1}\left( \Theta _s(t_1,t_2),s;H-1,\alpha \right) dL_{\alpha }(s)\right| ^{\theta }\right] \\&\quad = (t_2-t_1)^{\theta }\frac{2^{\theta }\Gamma ((1+\theta )/2)\Gamma (1-\theta /\alpha )}{\sqrt{\pi }\Gamma (1-\theta /2)}\left( \int _{-\infty }^{-\kappa }\left| f_{n-1}\left( \Theta _s(t_1,t_2),s;H-1,\alpha \right) \right| ^{\alpha }ds\right) ^{\theta /\alpha }\\&\quad \le (t_2-t_1)^{\theta }\frac{2^{\theta }\Gamma ((1+\theta )/2)\Gamma (1-\theta /\alpha )}{\sqrt{\pi }\Gamma (1-\theta /2)}\left( \int _{-\infty }^0\left| f_{n-1}\left( T,s;H-1,\alpha \right) \right| ^{\alpha }ds\right) ^{\theta /\alpha }, \end{aligned}$$

  uniformly in $\kappa (\uparrow +\infty )$, where we have applied the mean value theorem (8.1) for the first equality, the well known stable moment formula [21] for the second equality, and (3.8) for the last inequality. In order to apply the Kolmogorov- C̆ entsov theorem, the power $\theta$ of $(t_2-t_1)^{\theta }$ on the right hand side is required to be strictly greater than 1. Hence, the index $\alpha$ is required to be strictly greater than 1 as well. Then, we get tightness of the sequence $\{R(t;\kappa ):\,t\in [0,T]\}$ in $\kappa (\uparrow +\infty )$. Hence, by Lemma 16.2 and Theorem 16.3 of [11], the weak convergence holds in the space $\mathcal {C}([0,T];\mathbb {R})$ to the degenerate zero process. This implies the convergence in probability uniformly on [0, T] as well.

• (iii) This result is closely related to sample continuity (Theorem 4.1(iv)). By [19, Theorem 10.5.1], the left hand side of (6.11) convergences to $c_{\alpha }\int _{-\infty }^{-\kappa }\sup _{t\in T^*}|f_n(t,s;H,\alpha )|^{\alpha }ds,$ where $T^*$ denotes an arbitrary countable dense subset of [0, T]. For each $s<0$, $\sup _{t\in T^*}|f_n(t,s;H,\alpha )|=|f_n(T,s;H,\alpha )|$, which is in $L^{\alpha }(-\infty ,-\kappa )$.

• (iv) By [19, Corollary 4.4.6], the left hand side of (6.12) converges to the quantity $c_{\alpha }\int _{-\infty }^{-\kappa }\max _{k=1,\ldots ,N}|f_n(t_k,s;H,\alpha )|^{\alpha }ds.$ For $s<0$, $\max _{k=1,\ldots ,N}|f_n(t_k,s;H,\alpha )|=|f_n(T,s;H,\alpha )|$, which is in $L^{\alpha }(-\infty ,-\kappa )$. This gives the result.$\square$