2.1. Lévy processes

A Lévy process, $(X(t))=(X(t)\;;\;t\geq 0)$ , is a real-valued, infinitely divisible stochastic process with stationary and independent increments. The log-characteristic function, commonly known as the characteristic exponent (CE) or cumulant function, of any Lévy process is given by [Reference Eberlein16]

(2.1) \begin{equation} K(t;\,\theta) \,:\!=\,\ln\mathbb{E}\!\left(e^{i\theta X(t)}\right) = ti\theta a - t\frac{1}{2}b^{2}\theta^{2} + t\int_{\mathbb{R}^*}\left[e^{i\theta x} -1 - \mathbb{1}(|x|<1)ix\theta\right]Q(dx), \end{equation}

where $\mathbb{R}^{*} \,:\!=\, \mathbb{R}\backslash \{0\}$ . The term $\mathbb{1}(|x|<1)ix\theta$ is a centring term that ensures convergence of the CE for processes for which $\int_{|x|<1}xQ(dx)$ is divergent, though it can be omitted for processes with finite first absolute moment. The Lévy triplet $(a, b^{2}, Q)$ uniquely defines the Lévy process [Reference Eberlein16], with $a \in \mathbb{R}$ , $b \in [0,\infty)$ , and where the Lévy measure, Q(dx), is a Poisson-process intensity measure defining the distribution of jumps in the Lévy process, satisfying

(2.2) \begin{equation} \int_{\mathbb{R}^*}\big(1 \wedge x^{2}\big)Q(dx) < \infty, \end{equation}

where $(a\wedge b)$ denotes the minimum value of a and b. See e.g. [Reference Barndorff-Nielsen and Shephard5, Section 2.3] for more details.

2.2. Subordinators

A subordinator $(Z(t))_{t\geq0}$ is a particular case of a Lévy process with non-negative increments, such that its paths are almost surely (a.s.) increasing [Reference Wolpert and Ickstadt47]. The Lévy measure of any subordinator, $Q_{Z}(dz)$ , satisfies [Reference Barndorff-Nielsen and Shephard5]

(2.3) \begin{equation} \int_{(0,\infty)} (1\wedge z)Q_{Z}(dz) < \infty. \end{equation}

Observe that this is a stricter condition than that required for general Lévy processes in (2.2). Consequently, the Lévy triplet of a subordinator has $b^{2} = 0$ and no centring term is required [Reference Wolpert and Ickstadt47]. The current work will consider only subordinators without drift; thus the CE for such a subordinator (Z(t)) will always be of the form

(2.4) \begin{equation} K_Z(t;\,\theta)\,:\!=\, \ln\mathbb{E}\big[e^{i\theta Z(t)}\big] = t\int_{(0,\infty)}\left(e^{i\theta z} - 1\right)Q_{Z}(dz). \end{equation}

The mean and variance of Z(t), when they exist, may be obtained for all t from the Lévy measure:

\begin{equation*} \mathbb{E}[Z(t)] = t\int_{(0,\infty)}z Q_Z(dz), \quad {Var}[Z(t)] = t\int_{(0,\infty)}z^{2}Q_{Z}(dz). \end{equation*}

Most but not all of the processes we consider here will have finite first and second moments, because their Lévy densities decay exponentially for large jump sizes; see Sections 3.1.4 and 3.3.2 of [Reference Barndorff-Nielsen and Shephard5]. We will also be concerned here with so-called infinite-activity subordinators, which exhibit infinitely many jumps in any finite time interval: $Q_Z((0,\infty)) = \infty$ . Combined with (2.2), this implies the presence of infinitely many small jumps ( $|x|<1$ ) within finite time intervals.

2.3. Normal variance-mean processes

A normal variance-mean (NVM) process is defined as time-deformed Brownian motion, where jumps in a subordinator process (Z(t)) drive random time deformations of an independent Brownian motion (B(t)) as follows [Reference Barndorff-Nielsen and Shephard5]:

(2.5) \begin{equation} X(t) = \mu t + \mu_{W}Z(t) + \sigma_{W}B(Z(t)), \quad t\geq 0,\; \mu, \mu_{W} \in \mathbb{R}, \ \sigma_{W} \in (0,\infty). \end{equation}

Here (B(t)) is a standard one-dimensional Brownian motion and (Z(t)) is a subordinator process as in the previous section. We limit attention to the case $\mu=0$ without loss of generality. The parameter $\mu_{W}$ models the skewness of the jump distribution, with a fully symmetric process for $\mu_{W} = 0$ . The specification of $\left(Z(t)\right)$ , coupled with the choice of $\mu_{W}$ and $\sigma_{W}$ , allows for a broad family of heavy-tailed and skewed processes to be implemented. We can express the Lévy measure Q of any such process in terms of its subordinator’s Lévy measure $Q_Z$ :

(2.6) \begin{equation} Q(dx) = \int_{(0,\infty)}\mathcal{N}\big(dx;\,\mu_{W}z,\sigma_{W}^{2}z\big)Q_{Z}(dz), \end{equation}

where ${\mathcal N}(\cdot;\,\mu,\sigma^2)$ denotes the Gaussian law with mean $\mu$ and variance $\sigma^2$ . Finally, if $K_Z(t;\,\theta)$ is the Lévy–Khintchine exponent of Z(t) in (2.4), then the CE for the subordinated process is given by [Reference Cont and Tankov12, Section 4.4]

(2.7) \begin{align} K_{X}(t;\,\theta) = K_Z\!\left(t;\, {{\mu_{W}\theta +i \frac{1}{2}\sigma_{W}^{2}\theta^{2}}}\right) &= t\int_{(0,\infty)}\left[e^{\left(i\mu_{W}\theta - \frac{1}{2}\sigma_{W}^{2}\theta^{2}\right)z}-1\right]Q_{Z}(dz), \end{align}

from which we obtain the mean and variance directly in terms of the moments of the subordinator. Specifically, for $t\geq 0,$

(2.8) \begin{align} &\mathbb{E}[X(t)] = \mu_{W}\int_{(0,\infty)}zQ_{Z}(dz) = t\mu_{W}\mathbb{E}\!\left[Z(1)\right] \end{align}

and

(2.9) \begin{align} &{Var}[X(t)] = t\mu_{W}^{2}\int_{(0,\infty)}z^{2}Q_{Z}(dz) + t\sigma_{W}^{2}\int_{0}^{\infty}zQ_{Z}(dz) = t\mu_{W}^{2}{Var}[Z(1)] + t\sigma_{W}^{2}\mathbb{E}[Z(1)], \end{align}

when these expectations exist.

We will consider examples based on the normal-gamma, normal tempered stable, and generalised hyperbolic processes, which are obtained via Brownian motion subordinated to the gamma, tempered stable, and GIG processes, respectively, with Lévy measures shown in (2.10), (2.11), (2.12), respectively:

(2.10) \begin{align} &Q_{Z}(dz) = \nu z^{-1}\exp\!\left({-}\frac{1}{2}\gamma^{2}z\right)dz, \end{align}

(2.11) \begin{align} &Q_{Z}(dz) = Az^{-1-\kappa}\exp\!\left({-}\frac{1}{2}\gamma^{\frac{1}{\kappa}}z\right)dz, \end{align}

(2.12) \begin{align} &Q_{Z}(dz) = z^{-1}\exp \!\left({-}\frac{\gamma^{2}}{2}z\right)\left[\max(0, \lambda)+\frac{2}{\pi^{2}}\int_{0}^{\infty}\frac{1}{y\!\left|H_{|\lambda|}(y)\right|^{2}} \exp \!\left( -\frac{zy^{2}}{2\delta^{2}}\right) dy\right] dz. \end{align}

Appropriate ranges for the different parameters are specified in Section 6.

2.4. Generalised shot-noise representation

The shot-noise representation of a Lévy process is fundamental to the simulation methods in [Reference Godsill, Riabiz and Kontoyiannis23], [Reference Rosinski39], and [Reference Godsill and Kndap22]. Adopting the perspective of viewing a Lévy process as a point process defined on $[0, T] \times \mathbb{R}^{d}$ , Rosinski [Reference Rosinski39] considers the equivalent representation

(2.13) \begin{equation} N = \sum_{i=1}^{\infty}\delta_{V_{i}, H(Z_{i}, U_{i})}, \end{equation}

where $\delta_x$ denotes the Dirac measure at x, and $\left\{V_{i}\right\}$ is a sequence of independent and identically distributed (i.i.d.) uniforms, $V_{i} \sim U[0,T]$ , independent of $\left\{Z_{i}, U_{i}\right\}$ . Intuitively, $\{V_{i}\}$ represent the arrival times of the jumps $X_{i} = H(Z_{i}, U_{i})$ in the process, and $\{Z_i\}$ represent a non-increasing set of jump sizes drawn from the subordinator process.

For the NVM Lévy process, $H(\cdot)$ is related to the time-domain representation of the process in (2.5) via

(2.14) \begin{equation} H(Z_{i}, U_{i}) = \mu_{W}Z_{i} + \sigma_{W}\sqrt{Z_i}U_{i}, \quad \ U_{i} \overset{i.i.d.}{\sim} \mathcal{N}(0,1). \end{equation}

The ordered jumps $\{Z_i\}$ are typically simulated through generation of the ordered epochs $\{\Gamma_i\}$ of a standard Poisson point process, obtained by the partial sum of exponential random variables, $e_{i} \overset{i.i.d.}{\sim} {\textrm{Exp}}(1)$ : $\Gamma_{i} = \Gamma_{i-1} + e_{i}.$ Then we obtain the ith ordered subordinator jump through a function $Z_{i} = h(\Gamma_{i})$ as

\begin{equation*} Z(t) = \sum_{i=1}^{\infty}h(\Gamma_{i})\mathbb{1}(V_{i} \leq t), \end{equation*}

where the map $h(\cdot)$ can be expressed in terms of the upper tail of $Q_Z$ [Reference Barndorff-Nielsen and Shephard5, Section 3.4.1] as $h(\Gamma_{i}) = \inf \{z \in \mathbb{R} : Q_Z([z,\infty)) < \Gamma_i\},$ which may or may not be available in closed form, depending on the particular choice of $Q_Z$ . If it is not available, then rejection sampling or thinning methods can be employed, as in [Reference Godsill and Kndap22], which leads to suitable algorithms for all of the models considered here.

Substituting (2.13) into the Lévy–Khintchine representation (2.1), the Lévy process can be expressed as the convergent infinite series [Reference Rosinski39]

(2.15) \begin{equation} X(t) = \sum_{i=1}^{\infty}H(Z_i, U_{i})\mathbb{1}(V_{i} \leq t) -tb_{i}, \end{equation}

where $b_{i}$ is a compensator or centring term [Reference Rosinski39] that ensures the convergence of the series. A Lévy process need not be compensated if and only if

\begin{equation*} \int_{\mathbb{R}^*}(1 \wedge |x|)Q(dx) < \infty; \end{equation*}

see [Reference Wolpert and Ickstadt47, Reference Winkel46]. In view of (2.3), any subordinator satisfies this condition, and hence so does the corresponding NVM process; see Appendix C. Therefore, none of the Lévy processes studied in this paper requires compensating terms, and we take $b_{i} = 0 $ for all i throughout.

4.1. Preliminaries

Consider a subordinator (Z(t)) with Lévy measure $Q_Z$ . The corresponding process with jumps truncated at $\epsilon$ is denoted by $(Z_\epsilon(t))$ and has Lévy measure $Q_{Z_\epsilon}(B)=Q_Z(B\cap (0,\epsilon])$ for all Borel sets B. The corresponding residual NVM process, denoted by $(X_\epsilon(t))$ , has Lévy measure

(4.1) \begin{equation}Q_{X_\epsilon}(dx)=\int_{0}^{\epsilon}{\mathcal {N}}\big(dx|\mu_{W} z, \sigma_{W}^2z\big)Q_Z(dz),\end{equation}

and its moments can be obtained from (2.8) and (2.9):

\begin{equation*}\mathbb{E}[X_\epsilon(t)]=t\int_{-\infty}^\infty x Q_{X_\epsilon}(dx)=t\mu_{W} M^{(1)}_{Z_\epsilon}\end{equation*}

and

(4.2) \begin{equation} t\sigma_\epsilon^2=\mathbb{Var} \!\left[X_\epsilon(t)\right]=t\int_{-\infty}^\infty x^2 Q_{X_\epsilon}(dx)=t\Big[\mu_{W}^2M^{(2)}_{Z_\epsilon}+\sigma_{W}^2M^{(1)}_{Z_\epsilon}\Big], \end{equation}

where

(4.3) \begin{equation} M^{(n)}_{Z_\epsilon}=\int_0^\epsilon z^nQ_{Z_\epsilon}(dz)<\infty, \quad n\geq 1. \end{equation}

Note that these moments are all well defined and finite since $Q_{Z_\epsilon}$ satisfies (2.3) and for all $n\geq 1$ we have $\lim_{\epsilon\rightarrow0}M_{Z_{\epsilon}}^{(n)} = 0$ .

Note also that, for any $0< \epsilon \leq 1$ and $m<n$ ,

(4.4) \begin{equation}M^{(n)}_{Z_\epsilon}\leq \epsilon^{n-m} M^{(m)}_{Z_\epsilon}, \quad n\geq 1,\end{equation}

since $x^n\leq \epsilon^{n-m} x^m$ for $0<x\leq \epsilon$ . In particular this implies that

\begin{equation*}\sigma_\epsilon^2=\mu_{W}^2M^{(2)}_{Z_\epsilon}+\sigma_{W}^2M^{(1)}_{Z_\epsilon}\leq M^{(1)}_{Z_\epsilon}\left(\mu_{W}^2\epsilon+\sigma_{W}^2\right)\overset{\epsilon\rightarrow 0}{\rightarrow} 0\end{equation*}

and that

(4.5) \begin{equation}\frac{{\sigma_\epsilon}^4}{M^{(2)}_{Z_\epsilon}}=\frac{\left(\mu_{W}^2M^{(2)}_{Z_\epsilon}+\sigma_{W}^2M^{(1)}_{Z_\epsilon}\right)^2}{M^{(2)}_{Z_\epsilon}}\xrightarrow{\epsilon \rightarrow 0} \sigma_{W}^4\lim_{\epsilon\rightarrow 0}\frac{{M^{(1)}_{Z_\epsilon}}^2}{M^{(2)}_{Z_\epsilon}},\end{equation}

whenever the limit $\lim_{\epsilon\rightarrow 0}\frac{{M^{(1)}_{Z_\epsilon}}^2}{M^{(2)}_{Z_\epsilon}}$ exists.

4.2. Gaussianity of the limit of $X_{\epsilon}$

The following is our first main result; it gives conditions under which the residual process corresponding to an NVM Lévy process with jumps truncated at level $\epsilon$ converges to Brownian motion as $\epsilon\to 0$ . An analogous one-dimensional result for $\alpha$ -stable processes is established, along with corresponding convergence bounds, in [Reference Riabiz, Ardeshiri, Kontoyiannis and Godsill37].

Theorem 4.1. Consider a truncated NVM Lévy process $X_{\epsilon}=(X_{\epsilon}(t))$ with Lévy measure as in (4.1). Let the standardised process $Y_\epsilon$ be defined as $Y_\epsilon(t)=(X_\epsilon(t)-\mathbb{E}[X_\epsilon(t)])/\sigma_\epsilon$ , $t\geq 0$ . If

(4.6) \begin{align} \lim_{\epsilon \rightarrow 0}\frac{M_{Z_\epsilon}^{(2)}}{{M_{Z_\epsilon}^{(1)}}^2} = 0, \end{align}

then $Y_\epsilon$ converges weakly to a standard Brownian motion $W=(W(t))$ in D[0,1] with the topology of uniform convergence.

Proof. The main content of the proof is in establishing the claim that $X_\epsilon(1)$ , properly standardised, converges to a Gaussian under (4.6). For that, it suffices to show that the CE of the standardised version of $X_\epsilon(1)$ converges pointwise to $-u^2/2$ . The CE of $X_\epsilon(1)$ is given as in (2.7), by

\begin{align*} \phi^\epsilon_{X}(u)=\int_{\mathbb{R}^*} (\exp(iux)-1)Q_X^\epsilon(dx)=\int_0^\epsilon\left[\exp\Big(iu\mu_{W}z-\frac{1}{2}{u^2}\sigma_{W}^2z\Big)-1\right]Q_Z(dz),\end{align*}

where $Q_X^\epsilon(dx)$ is the Lévy measure for $X_\epsilon$ , having the same structure as (4.1). The CE for $\tilde{X}_\epsilon(t)={X}_\epsilon(t)-\mathbb{E}[{X}_\epsilon(t)]$ is then given by

\begin{align*} \phi_{\tilde{X}}^{\epsilon}(u) = \phi_{X}^{\epsilon}(u) - iu\int_{\mathbb{R}^*}xQ_{X}^{\epsilon}(dx) = \int_0^\epsilon\left[\exp\!\left(iu\mu_{W}z-\frac{1}{2}u^2\sigma_{W}^2z\right)-1-iu\mu_{W}z\right]Q_{Z}(dz).\end{align*}

Note that $X_\epsilon$ has finite mean, $\int_{\mathbb{R}^*}x Q_X^\epsilon(dx)=\mu_{W}M_{Z\epsilon}^{(1)}<\infty$ , by the finiteness of $M_{Z\epsilon}^{(1)}$ in (4.3). Now, if we scale the centred process, $\tilde{X}_{\epsilon}$ , to have unit variance at $t=1$ , i.e., if we let $Y_\epsilon(t)=\tilde{X}_\epsilon(t)/\sigma_\epsilon,$ then the CE for $Y_\epsilon(1)$ becomes

\begin{align*} \phi^\epsilon_Y(u) = \phi^\epsilon_{\tilde{X}}(u/\sigma_\epsilon)&= \int_{0}^{\epsilon}\left[\exp\!\left(v\frac{z}{\sigma_\epsilon^2}\right)-1-iu\mu_{W}\frac{z}{\sigma_\epsilon}\right]Q_Z(dz)\\ &=\int_0^\epsilon\left[\exp\!\left(v\frac{z}{\sigma_\epsilon^2}\right)-1-v\frac{z}{\sigma_\epsilon^{2}}-\frac{1}{2}u^2\sigma_{W}^2\frac{z}{\sigma_\epsilon^2}\right]Q_Z(dz)\\ &= \int_{0}^{\epsilon}\left[\exp\!\left(v\frac{z}{\sigma_\epsilon^2}\right)-1-v\frac{z}{\sigma_\epsilon^{2}}\right]Q_Z(dz)-\frac{1}{2}u^2\sigma_{W}^2\frac{M^{(1)}_{Z_\epsilon}}{\sigma_\epsilon^2}\\ &=\varphi_\epsilon(u)-\frac{1}{2}u^2\psi_\epsilon,\end{align*}

where $v={iu\mu_{W}}{\sigma_\epsilon}-{u^2\sigma_{W}^2}/2$ , $\varphi_\epsilon(u)=\int_{0}^{\epsilon}\big[\exp\big(v\frac{z}{\sigma_\epsilon^2}\big)-1-v\frac{z}{\sigma_\epsilon^{2}}\big]Q_Z(dz)$ , and $\psi_\epsilon=\sigma_{W}^2\frac{M^{(1)}_{Z_\epsilon}}{\sigma_\epsilon^2}$ .

Consider the difference between $\exp(\phi^\epsilon_Y(u))$ and the CF of the standard normal:

\begin{align*} e_\epsilon(u)&\,:\!=\,\exp\!\left[\varphi_\epsilon(u)-\frac{1}{2}u^2\psi_\epsilon\right]-\exp\!\left({-}\frac{u^2}{2}\right)\\ &=\exp\!\left({-}\frac{u^2}{2}\right)\left\{\exp\!\left[\varphi_\epsilon(u)+\frac{1}{2}u^2(1-\psi_\epsilon)\right]-1\right\}.\end{align*}

First we establish bounds on $\varphi_\epsilon$ and $1-\psi_\epsilon$ . For $1-\psi_\epsilon$ , from (4.2) and (4.4) we have

\begin{equation*}1-\psi_\epsilon=\frac{\mu_{W}^2M^{(2)}_{Z_\epsilon}}{{\sigma_\epsilon}^2}=\frac{\mu_{W}^2M^{(2)}_{Z_\epsilon}}{\mu_{W}^2M^{(2)}_{Z_\epsilon}+\sigma_{W}^2M^{(1)}_{Z_\epsilon}}=\frac{\mu_{W}^2}{\mu_{W}^2+\sigma_{W}^2M^{(1)}_{Z_\epsilon}/M^{(2)}_{Z_\epsilon}};\end{equation*}

therefore, since clearly $1-\psi_\epsilon\geq 0$ ,

(4.7) \begin{equation}0\leq 1-\psi_\epsilon\leq \frac{\mu_{W}^2}{\mu_{W}^2+\sigma_{W}^2/\epsilon}\xrightarrow[]{\epsilon\rightarrow 0}0.\end{equation}

For $\varphi_\epsilon$ , first recall that for any complex z with negative real part,

(4.8) \begin{align} \Bigg|\exp(z)-\sum_{i=0}^{n}z^n/n!\Bigg|\leq |z|^{n+1}/(n+1)!, \end{align}

and so

(4.9) \begin{align} |\varphi_\epsilon(u)|&=\left|\int_0^\epsilon\big(\!\exp\big(vz/\sigma_\epsilon^2\big)-1-vz/\sigma_\epsilon^2\big)Q(dz)\right|\nonumber\\ &\leq \int_0^\epsilon \left|\big(\!\exp\big(vz/\sigma_\epsilon^2\big)-1-vz/\sigma_\epsilon^2\big)\right|Q(dz)\nonumber\\ &\leq \int_0^\epsilon \frac{1}{2}|v|^2\frac{z^2}{\sigma_\epsilon^{4}}Q(dz) \end{align}

(4.10) \begin{align}=\frac{1}{2}|v|^2 \frac{M^{(2)}_{Z_\epsilon}}{\sigma_\epsilon^4}.\end{align}

The final integral is well defined by (4.3). Since $|v|^2 = {u^2\mu_{W}^2}{\sigma_\epsilon^2}+{u^4\sigma_{W}^4}/4$ , using (4.5) and (4.7), we have

\begin{align*} \lim_{\epsilon\rightarrow 0}|v|^2\frac{M^{(2)}_{Z_\epsilon}}{\sigma_\epsilon^4}&= \lim_{\epsilon\rightarrow 0}\left[u^{2}\frac{\mu_{W}^{2}M_{Z_{\epsilon}}^{(2)}}{\sigma_{\epsilon}^{2}}+\frac{1}{4}u^{4}\sigma_{W}^{4}\frac{M_{Z_{\epsilon}}^{(2)}}{\sigma_{\epsilon}^{4}}\right]\\ &=\frac{1}{4}u^{4}\sigma_{W}^4\lim_{\epsilon\rightarrow 0}\frac{M^{(2)}_{Z_\epsilon}}{\Big(\mu_{W}^2M^{(2)}_{Z_\epsilon}+\sigma_{W}^2M^{(1)}_{Z_\epsilon}\Big)^2}\\ &=\frac{1}{4}u^4\lim_{\epsilon\rightarrow 0} \frac{M^{(2)}_{Z_\epsilon}}{\big(M^{(1)}_{Z_\epsilon}\big)^2},\end{align*}

where we used the fact that $M^{(2)}_{Z_\epsilon}/M^{(1)}_{Z_\epsilon} \leq \epsilon \rightarrow 0$ from (4.4). Combining the bounds in (4.7) and (4.10), we have

\begin{align*} \Big|\varphi_\epsilon(u)+\frac{1}{2}u^2(1-\psi_\epsilon)\Big|&\leq \frac{1}{2}|v|^2 \frac{M^{(2)}_{Z_\epsilon}}{\sigma_\epsilon^4}+\frac{1}{2}u^2\frac{\epsilon\mu_{W}^2}{\epsilon\mu_{W}^2+\sigma_{W}^2}\xrightarrow{\epsilon\rightarrow 0} \frac{u^4}{8}\lim_{\epsilon\rightarrow 0} \frac{M^{(2)}_{Z_\epsilon}}{{M^{(1)}_{Z_\epsilon}}^2}.\end{align*}

This bound is finite for any $|u|<\infty$ by the properties of $\sigma_\epsilon$ and $M^{(2)}_{Z_\epsilon}$ , and hence, under (4.6),

\begin{align*} |e_\epsilon(u)| &=\left|\exp\!\left({-}\frac{u^2}{2}\right)\left\{\exp\!\left[\varphi_\epsilon(u)+\frac{1}{2}u^2(1-\psi_\epsilon)\right]-1\right\}\right|\\ &\leq \exp\!\left({-}\frac{u^2}{2}\right)\left(\exp\!\left|\varphi_\epsilon(u)+\frac{1}{2}u^2(1-\psi_\epsilon)\right|-1\right) \xrightarrow[]{\epsilon\rightarrow 0} 0.\end{align*}

This proves the claimed pointwise convergence $\exp(\phi^\epsilon_Y(u))\rightarrow \exp({-}u^2/2)$ as $\epsilon\to 0$ .

Next we argue that the condition (4.6) is in fact sufficient for the process-level convergence claimed in the theorem. Note that the result of the claim also implies uniform convergence of the relevant characteristic functions on compact intervals [Reference Kallenberg26, Theorem 5.3, p. 86]. Now, it is straightforward, from the definition of a Lévy process, that the increments $Y_\epsilon(t)-Y_\epsilon(s)$ converge in distribution to the corresponding (Gaussian) increments of a Brownian motion, for all $s < t$ . We proceed to verify the requirement (III) in [Reference Pollard36, Theorem V.19], i.e. that given $\delta > 0$ , there are $\alpha > 0$ , $\beta>0$ , and $\epsilon_0 > 0$ such that ${\mathbb P}\{|Y_\epsilon(t) - Y_\epsilon(s)| \leq \delta\} \geq \beta$ whenever $|t - s| < \alpha$ and $\epsilon < \epsilon_0$ . To see this, note that the Lévy process $(Y_\epsilon(t) : t \in [0, 1])$ is centred, so for any $1 \geq t \geq s \geq 0$ , by Chebyshev,

\begin{align*}{\mathbb P}\{|Y_\epsilon(t) - Y_\epsilon(s)| > \delta\}&={\mathbb P}\{|Y_\epsilon(t - s)| > \delta\}\\&\leq \frac{1}{\delta^2} {\textrm{Var}}(Y_\epsilon(t - s))\\&=\frac{(t - s){\textrm{Var}}(Y_\epsilon(1))}{\delta^2}= \frac{(t - s)}{\delta^2}.\end{align*}

By taking $\delta\in (0, 1)$ , we can choose any $\epsilon_0$ and $\alpha = \delta^3$ . Then, ${\mathbb P}\{|Y_\epsilon(t) - Y_\epsilon(s)| > \delta\}=1-{\mathbb P}\{|Y_\epsilon(t) - Y_\epsilon(s)| \leq \delta\}$ , so we can set $\beta=1-\delta$ , and we conclude that $(Y_{\epsilon}(t)) \xrightarrow{\epsilon \rightarrow 0}{} (W(t))$ in D[0,1], as claimed.

Next we show that some conditions are in fact necessary for the Gaussian limit in Theorem 4.1 to hold.

Theorem 4.2. Consider a truncated NVM Lévy process $X_{\epsilon}=(X_{\epsilon}(t))$ , and define the associated standardised process $Y_\epsilon$ as in Theorem 4.1. If the condition (4.6) does not hold and, moreover,

(4.11) \begin{align}L_1\,:\!=\,\liminf_{\epsilon \rightarrow 0}\frac{M_{Z_\epsilon}^{(2)}}{{M_{Z_\epsilon}^{(1)}}^2} >0\quad \mbox{and}\quad L_2\,:\!=\,\sigma_W^6\limsup_{\epsilon\rightarrow 0}\frac{M^{(3)}_{Z_\epsilon}}{\sigma_\epsilon^6}>0,\end{align}

then $Y_\epsilon(1)$ does not converge to ${\mathcal N}(0,1)$ in distribution as $\epsilon\to 0$ .

Proof. Suppose the conditions in (4.11) hold. We will assume that $Y_\epsilon(1)$ converges to ${\mathcal N}(0,1)$ in distribution as $\epsilon\to 0$ , and derive a contradiction.

In the notation of the proof of Theorem 4.1, expanding the exponential series in (4.9) for one more term than before and using (4.8) yields

\begin{align*} \int_0^\epsilon\big(\exp\big(vz/\sigma_\epsilon^2\big)-1-vz/\sigma_\epsilon^2-v^2z^2/\big(2\sigma_\epsilon^4\big)\big)Q(dz)=D_\epsilon(v), \end{align*}

with $|D_\epsilon(v)|\leq |v|^3M^{(3)}_{Z_\epsilon}/(3!\sigma_\epsilon^6)$ . Hence, rearranging and integrating, we obtain

(4.12) \begin{equation}\varphi_\epsilon(u)=\int_0^\epsilon\big(\!\exp\big(vz/\sigma_\epsilon^2\big)-1-vz/\sigma_\epsilon^2\big)Q(dz)=F_\epsilon(v)+D_\epsilon(v), \end{equation}

where $v={iu\mu_{W}}{\sigma_\epsilon}-{u^2\sigma_{W}^2}/2$ as before and $F_\epsilon(v)\,:\!=\,v^2 M^{(2)}_{Z_\epsilon}/\big(2\sigma_\epsilon^4\big).$ The assumption that $Y_\epsilon(1)$ converges to ${\mathcal N}(0,1)$ implies that $|\varphi_\epsilon(u)|\to 0$ for all u.

We consider several cases. First, we note that $F_\epsilon(v)$ cannot converge to zero for any $u\neq 0$ , since, by (4.11),

(4.13) \begin{align} \liminf_{\epsilon\rightarrow 0} {|}F_\epsilon(v){|} =\frac{1}{4}u^4\sigma_{W}^4\liminf_{\epsilon\rightarrow 0}\frac{M^{(2)}_{Z_\epsilon}}{\sigma_\epsilon^4} =\frac{1}{4}u^4\sigma_{W}^4\liminf_{\epsilon \rightarrow 0}\frac{M^{(2)}_{Z_\epsilon}}{{M^{(1)}_{Z_\epsilon}}^2} =\frac{1}{4}u^4\sigma_{W}^4 L_1>0. \end{align}

Second, we note that $D_\epsilon(v)$ also cannot converge to zero, because then $F_\epsilon(v)$ would need to converge to zero as well. The only remaining case is if, for every u, neither $F_\epsilon(v)$ nor $D_\epsilon(v)$ vanishes as $\epsilon\to0$ , while their sum does vanish. By (4.11) we have

\begin{equation*}\limsup_{\epsilon\to 0} |D_\epsilon(v)|\leq \limsup_{\epsilon\to 0}|v|^3\frac{M^{(3)}_{Z_\epsilon}}{3!\sigma_\epsilon^6}={\frac{1}{48}u^6L_2}.\end{equation*}

On the other hand, by (4.12) and (4.13) we have

\begin{equation*}\liminf_{\epsilon\to 0} |D_\epsilon(v)|=\liminf_{\epsilon\to 0} \Big| |\varphi_\epsilon(v)| - F_\epsilon(v)\Big|=\liminf_{\epsilon\to 0} |F_\epsilon(v)|=\frac{1}{4}u^4\sigma_{W}^4 L_1.\end{equation*}

Therefore,

\begin{equation*}{\frac{1}{48}u^6L_2}\geq\limsup_{\epsilon\to 0} |D_\epsilon(v)|\geq \liminf_{\epsilon\to 0} |D_\epsilon(v)|=\frac{1}{4}\sigma_{W}^4 L_1u^4.\end{equation*}

Since $L_1,L_2>0$ , this is clearly violated for u small enough, providing the desired contradiction and completing the proof.

Corollary 4.1. Suppose the subordinator Lévy measure $Q_{Z}(dz)$ admits a density $Q_{Z}(z)$ , which, for some $E>0$ , satisfies $Q_{Z}(z) > 0$ for all $z \in (0, E]$ . Then the condition (4.6) in Theorem 4.1 is equivalent to

\begin{equation*} \lim_{\epsilon\rightarrow0}\epsilon Q_{Z}(\epsilon) = +\infty.\end{equation*}

Proof. A straightforward application of L’Hôpital’s rule twice gives

\begin{align*} \lim_{\epsilon\rightarrow0} \frac{M^{(2)}_{Z_\epsilon}}{M^{(1)^{2}}_{Z_\epsilon}} = \lim_{\epsilon\rightarrow0}\frac{\int_{0}^{\epsilon}z^{2}Q_{Z}(z)dz}{\left(\int_{0}^{\epsilon}zQ_{Z}(z)dz\right)^{2}} = \lim_{\epsilon\rightarrow0} \frac{\epsilon}{2M^{(1)}_{Z_\epsilon}} = \lim_{\epsilon\rightarrow0}\frac{1}{2\epsilon Q_{Z}(\epsilon)},\end{align*}

where the positivity of $Q_Z(z)$ in (0, E] was required to ensure that the derivative of the denominator is non-zero for all $\epsilon$ small enough.

6.1. Normal-gamma process

The subordinator of the normal-gamma (NG) process is a gamma process, with parameters $\nu, \gamma>0$ and with Lévy density

\begin{equation*} Q_{Z}(z) = \nu z^{-1} \exp\!\left({-}\frac{1}{2}\gamma^{2}z\right), \quad z>0. \end{equation*}

Here, in view of Corollary 4.1,

\begin{align*} \lim_{\epsilon\rightarrow0}\frac{M_{Z_\epsilon}^{(2)}}{M_{Z_\epsilon}^{(1)^2}}= \lim_{\epsilon\rightarrow0}\frac{1}{2\epsilon Q_{Z}(\epsilon)} = \frac{1}{2\nu} > 0,\end{align*}

and also,

\begin{align*} \lim_{\epsilon\to0}\frac{M_{Z_\epsilon}^{(3)}}{\sigma_\epsilon^6}= \frac{1}{3\nu^{2}\sigma_{W}^{6}} >0.\end{align*}

Therefore, $L_1$ and $L_2$ in Theorem 4.2 are both non-zero, so we expect the residuals of the NG process not to be approximately normally distributed.

Furthermore, since $M^{(n)}_{Z_\epsilon}=\frac{\nu}{b^n}\gamma(n,b\epsilon)$ , where $b=\gamma^{2}/2$ , we have from (5.1) that

\begin{align*} E_\epsilon \leq C \sigma_{W}^{3}\Phi\!\left({-}\frac{3}{2}, \frac{1}{2};\, -\frac{\mu_{W}^{2}}{2\sigma_{W}^{2}}\epsilon\right)\frac{\frac{\nu}{b^{3/2}}\gamma(3/2,b\epsilon)}{\Big(\mu_W^2\frac{\nu}{b^2}\gamma(2,b\epsilon)+\sigma_W^2\frac{\nu}{b}\gamma(1,b\epsilon)\Big)^{3/2}}={\mathcal O}(1),\,\,{ as \, \epsilon\to 0}, \end{align*}

where we have obtained the asymptotic behaviour using $\gamma(s, x) = x^s\,\sum_{k=0}^\infty \frac{({-}x)^k}{k!(s+k)}$ (for positive, real s and x) and (5.7). Hence, as expected, the bound on the distance from Gaussianity, $E_\epsilon$ , does not tend to zero as $\epsilon\to 0$ .

In order to verify this empirically, we generate a random sample of $M=10^4$ residual NG paths with parameters $\mu=0$ , $\mu_w=1$ , $\sigma_W=\nu=2$ , and $\gamma=\sqrt{2}$ , and compare the empirical distribution of the values of the residual at time $t=1$ with a standard normal distribution by standardising the residual values to have zero mean and unit variance. As expected, the resulting histograms are not approximately Gaussian. Figures 4 and 5 show that the distribution of the residual (with truncation level $\epsilon=10^{-6}$ ) is in fact leptokurtic and heavier-tailed than the standard normal. Further simulations confirmed this empirical observation even for smaller truncation levels $\epsilon$ .

Figure 4. Histogram of $M=10^5$ NG residual path values at $t=1$ . The smooth curve represents the standard normal density.

Figure 5. Q–Q plot of $M=10^4$ NG residual path values at $t=1$ .

6.2. Normal tempered stable process

The subordinator for the normal tempered stable (NTS) process is the tempered stable (TS) process TS $(\kappa, \delta, \gamma)$ , for $ \kappa \in (0,1), \ \delta > 0, \ \gamma \geq 0$ , which has a Lévy density

\begin{equation*} Q_{Z}(z) = Az^{-1-\kappa}\exp\!\left({-}\frac{1}{2}\gamma^{\frac{1}{\kappa}}z\right), \quad z > 0, \end{equation*}

where $A = \delta\kappa 2^{\kappa}\Gamma^{-1}(1-\kappa)$ and $\Gamma^{-1}(\cdot)$ is the reciprocal of the gamma function. Here,

\begin{equation*} \lim_{\epsilon\rightarrow 0}\epsilon Q_{Z}(\epsilon) = \lim_{\epsilon\rightarrow 0}A\epsilon^{-\kappa}\exp\!\left({-}\frac{1}{2}\gamma^{\frac{1}{\kappa}}\epsilon\right) = +\infty, \end{equation*}

since $\kappa\in(0,1)$ . Therefore, in view of Corollary 4.1 and Theorem 4.1, the residuals are expected to be approximately Gaussian. Moreover, in this case we can derive a bound on the corresponding marginal convergence rate.

Lemma 6.1. For $\epsilon\in(0,1)$ , let $(Y_{\epsilon}(t))$ denote the standardised truncated process associated to an NVM process subordinated to the residual TS process ${\textrm{TS}}(\kappa, \delta, \gamma)$ . If $B\sim\mathcal{N}(0,1)$ , then the Kolmogorov distance $E_\epsilon$ between $Y_{\epsilon}(1)$ and B satisfies

(6.1) \begin{align} E_{\epsilon} \leq \frac{0.7975\times 2^{\frac{3}{2}}\sqrt{\Gamma(1-\kappa)}}{\sqrt{\delta\kappa\pi\gamma}}\times \Phi_{\epsilon}\gamma\!\left(1-\kappa,\frac{1}{2}\epsilon\gamma^{\frac{1}{\kappa}}\right)^{-\frac{3}{2}} \gamma\!\left(\frac{3}{2}-\kappa, \frac{1}{2}\gamma^{\frac{1}{\kappa}}\epsilon\right), \end{align}

where

\begin{align*} \Phi_{\epsilon} = \Phi\!\left({-}\frac{3}{2}, \frac{1}{2};\, -\frac{\mu_{W}^{2}}{2\sigma_{W}^{2}}\epsilon\right), \end{align*}

and $\Phi(a, b;\, m), \gamma(s,x)$ are the Kummer confluent hypergeometric function and the incomplete lower gamma function, respectively. Furthermore, as $\epsilon\rightarrow 0$ we have

(6.2) \begin{equation} E_{\epsilon} \leq \frac{0.7975\times 2^{\frac{3}{2}-\frac{\kappa}{2}}(1-\kappa)^\frac{3}{2}\sqrt{\Gamma(1-\kappa)}}{\big(\frac{3}{2}-\kappa\big)\sqrt{\delta\kappa\pi}}\epsilon^{\frac{\kappa}{2}}+\mathcal{O}\big(\epsilon^{1+\frac{\kappa}{2}}\big). \end{equation}

Proof. From Theorem 5.1 we obtain $E_{\epsilon} \leq C \sigma_{W}^{3}\Phi_{\epsilon}M^{\big(\frac{3}{2}\big)}_{Z_\epsilon}/\sigma_{\epsilon}^{3}$ , and from (2.9) it follows that

\begin{equation*} \sigma_{\epsilon}^{2} = \mu_{W}^{2}M_{Z_{\epsilon}}^{(2)} + \sigma_{W}^{2}M_{Z_{\epsilon}}^{(1)} \geq \sigma_{W}^{2}M_{Z_{\epsilon}}^{(1)}, \end{equation*}

where the residual first moment is given by

\begin{equation*} M^{(1)}_{Z_{\epsilon}} = \int_{0}^{\epsilon}zAz^{-1-\kappa}\exp\!\left({-}\frac{1}{2}\gamma^{\frac{1}{\kappa}}z\right)dz=A \gamma^{\frac{\kappa-1}{\kappa}}2^{1-\kappa}\gamma\!\left(1-\kappa,\frac{1}{2}\epsilon\gamma^{\frac{1}{\kappa}}\right). \end{equation*}

To find $M_{Z_{\epsilon}}^{\left(\frac{3}{2}\right)}$ , substitute the Lévy density of the TS process described earlier,

\begin{align*} M_{Z_{\epsilon}}^{\left(\frac{3}{2}\right)} &= A\int_{0}^{\epsilon}z^{\frac{1}{2}-\kappa}e^{-\frac{1}{2}\gamma^{\frac{1}{\kappa}}z}dz =A\!\left(\frac{1}{2}\gamma^{\frac{1}{\kappa}}\right)^{\kappa - \frac{3}{2}}\gamma\!\left(\frac{3}{2}-\kappa, \frac{1}{2}\gamma^{\frac{1}{\kappa}}\epsilon\right). \end{align*}

Substituting this in (5.5) and noting that $A = \delta \kappa 2^{\kappa}\Gamma^{-1}(1-\kappa)$ and $C = 0.7975 \times 2\sqrt{2/\pi}$ yields

\begin{align*} E_{\epsilon} \leq 0.7975\times \frac{2^{\frac{3}{2}}\sqrt{\Gamma(1-\kappa)}}{\sqrt{\delta\kappa\pi\gamma}}\Phi_{\epsilon}\gamma\!\left(1-\kappa,\frac{1}{2}\epsilon\gamma^{\frac{1}{\kappa}}\right)^{-\frac{3}{2}}\gamma\!\left(\frac{3}{2}-\kappa, \frac{1}{2}\gamma^{\frac{1}{\kappa}}\epsilon\right), \end{align*}

as claimed. Finally, recalling the series expansion for $\gamma(s,x)$ from the previous section, the asymptotic expression in (5.2) leads to (6.2).

Next we examine the empirical accuracy of the Gaussian approximation in this case. Figures 6 and 7 show the empirical distribution of the residual NTS process at $t=1$ , with parameter values $\mu=0$ , $\mu_W=\delta=1$ , $\sigma_W=2$ , $\kappa=1/2$ , $\gamma=1.35$ , and truncation level $\epsilon=10^{-6}$ .

Figure 6. Histogram of $M = 10^5$ NTS residual path values at $t=1$ . The smooth curve represents the standard normal density.

Figure 7. Q–Q plot of $M = 10^4$ NTS residual path values at $t=1$ .

With the same parameter values, Figure 8 shows the behaviour of the bound in (6.1) and the first term in the asymptotic bound (6.2).

Figure 8. Plot of the finite- $\epsilon$ bound in (6.1) and the first term in the asymptotic bound (6.2) for the approximation error $E_\epsilon$ in Lemma 6.1.

6.3. Generalised hyperbolic process

Finally, we consider the general class of generalised hyperbolic (GH) processes. The subordinator in this case is the generalised inverse Gaussian (GIG) process GIG( $\lambda, \delta, \gamma$ ), with constraints on parameter values as detailed in [Reference Godsill and Kndap22], and with Lévy density given by [Reference Godsill and Kndap22]

\begin{equation*} Q_{Z}(z) = \frac{\exp \!\left({-}z\gamma^{2}/2\right)}{z} \left[\max(0, \lambda)+\frac{2}{\pi^{2}}\int_{0}^{\infty}\frac{1}{y\big|H_{|\lambda|}(y)\big|^{2}} \exp \Big( -\frac{zy^{2}}{2\delta^{2}}\Big) dy\right]\mathbb{1}(z > 0), \end{equation*}

where $H_{v}(z)$ is the Hankel function of real order v. A direct verification of the sufficient condition in Corollary 4.1 is readily obtained as

\begin{align*} \lim_{\epsilon\rightarrow0}\epsilon Q_{Z}(\epsilon) &= \lim_{\epsilon\rightarrow 0}\exp\Big({-}\frac{\gamma^{2}}{2}\epsilon\Big)\max(0,\lambda) + \frac{2}{\pi^{2}}\lim_{\epsilon\rightarrow0}\int_{0}^{\infty}\frac{1}{y\big|H_{|\lambda|}(y)\big|^{2}} \exp\Big({-}\frac{\epsilon\gamma^{2}}{2} -\frac{\epsilon y^{2}}{2\delta^{2}}\Big) dy \\ &= \max(0,\lambda) + \frac{2}{\pi^{2}}\int_{0}^{\infty}\frac{1}{y\big|H_{|\lambda|}(y)\big|^{2}}dy\\ &= +\infty, \end{align*}

since ${y\big|H_{\nu}(y)\big|^{2}}$ is non-zero for $z\in [z_1,\infty)$ where

\begin{equation*}z_1 = \left(\frac{ 2^{1-2\nu}\pi}{\Gamma^2(\nu)}\right)^{1/(1-2\nu)};\end{equation*}

see e.g. [Reference Godsill and Kndap22, Theorem 2] and [Reference Watson45]. Once again, in view of Corollary 4.1 and Theorem 4.1, the residuals are expected to be approximately Gaussian. Indeed, in this case we can derive the following bound on the corresponding marginal convergence rate.

Lemma 6.2. For $\epsilon\in(0,1)$ , let $Y_{\epsilon}(t)$ denote the standardised truncated process associated to an NVM process subordinated to the residual ${\textrm{GIG}}(\lambda, \delta, \gamma)$ process. If $B\sim\mathcal{N}(0,1)$ , then for any $z_0\in(0,\infty)$ , the Kolmogorov distance $E_\epsilon$ between $Y_{\epsilon}(1)$ and B can be bounded as

(6.3) \begin{align} E_{\epsilon} &\leq \frac{{0.7975\Phi_\epsilon\gamma^{3/2}}}{{\textrm{erf}}\left(\gamma\frac{\sqrt{\epsilon}}{\sqrt{2}}\right)^{\frac{3}{2}}} \biggl( \frac{2\max(0,\lambda)}{{\tilde{\pi}}(b\delta)^{3/2}}\gamma\!\left(\frac{3}{2}, b\epsilon\right) + \frac{2^{|\lambda|{+1}}\delta^{2|\lambda|-\frac{3}{2}}\Gamma(|\lambda|)} {{\pi^{2}\tilde{\pi}} H_{0}z_{0}^{2|\lambda|-1}b^{3/2-|\lambda|}}\times \gamma\!\left(\frac{3}{2}-|\lambda|, b\epsilon\right) \nonumber\\ &\qquad \qquad\qquad\qquad \qquad\qquad + \frac{1}{{\tilde{\pi}^{4}}H_{0}b\sqrt{\delta}}\gamma\!\left(1, b\epsilon\right)\biggr),\qquad{for} \;|\lambda| \leq \frac{1}{2}, \nonumber\\[4pt] E_{\epsilon} & \leq \frac{{0.7975\Phi_\epsilon\left(\gamma H_0\right)^{3/2}}}{{\textrm{erfc}}\left( \frac{z_{0}}{\delta\sqrt{2}}\sqrt{\epsilon}\right)^{\frac{3}{2}}{\textrm{erf}}(\gamma\frac{\sqrt{\epsilon}}{\sqrt{2}})^{\frac{3}{2}}} \biggl( {\frac{\max(0,\lambda)\pi}{(b\delta)^{\frac{3}{2}}}} \times \gamma\!\left(\frac{3}{2}, b\epsilon\right) \nonumber\\[4pt] &\qquad \qquad\qquad\qquad \qquad\qquad + {\frac{\tilde{\pi}}{b\sqrt{\delta}}} \times \gamma\!\left(1, b\epsilon\right)\biggr), \qquad {for}\;|\lambda| > \frac{1}{2}, \end{align}

with $b = \frac{\gamma^{2}}{2}$ , $\tilde{\pi} = \sqrt{\frac{\pi}{2}}$ , $H_{0} = z_{0}\big|H_{|\lambda|}(z_{0})\big|^{2},$ and

\begin{align*} \Phi_{\epsilon} = \Phi\!\left({-}\frac{3}{2}, \frac{1}{2};\, -\frac{\mu_{W}^{2}}{2\sigma_{W}^{2}}\epsilon\right), \end{align*}

where, using the standard definitions ${{\textrm{erf}}} (x) = \frac{2}{\sqrt\pi}\int_0^x e^{-t^2}\,\textrm{dt}$ and ${{\textrm{erfc}}} (x)=1-{{\textrm{erf}}} (x)$ , we denote by $\Phi(a, b;\, m)$ , $\gamma(s,x)$ , ${\textrm{erf}}(x)$ , and ${\textrm{erfc}}(x)$ the Kummer confluent hypergeometric function, the incomplete lower gamma function, the error function, and the complementary error function, respectively. Furthermore, as $\epsilon\rightarrow 0$ we have

(6.4) \begin{align} E_{\epsilon} & \leq {\frac{0.7975}{\tilde{\pi}^{\frac{5}{2}}bH_{0}\sqrt{\delta}}}\epsilon^{\frac{1}{4}}+\mathcal{O}\big(\epsilon^{\frac{5}{4}}\big), \qquad {for}\;|\lambda| \leq \frac{1}{2}, \nonumber \\ E_{\epsilon} &\leq {\frac{0.7975 \tilde{\pi}^{\frac{5}{2}}H_{0}^{\frac{3}{2}}}{b\sqrt{\delta}}}\epsilon^{\frac{1}{4}} + \mathcal{O}\big(\epsilon^{\frac{5}{4}}\big), \qquad {for}\;|\lambda| > \frac{1}{2}. \end{align}

Proof. Recalling the definition of the Jaeger integral as

\begin{equation*}J(z)=\int_0^\infty\frac{e^{-\frac{x^2z}{2\delta^2}}}{x|H_{|\lambda|}(x)|^2}dx,\end{equation*}

we have from the bounds obtained in Appendix A, for $|\lambda|\leq 1/2$ ,

\begin{equation*} J(z) \geq {\delta{\left(\frac{\pi}{2}\right)}^{3/2}z^{-\frac{1}{2}}}, \end{equation*}

and for $|\lambda|>1/2$ ,

\begin{equation*} J(z) \geq \frac{\delta^{2|\lambda|}2^{|\lambda|-1}}{H_{0}z_{0}^{2|\lambda|-1}}z^{-|\lambda|}\gamma\!\left(|\lambda|, \frac{z_{0}^{2}}{2\delta^{2}}z\right)+\frac{\delta}{H_{0}\sqrt{2}}z^{-\frac{1}{2}}\Gamma\!\left(\frac{1}{2},\frac{z_{0}^{2}}{2\delta^{2}}z\right). \end{equation*}

Noting that the variance of the GH process satisfies $\sigma_{\epsilon}^{2} = \mu_{W}^{2}M^{(2)}_{Z_{\epsilon}} + \sigma_{W}^{2}M^{(1)}_{Z_{\epsilon}} \geq \sigma_{W}^{2}M^{(1)}_{Z_{\epsilon}}$ , for $|\lambda|\leq \frac{1}{2}$ we obtain

\begin{align*} M^{(1)}_{Z_{\epsilon}} &= \int_{0}^{\epsilon}\exp \!\left({-}\frac{\gamma^{2}z}{2}\right)\left[\max(0,\lambda) + \frac{2}{\pi^{2}}J(z)\right]dz\\ &\geq \frac{2}{\pi^{2}}\int_{0}^{\epsilon}\exp \!\left({-}\frac{\gamma^{2}z}{2}\right)J(z)dz\\ &\geq {\frac{\delta}{\gamma}{\textrm{erf}}\left(\frac{\gamma\sqrt{\epsilon}}{\sqrt{2}}\right)}, \end{align*}

and similarly for $|\lambda|>\frac{1}{2}$ we obtain

\begin{align*} M^{(1)}_{Z_{\epsilon}} &\geq \frac{2}{\pi^{2}}\int_{0}^{\epsilon}e^{-\frac{z\gamma^{2}}{2}}\left[\frac{\delta^{2|\lambda|}2^{|\lambda|-1}}{H_{0}z_{0}^{2|\lambda|-1}}z^{-|\lambda|}\gamma\!\left(|\lambda|, \frac{z_{0}^{2}}{2\delta^{2}}z\right)+\frac{\delta}{H_{0}\sqrt{2}}z^{-\frac{1}{2}}\Gamma\!\left(\frac{1}{2},\frac{z_{0}^{2}}{2\delta^{2}}z\right)\right]dz\nonumber\\ &\geq \frac{2}{\pi^{2}}\int_{0}^{\epsilon}e^{-\frac{z\gamma^{2}}{2}}\frac{\delta}{H_{0}\sqrt{2}}z^{-\frac{1}{2}}\Gamma\!\left(\frac{1}{2},\frac{z_{0}^{2}}{2\delta^{2}}z\right)dz\\ &\geq \frac{2\delta}{H_{0}\pi\gamma}{\textrm{erfc}}\left(\frac{z_{0}}{\delta\sqrt{2}}\sqrt{\epsilon}\right){\textrm{erf}}\left(\frac{\gamma\sqrt{\epsilon}}{\sqrt{2}}\right). \end{align*}

Therefore, we have the following bound on the variance:

\begin{align*} \sigma_{\epsilon}^{2} \geq \begin{cases} {\frac{\sigma_{W}^{2}\delta}{\gamma}}{\textrm{erf}}\left(\frac{\gamma\sqrt{\epsilon}}{\sqrt{2}}\right), &\mbox{for}\;|\lambda| \leq \frac{1}{2},\\[4pt] \frac{\sigma_{W}^{2}\delta}{H_{0}\tilde{\pi}^{2}\gamma}{\textrm{erfc}}\left( \frac{z_{0}}{\delta\sqrt{2}}\sqrt{\epsilon}\right) {\textrm{erf}}\left(\frac{\gamma\sqrt{\epsilon}}{\sqrt{2}}\right), &\mbox{for}\;|\lambda| > \frac{1}{2}. \end{cases} \end{align*}

Finding $M_{Z_{\epsilon}}^{\left(\frac{3}{2}\right)}$ in line with Theorem 5.1, we have

\begin{align*} M_{Z_{\epsilon}}^{\left(\frac{3}{2}\right)} &= \int_{0}^{\epsilon}z^{\frac{3}{2}}Q_{Z}(dz)\\ &=\max(0,\lambda) \int_{0}^{\epsilon}z^{\frac{1}{2}}\exp\!\left({-}\frac{\gamma^{2}}{2}z\right)dz +\frac{2}{\pi^{2}}\int_{0}^{\epsilon}z^{\frac{1}{2}}\exp\!\left({-}\frac{\gamma^{2}}{2}z\right)J(z)dz \\ &= \mathcal{S}_{1} + \mathcal{S}_{2}. \end{align*}

Writing $b = \frac{\gamma^{2}}{2}$ , we have

\begin{equation*} \mathcal{S}_{1} = \frac{\max(0,\lambda)}{b\sqrt{b}}\gamma\!\left(\frac{3}{2}, b\epsilon\right); \end{equation*}

for $|\lambda| > \frac{1}{2}$ we have ${J(z)\leq \delta{\left(\frac{\pi}{2}\right)}^{3/2}z^{-\frac{1}{2}}}$ and

\begin{equation*} \mathcal{S}_{2}\leq \frac{2}{\pi^{2}}\delta \!\left(\frac{\pi}{2}\right)^{\frac{3}{2}}\int_{0}^{\epsilon}\exp\!\left({-}bz\right)dz = \frac{\delta}{\sqrt{2\pi} b}\gamma\!\left(1, b\epsilon\right), \end{equation*}

and for $|\lambda| \leq \frac{1}{2}$ ,

\begin{align*} {\mathcal S}_{2} &\leq \frac{2^{|\lambda|}\delta^{2|\lambda|}}{\pi^{2}H_{0}z_{0}^{2|\lambda|-1}}\int_{0}^{\epsilon}z^{\frac{1}{2}-|\lambda|}e^{-bz}\gamma\!\left(|\lambda|, \frac{z_{0}^{2}}{2\delta^{2}}z\right)dz + \frac{\sqrt{2}\delta}{\pi^{2}H_{0}}\int_{0}^{\epsilon}e^{-bz}\Gamma\!\left(\frac{1}{2}, \frac{z_{0}^{2}}{2\delta^{2}}z\right)dz\\ &\leq \frac{2^{|\lambda|}\delta^{2|\lambda|}\Gamma(|\lambda|)}{\pi^{2}H_{0}z_{0}^{2|\lambda|-1}}\int_{0}^{\epsilon}z^{\frac{1}{2}-|\lambda|}e^{-bz}dz + \frac{\delta}{\pi\tilde{\pi}H_{0}}\int_{0}^{\epsilon}e^{-bz}dz\\ &\leq \frac{2^{|\lambda|}\delta^{2|\lambda|}\Gamma(|\lambda|)}{\pi^{2}H_{0}z_{0}^{2|\lambda|-1}b^{\frac{3}{2}-|\lambda|}}\gamma\!\left(\frac{3}{2}-|\lambda|, b\epsilon\right) + \frac{\delta}{\pi\tilde{\pi}H_{0}b}\gamma\!\left(1, b\epsilon\right). \end{align*}

Combining the above bounds, for $|\lambda|\leq 1/2$ ,

\begin{equation*} M_{Z_{\epsilon}}^{\left(\frac{3}{2}\right)} \leq \frac{\max(0,\lambda)}{b\sqrt{b}}\gamma\!\left(\frac{3}{2}, b\epsilon\right) + \frac{2^{|\lambda|}\delta^{2|\lambda|}\Gamma(|\lambda|)}{\pi^{2}H_{0}z_{0}^{2|\lambda|-1}b^{\frac{3}{2}-|\lambda|}}\gamma\!\left(\frac{3}{2}-|\lambda|, b\epsilon\right) + \frac{\delta}{\pi\tilde{\pi}H_{0}b}\gamma\!\left(1, b\epsilon\right), \end{equation*}

and for $|\lambda|>1/2,$

\begin{equation*} M_{Z_{\epsilon}}^{\left(\frac{3}{2}\right)}\leq \frac{\max(0,\lambda)}{b\sqrt{b}}\gamma\!\left(\frac{3}{2}, b\epsilon\right)+ {\frac{\delta}{\sqrt{2\pi}b}}\gamma\!\left(1, b\epsilon\right). \end{equation*}

Finally, substituting these bounds for $M_{Z_{\epsilon}}^{\left(\frac{3}{2}\right)}$ into (5.1), we obtain the bounds as stated in (6.3). The series expansions of the gamma and hypergeometric functions then lead to the asymptotic expansion (6.4).

Once again, we examine the validity of the Gaussian approximation empirically. Figures 9 and 10 show the empirical distribution of the residual GH process at time $t = 1$ , with parameter values $\mu = 0$ , $\mu_W =1$ , $\sigma_W = 2$ , $\delta=1.3$ , $\gamma=\sqrt{2}$ , $\lambda=0.2$ , and truncation level $\epsilon=10^{-6}$ .

Figure 9. Histogram of $M = 10^5$ GH residual path values at $t=1$ . The smooth curve represents the standard normal density.

Figure 10. Q–Q plot of $M = 10^4$ GH residual path values at $t=1$ .

With the same parameter values, Figure 11 shows behaviour of the bound in (6.3) and the first term in the asymptotic bound (6.4).

Figure 11. Plot of the finite- $\epsilon$ bound in (6.3) and the first term in the asymptotic bound (6.4) for the approximation error $E_\epsilon$ in Lemma 6.2.

Note that the bounds (6.3) and (6.4) are discontinuous at $|\lambda| = \frac{1}{2}$ . This discrepancy is likely due to the upper bound for $M_{Z_{\epsilon}}^{(1)}$ in the case $|\lambda| > \frac{1}{2}$ . Although we do expect this could be improved, obtaining such refined bounds is beyond the scope of this paper.

7.1. Shot-noise representation of SDE

In order to apply the Lévy state space model to NVM Lévy processes, we first establish their representation as generalised shot-noise series. Theorem 7.1 gives the result for a general integrand.

Theorem 7.1. Let (X(u)) be an NVM process generating the filtration $(\mathcal{F}_{t})$ . Suppose $\boldsymbol{f}_{t} : [0, \infty) \rightarrow \mathbb{R}^{P}$ is an $L_{2}$ deterministic function, and let $\boldsymbol{I}({\textbf{f}_{t}})$ denote the integral

\begin{equation*}\boldsymbol{I}({\boldsymbol{f}_{t}}) = \int_{0}^{T}\boldsymbol{f}_{t}(u)dX(u), \qquad 0\leq t\leq T. \end{equation*}

Then $\boldsymbol{I}({\boldsymbol{f}_{t}})$ admits the series representation

\begin{equation*} \boldsymbol{I}({\boldsymbol{f}_{t}}) = \sum_{i=1}^{\infty}X_i\boldsymbol{f}_{t}(TV_{i})\mathbb{1} \!\left( V_{i} \leq \frac{t}{T}\right), \end{equation*}

where

\begin{align*} X_i=\mu_{W} Z_i+\sigma_{W}\sqrt{Z_i}U_{i}, \end{align*}

and $\mu_{W} \in \mathbb{R}, \ \sigma_{W} \in (0,\infty)$ are the variance-mean mixture parameters, $V_{i} \overset{i.i.d.}{\sim} \mathcal{U}[0,1]$ are normalised jump times, $Z_i$ are the jumps of the subordinator process arranged in non-increasing order, and $U_i \overset{i.i.d.}{\sim} \mathcal{N}(0, 1)$ .

Proof. Arguing as in [Reference Rosinski39, Section 7], we can extend (2.14) to any NVM process defined on $t\in[0, T]$ , with $\tilde{V}_i = TV_i\sim\mathcal{U}[0,T]$ , to obtain, for all $u\in[0,T]$ ,

\begin{align*} X(u) &= \sum_{i=1}^{\infty}\left[\mu_{W}Z_{i}+\sigma_{W}\sqrt{Z_{i}}U_{i}\right]\mathbb{1}(\tilde{V}_{i}\leq u)\\ &= \mu_{W}\sum_{i=1}^{\infty}Z_{i}\mathbb{1}(\tilde{V}_{i}\leq u) + \sigma_{W}\sum_{i=1}^{\infty}\sqrt{Z_{i}}U_{i}\mathbb{1}(\tilde{V}_{i}\leq u)\\ &= \mu_{W}M(u) + \sigma_{W}S(u), \end{align*}

where M(u) is a subordinator Lévy process and S(u) a symmetric Gaussian mixture process. Therefore,

\begin{equation*} dX(u) = \mu_{W}dM(u) + \sigma_{W}dS(u), \end{equation*}

and hence we obtain the following representation for $\boldsymbol{I}({\boldsymbol{f}_{t}})$ :

\begin{equation*} \boldsymbol{I}({\boldsymbol{f}_{t}}) = \mu_{W}\int_{0}^{T}\boldsymbol{f}_{t}(u)dM(u) + \sigma_{W}\int_{0}^{T}\boldsymbol{f}_{t}(u)dS(u). \end{equation*}

From [Reference Barndorff-Nielsen and Shephard3, Corollary 8.2], a stochastic integral with respect to a Lévy subordinator admits the a.s. generalised shot-noise representation

\begin{equation*} \int_{0}^{T}\boldsymbol{f}_{t}(u)dM(u) = \sum_{i=1}^{\infty}Z_{i}\boldsymbol{f}_{t}(\tilde{V}_{i})\mathbb{1}(\tilde{V}_{i} \leq t). \end{equation*}

Similarly, Rosinski proves in [Reference Rosinski38, Section 4] the a.s. representation of stochastic integrals with respect to Lévy processes of type G. These are symmetric normal variance mixture processes such as the symmetric Student-t, symmetric $\alpha$ -stable, and Laplace which are special cases of NVM Lévy processes having $\mu_W=0$ , in the form

\begin{equation*} \int_{0}^{T}\boldsymbol{f}_{t}(u)dS(u) = \sum_{i=1}^{\infty}\sqrt{Z_{i}}U_{i}\boldsymbol{f}_{t}(\tilde{V}_{i})\mathbb{1}(\tilde{V}_{i} \leq t). \end{equation*}

Combining the last three expressions proves the claimed result.

Applying the result of the theorem to X(t) in (7.1), with $\boldsymbol{f}_{t}(u) = e^{\boldsymbol{A}(t-u)}\boldsymbol{h}\mathbb{1}(s\leq u \leq t),$ yields

\begin{equation*} \boldsymbol{I}\left(\boldsymbol{f}_{t}\right) = \sum_{i=1}^{\infty}\left[\mu_{W}Z_{i}+\sigma_{W}\sqrt{Z_{i}}U_{i}\right]e^{\boldsymbol{A}(t-\tilde{V}_{i})}\boldsymbol{h}{\mathbb{1}(\tilde{V}_i\in (s,t ])} \end{equation*}

with $\tilde{V}_{i} \overset{i.i.d.}{\sim} \mathcal{U}(0,T]$ as before.