2. Double beta subordinators and double hypergeometric Lévy processes

Let us start by introducing some notation. As already alluded to above, ${_2F_1}$ stands for the Gauss hypergeometric function. However, we will, more generally, use ${_pF_q}$ for the generalised hypergeometric function:

\begin{equation*}{}_{p}F_{q} \left[\begin{array}{l} {a_1\cdots a_p}\\[2pt] {b_1\cdots b_q}\end{array}\,;\ {z}\right] s=\sum _{k=0}^{\infty } \frac{\left(a_1\right)_k\ldots \left(a_p\right)_k}{\left(b_1\right)_k\ldots \left(b_q\right)_k} \frac{z^k}{k!},\end{equation*}

where $(a)_k=a(a+1)\cdots(a+k-1)$ denotes the rising factorial power and is frequently called the Pochhammer symbol in the theory of special functions.

Next, suppose that A and B are two countably infinite subsets of $\mathbb{R}$, bounded from above and discrete. Then B is said to interlace with A if $\max A\geq \max B$ and if for the order-preserving enumerations $(a_n)_{n\in \mathbb{Z}_{\leq 0}}$ and $(b_n)_{n\in \mathbb{Z}_{\leq 0}}$ of A and B, respectively, we have $a_{n-1}<b_n<a_n$ for all $n\in \mathbb{Z}_{\leq 0}$, the non-positive integers. If A and B interlace, then necessarily they are disjoint.

Below we give our first key result of this paper. It identifies a class of subordinators having a Laplace transform of the form (1.3). Before stating it, recall that (i) a real meromorphic function is a meromorphic function that maps the real line excluding the poles into the real line, (ii) a Pick (or Nevanlinna–Pick) function is an analytic map defined on a subset of $\mathbb{C}$ containing the open upper complex half-plane and that maps the latter into itself [Reference Schilling, Song and Vondraček27, Definition 6.7], and (iii) the non-constant complete Bernstein functions are precisely the Pick functions that are non-negative on $(0,\infty)$ [Reference Schilling, Song and Vondraček27, Theorem 6.9] (the latter omits the qualification ‘non-constant’, but without it, it is clearly false).

We say that a subordinator S having the Laplace exponent $\mathtt{B}(\alpha, \beta, \gamma, \delta;\cdot)$ as in (1.3) is a double beta subordinator. The class of quadruples $(\alpha,\beta,\gamma,\delta)\in [0,\infty)^4$ satisfying conditions (i)–(iii) of Theorem 2.1 is denoted $\mathfrak{G}$. Note, we obtain the beta subordinators of (1.2) as a special case of double beta subordinators in the class $\mathfrak{G}$ by taking $\alpha=\gamma=0$ and substituting $\beta\rightarrow \beta+\gamma $, $\delta\rightarrow \beta$ in (1.3) (it satisfies (i) with $k=0$). The class of those quadruples $(\alpha,\beta,\gamma,\delta)\in (0,\infty)^4$ that satisfy the same conditions (i)–(iii) except that $\leq$ is replaced by $<$ is denoted $\mathfrak{G}^\circ$; it identifies precisely those quadruples $(\alpha,\beta,\gamma,\delta)\in (0,\infty)^4$ for which $\{\alpha-\beta,\gamma-\delta,\gamma-\alpha,\delta-\alpha, \gamma-\beta,\delta-\beta\}\cap \mathbb{Z}=\emptyset$ and that make $\mathtt{B}(\alpha, \beta, \gamma, \delta;\cdot)$ of (1.3) into a non-constant complete Bernstein function.

We should point out that our focus in Theorem 2.1 on double beta subordinators corresponding to (1.3) with $(\alpha,\beta,\gamma,\delta)\in \mathfrak{G}^\circ$, rather than more generally $(\alpha,\beta,\gamma,\delta)\in \mathfrak{G}$, is a matter of convenience: to describe the resulting subordinators for all the possible constellations that $\mathfrak{G}$ admits would be prohibitive in scope. For instance, as already remarked, $\mathfrak{G}$ includes the beta subordinators, but also the trivial (zero) subordinators killed at an independent exponential random time, etc. Besides, the conditions ‘$\alpha,\beta,\gamma,\delta>0$ and $\{\alpha-\beta,\gamma-\delta,\gamma-\alpha,\delta-\alpha, \gamma-\beta,\delta-\beta\}\cap \mathbb{Z}=\emptyset$’ serve to ensure that there are no ‘cancellations or reductions’ of the gamma factors in (1.3), i.e. $\mathtt{B}(\alpha, \beta, \gamma, \delta;\cdot)$ is ‘truly’ a quotient of four gammas. On the other hand, our insistence on the interlacing property, beyond merely demanding that $\alpha,\beta,\gamma,\delta>0$ and $\{\alpha-\beta,\gamma-\delta,\gamma-\alpha,\delta-\alpha, \gamma-\beta,\delta-\beta\}\cap \mathbb{Z}=\emptyset$, i.e. on the ‘non-constant complete Bernstein’ property, is more than just convenience: it is not clear to us how to establish the Bernstein property of $\mathtt{B}(\alpha, \beta, \gamma, \delta;\cdot)$ from (1.3) when the complete Bernstein property fails (if this can in fact happen; see the comments below Proposition 4.1).

From Vigon’s theory of philanthropy [Reference Vigon29], and in particular the fact that double beta subordinators corresponding to $\mathfrak{G}$ are completely monotone, thereby having associated Lévy measures that are absolutely continuous with non-increasing densities, we can now define a new class of Lévy processes from the double beta subordinator class as follows; see [Reference Kyprianou20, Section 6.6].

In light of the above, we call a Lévy process, $\xi$, of the type introduced in the preceding corollary a double hypergeometric Lévy process. The following proposition gathers some basic properties of our new class of double hypergeometric Lévy processes (and accordingly justifies the name ‘double hypergeometric’).

The exclusion of the parameters corresponding to $\mathfrak{G}{\setminus}\mathfrak{G}^\circ$ in the statement of Proposition 2.1(i) is not without cause: for instance, the functions 1 and $\textrm{id}_{[0,\infty)}$ are of the double beta class, but they do not yield a meromorphic Lévy process, at least not in the sense of [Reference Kuznetsov, Kyprianou and Pardo16]; similarly, the expression of Proposition 2.1(i) for the Lévy density does not hold true across the whole class of double hypergeometric Lévy processes corresponding to $\mathfrak{G}$.

In the next section we shall briefly discuss examples of how the double hypergeometric Lévy process emerges in the setting of certain self-similar Markov processes. As we will see in the exposition there, for the theory of the latter processes, an important quantity of interest is the exponential functional of Lévy processes. Below we show that it is possible to characterise the exponential functional of double hypergeometric Lévy processes via an explicit Mellin transform. Our method is based on the ‘verification’ result of [Reference Kuznetsov and Pardo19, Section 8.1]. In the following we use $\textbf{P}$ to denote the law of the double hypergeometric Lévy process.

Proposition 2.2. Let $\{(\alpha,\beta,\gamma,\delta),(\hat{\alpha},\hat{\beta},\hat{\gamma},\hat{\delta})\}\subset \mathfrak{G}$, and $\xi$ be a double hypergeometric Lévy process with these parameters as in Corollary 2.1. Assume $\hat{\gamma}\leq \hat{\delta}$, $\hat{\alpha}\leq \hat{\beta}$, and $0<\hat{\gamma}<\hat{\alpha}$ (in fact the very last condition is automatic if even $(\hat{\alpha},\hat{\beta},\hat{\gamma},\hat{\delta})\in \mathfrak{G}^\circ$, while the first two are without loss of generality). Then, for a given ${c}\in (0,\infty)$, with $\zeta$ being the lifetime of $\xi$, for $s\in \mathbb{C}$ with $\textrm{Re}(s)\in (0,1+\hat{\gamma}{c})$,

(2.3) \begin{align}\textbf{E}\left[\left(\int_0^\zeta \textrm{e}^{-\xi_t/{c}}\,\textrm{d}t\right)^{s-1}\right]=C\Gamma(s)\frac{G({c} \gamma+s;{c})G({c} \delta+s;{c})}{G({c}\alpha+s;{c})G({c} \beta+s;{c})}\frac{G(1+{c}\hat{\alpha}-s;{c})G(1+{c} \hat{\beta}-s;{c})}{G(1+{c} \hat{\gamma}-s;{c})G(1+{c} \hat{\delta}-s;{c})},\end{align}

where C is a normalisation constant such that the left- and right-hand sides agree at $s= 1$ (i.e., both are equal to unity), and G is Barnes’ double gamma function (see [Reference Kuznetsov and Pardo19, p. 121], and further references therein, for its definition and basic properties).

As indicated in the introduction, before moving to the proofs of the main results in this section, we will first look at how double hypergeometric Lévy processes appear naturally in the setting of a special family of self-similar Markov processes. Our first step in this direction is to briefly introduce the latter processes.

3. Self-similar Markov processes

Let $Y=(Y_s, s\geq 0)$ be a pssMp with self-similarity index $\alpha\in (0,\infty)$ and probabilities $(\mathsf{P}_y, y>0)$. Then Y enjoys a bijection with the class of Lévy processes (as presented in the introduction of this article) via the Lamperti transform. Lamperti’s result gives a bijection between the class of pssMps and the class of Lévy processes, possibly killed at an independent exponential time with cemetery state $-\infty$, such that, under $\mathsf{P}_y$, $y>0$, with the convention $\xi_\infty=-\infty$,

(3.1) \begin{equation}Y_t=\exp(\xi_{\varphi_t}),\qquad t\in [0,\infty),\end{equation}

where $\varphi_t=\inf\{s>0\,:\, \int_0^s \exp(\alpha \xi_u) \, \textrm{d} u >t \}$ and the Lévy process $\xi$ is started in $\log y$.

In recent work, e.g. [Reference Chaumont, Pantí and Rivero8-Reference Dereich, Döring and Kyprianou10, Reference Graversen and Vuolle-Apiala12, Reference Kiu15, Reference Kuznetsov, Kyprianou, Pardo and Watson18, Reference Vuolle-Apiala and Graversen30], effort has been invested in extending the theory of pssMps to the setting of $\mathbb{R}$. Analogously to Lamperti’s representation, for a real self-similar Markov process (rssMp), say Y, there is a Markov additive process $((\xi_t,J_t), t\geq 0)$ on $\mathbb{R}\times \{-1,1\}$ such that, again under the convention $\xi_\infty=-\infty$,

(3.2) \begin{align}Y_t = J_{\varphi_t}\exp\bigl( \xi_{\varphi_t}\bigr) ,\qquad t\in [0,\infty),\end{align}

where $\varphi_t=\inf\{s>0 \,:\, \int_0^s \exp(\alpha \xi_u) \, {\textrm{d}} u >t\}$ and $(\xi_0,J_0)=({\kern-1.5pt}\log |Y_0|,[Y_0])$ with

\begin{equation*}[z]=\begin{cases} 1 &\mbox{ if } z>0, \\[3pt] -1 &\mbox{ if }z<0.\end{cases}\end{equation*}

The representation (3.2) is known as the Lamperti–Kiu transform. Here, by a Markov additive process (MAP) we mean the regular strong Markov process with probabilities $\textbf{P}_{x,i}$, $x\in\mathbb{R}$, $i\in\{-1,1\}$, such that $(J_t, t\geq 0)$ is a continuous-time Markov chain on $\{-1,1\}$ (called the modulating chain) and, for any $i\in \{-1,1\}$ and $t\geq 0$, given $\{J_t=i\}$, the pair $(\xi_{t+s}-\xi_t,J_{t+s})_{s\geq 0}$ is independent of the past up to t and has the same distribution as $(\xi_s, J_s)_{s\geq 0}$ under $\textbf{P}_{0,i}$. If the MAP is killed, then $\xi$ is sent to the cemetery state $-\infty$. All background results for MAPs that relate to the present article can be found in the appendix of [Reference Dereich, Döring and Kyprianou10].

The analogue of the characteristic exponent of a Lévy process for the above MAP is provided by the matrix-valued function ${\boldsymbol{\Psi}}$ such that, for all $t\in [0,\infty)$, $\{i,j\}\subset \{1,-1\}$, $\theta\in \mathbb{R}$, we have $\textbf{E}_{0,i}[\textrm{e}^{\textrm{i}\theta \xi_t};J_t=j]=(\textrm{e}^{{\boldsymbol{\Psi}}(\theta)t})_{ij}$ and ${\boldsymbol{\Psi}}(\theta)= - \textrm{diag}(\Psi_1(\theta),\Psi_{-1}(\theta))+Q\circ G(\theta)$, where Q is the generator matrix of J; G is the matrix of the characteristic functions of the extra jumps that are inserted into the path of the MAP $\xi$ when J transitions, the diagonal elements of G being set to unity; and $\Psi_1$ and $\Psi_{-1}$ are the characteristic exponents of the Lévy processes that govern the evolution of the MAP component $\xi$ when it is in states 1 and $-1$, respectively. The symbol $\circ$ denotes element-wise multiplication (Hadamard product).

The mechanism behind the Lamperti–Kiu representation is thus simple. The modulation J governs the sign of Y and, on intervals of time for which there is no change in sign, the Lamperti–Kiu representation effectively plays the role of the Lamperti representation of a pssMp, multiplied with $-1$ if Y is negative. In a sense, the MAP formalism gives a concatenation of signed Lamperti representations between times of sign change. Typically we can assume the Markov chain J to be irreducible, as otherwise the corresponding self-similar Markov process only switches signs at most once and can therefore be treated using the theory of pssMp.

4. Richocheted stable processes

Let us now turn to the setting of stable processes and path perturbations thereof that fall under the class of self-similar Markov processes and give rise to double hypergeometric Lévy processes in their Lamperti representation.

Let $(X_t,t\geq 0)$ with probabilities $(\mathbb{P}_x,x\in \mathbb{R})$ (as usual $\mathbb{P}\,:\!=\,\mathbb{P}_0$) be a strictly $\alpha$-stable process (henceforth just written ‘stable process’), that is to say an unkilled Lévy process which also satisfies the scaling property that under $\mathbb{P}$, for every $c > 0$, the process $(cX_{t c^{-\alpha}})_{t \ge 0}$ has the same law as X. It is known that $\alpha$ necessarily belongs to (0, 2], and the case $\alpha = 2$ corresponds to Brownian motion, which we exclude. The Lévy–Khintchine representation of such a process is as follows: $\sigma = 0$, $\Pi$ is absolutely continuous with density given by $\pi(x)\,:\!=\, c_+ x^{-(\alpha+1)} \textbf{1}_{(x > 0)} + c_- {|x|}^{-(\alpha+1)} \textbf{1}_{(x < 0)}$, $x \in \mathbb{R}$, where $c_+,\, c_- \ge 0$.

For consistency with the literature that we shall appeal to in this article, we shall always parametrise our $\alpha$-stable process such that

\begin{equation*} c_+ = \Gamma(\alpha+1) \frac{\sin(\pi \alpha \rho)}{\pi} \quad \text{and} \quad c_- = \Gamma(\alpha+1) \frac{\sin(\pi \alpha \hat\rho)}{\pi },\end{equation*}

where $\rho = \mathbb{P}(X_{1} \ge 0)$ is the positivity parameter, and $\hat\rho = 1-\rho$. Note when $\alpha=1$, then $c_+=c_-$. Moreover, we may also identify the exponent as taking the form

(4.1) \begin{equation}\Psi(\theta) = |\theta|^\alpha (\textrm{e}^{\pi\textrm{i}\alpha(\frac{1}{2} -\rho)} \textbf{1}_{(\theta>0)} + \textrm{e}^{-\pi\textrm{i}\alpha (\frac{1}{2} - \rho)}\textbf{1}_{(\theta<0)}), \qquad \theta\in\mathbb{R}.\end{equation}

With this normalisation, we take the point of view that the class of stable processes is parametrised by $\alpha$ and $\rho$; the reader will note that all the quantities above can be written in terms of these parameters. We shall restrict ourselves a little further within this class by excluding the possibility of having only one-sided jumps. In particular, this rules out the possibility that X is a subordinator or the negative of a subordinator, which occurs when $\alpha\in(0,1)$ and either $\rho =1$ or 0.

For this class of two-sided jumping stable processes, it is well known that, when $\alpha\in(0,1]$, the process never hits the origin (irrespective of the point of issue) and $\lim_{t\to\infty}|X_t| = \infty$. When $\alpha\in(1,2)$, the process hits the origin with probability 1 (for all points of issue). In both cases, the process will cross the origin infinitely often.

In [4, Section 6] the following example of a positive self-similar Markov process, say $Y^*=(Y^*_t,t\geq 0)$, called a ricocheted stable process, was introduced in the context of the theory of planar maps. Fix a $\mathfrak{p}\in [0,1]$ and let $\tau^-_0 = \inf\{t>0\,:\, X_t<0\}$. The stochastic dynamics of $Y^*$ are as follows. From its point of issue in $(0,\infty)$, $Y^*$ evolves as an X until its first passage into $({-}\infty,0)$, i.e. $\tau^-_0$. At that time an independent coin is flipped with probability $\mathfrak{p}$ of heads. If heads is thrown, then the process $Y^*$ is immediately transported to $-X_{\tau^-_0}$ (i.e. it is ‘ricocheted’ across 0). If tails is thrown, then it is sent to 0 and the process is killed. In the event $Y^*$ is ricocheted, it continues to evolve as an independent copy of X, flipping a new coin on the first pass into $({-}\infty,0)$, and so on.

The process $Y^*$ can be viewed as a way of resurrecting (with probability $\mathfrak{p}$) the process $(X_t\textbf{1}_{(t<\tau^-_0)}, t\geq 0)$ every time it is about to die; indeed, it degenerates to the latter process when $\mathfrak{p}=0$. In this respect the ricocheted process is related to the forthcoming work of [Reference Caballero, Chaumont and Rivero6]. This paper highlights the interesting phenomenon as to whether such resurrected processes will eventually continuously absorb at the origin or not.

As alluded to above, ricocheted processes were considered in [Reference Budd4]. In particular, it was shown in [Reference Budd4, Proposition 11] that the characteristic exponent of the Lévy process associated to $Y^*$ via the Lamperti transform takes the shape

(4.2) \begin{equation}\Psi^*(\theta)=\frac{\Gamma(\alpha-\textrm{i}\theta)\Gamma(1+\textrm{i}\theta)}{\pi}\left[\sin(\pi(\alpha\hat{\rho} -\textrm{i}\theta))-\mathfrak{p}\sin(\pi\alpha\hat{\rho})\right],\qquad \theta\in \mathbb{R}.\end{equation}

Moreover, by setting

\begin{equation*}\sigma\,:\!=\,\frac{1}{2}-\alpha\hat{\rho}\in \left({-}\frac{1}{2},\frac{1}{2}\right)\text{ and }b\,:\!=\,\frac{1}{\pi}\arccos(\mathfrak{p}\cos(\pi\sigma))\in \left[\vert\sigma\vert,\frac{1}{2}\right],\end{equation*}

we may also write

(4.3) \begin{equation}\Psi^*(\theta)=2^{\alpha}\dfrac{\Gamma \big(\frac{1+\textrm{i}\theta}{2}\big)\Gamma\big(\frac{2+\textrm{i}\theta}{2}\big)}{\Gamma\big( \frac{\sigma+b+\textrm{i}\theta}{2}\big)\Gamma\big(\frac{\sigma-b+\textrm{i}\theta+2}{2}\big)}\times\frac{\Gamma \big(\frac{\alpha-\textrm{i}\theta}{2}\big)\Gamma\big(\frac{1+\alpha-\textrm{i}\theta}{2}\big)}{\Gamma\big( \frac{b-\sigma-\textrm{i}\theta}{2}\big)\Gamma\big( \frac{2-\sigma-b-\textrm{i}\theta}{2}\big)},\qquad \theta\in \mathbb{R}.\end{equation}

With this in hand we now see an apparent relation with Corollary 2.1.

There are a number of remarks that are worth making at this point. First, $b\in \{\sigma,-\sigma\}$ occurs if and only if $\mathfrak{p}=1$. In that case we see that $\xi^*$ has infinite lifetime; moreover, by computing $\Psi^{*\prime}(0)$, we see that $Y^*$ is absorbed at the origin in finite time, oscillates between 0 and $\infty$, or drifts to $\infty$, according as $\sigma<0$, $\sigma=0$, or $\sigma>0$ (equivalently $\alpha\hat\rho>1/2$, $\alpha\hat\rho=1/2$, or $\alpha\hat\rho<1/2$), which resonates with the work on resurrected Lévy processes in [Reference Caballero, Chaumont and Rivero6].

Second, when $\mathfrak{p}=0$, and hence $b=\frac{1}{2}$, we recover the Wiener–Hopf factorisation of the Lamperti stable Lévy process corresponding to killing the stable process on hitting $({-}\infty,0)$ (in this case the condition $\alpha\in [1-\sigma-b,b-\sigma+1]$ is met).

Third, we can easily verify that the condition $\alpha\in [1-\sigma-b,b-\sigma+1]$ is a necessary and sufficient condition needed to ensure the ‘interlacing’ property of Theorem 2.1. Despite the exponent $\Psi^*$ falling outside of the double hypergeometric class when the aforementioned condition fails, we nonetheless conjecture that (4.3) is in fact the correct factorisation for the whole of the parameter regime. By way of example, taking, e.g., $\alpha=21/20$, $\rho=5/21$, and $\mathfrak{p}=0.9$, then the condition $\alpha\in [1-\sigma-b,b-\sigma+1]$ fails, and indeed numerically it is seen that the second factor in (4.3) does not correspond to a Pick function; still, an inspection of the first few derivatives suggests that it nevertheless corresponds to a Bernstein function.

Finally, we note that, in the spirit of [Reference Caballero and Chaumont5] and [Reference Budd4], two more characteristic exponents of Lévy processes appear which are obtained by a simple Esscher transform once we note that $\textrm{i}(b+\sigma)$ and $-\textrm{i}(b-\sigma)$ are roots of $\Psi^*$. Indeed, again provided $\alpha\in [1-\sigma-b,b-\sigma+1]$, we also have that $\Psi ^*(z+\textrm{i}(b+\sigma))$ and $\Psi ^*(z-\textrm{i}(b-\sigma))$ fall into the double hypergeometric class.

Now let $T_0^*$ be the first hitting time of 0 by the process $Y^*$, and let $\zeta^*$ be the lifetime of $\xi^*$. By the Lamperti transform we may express

\begin{equation*}T_0^*=\int_0^{\zeta^*}\textrm{e}^{\alpha \xi_s^*} \, \textrm{d} s=2^{-\alpha}\int_0^{2^\alpha\zeta^*}\textrm{e}^{-({-}2\xi^*_{2^{-\alpha}u})/(2/\alpha)} \, \textrm{d} u,\end{equation*}

so that Proposition 2.2 (coupled with Remark 2.1) yields the Mellin transform of $T_0^*$ provided $0<b-\sigma<\alpha$. We leave making the eventual expression explicit to the interested reader. We do, however, give the following detail.

Proof. The various (a.s.; we have in some places omitted this qualification for brevity) equalities of random variables follow from the Lamperti transformation. The equalities in law are part of the statement of the Wiener–Hopf factorisation of $\xi^*$. To see (i), let $\eta^*$ be the lifetime of the increasing ladder heights subordinator $H^{+*}$ of $\xi^*$ and denote by $\Phi^{+*}$ the Laplace exponent of $H^{+*}$. Then $\eta^*\sim\textrm{Exp}(\Phi^{+*}(0))$ and $H^{+*}_{\eta^*-}=\overline{\xi^*}_{\zeta^*-}$. When $\Phi^{+*}(0)=0$, i.e. $b=\sigma$, this renders $\zeta^*=\overline{\xi^*}_{\zeta^*-}=\infty$ a.s., hence, by the Lamperti transform, a.s. $T_0^*=\overline{\xi^*}_{\zeta^*-}=\infty$. Otherwise, when $\Phi^{+*}(0)>0$, it then follows from the statements surrounding the Wiener–Hopf factorisation of Lévy processes that $\textbf{E}[\textrm{e}^{-\theta H^{+*}_{\zeta^*-}}]=\frac{\Phi^{+*}(0)}{\Phi^{+*}(\theta)}$ for $\theta\in [0,\infty)$. The argument for (ii) is similar.

We can also push the notion of a ricocheted stable process into the realms of rssMp, but before doing so we present another interesting example of a pssMp due to Alex Watson [Reference Watson31], $Y^\natural=(Y^\natural_t,t\ge 0)$, associated to the symmetric stable process X, and whose corresponding Lévy process in the Lamperti transform also lies in the double hypergeometric class. Indeed, fix a $\mathfrak{q}\in [0,1)$, take $\rho=1/2$ and let $\tau^-_0 $ be as before. The stochastic dynamics of $Y^\natural$ are as follows. From its point of issue in $(0,\infty)$, $Y^\natural$ evolves as X until its first passage into $({-}\infty,0)$. At that time an independent coin is flipped with probability $\mathfrak{q}$ of heads. If heads is thrown, then the process $Y^\natural$ is immediately transported to $X_{\sigma_0}$, where $\sigma_0=\inf\{s\ge \tau_0\,:\, X_s>0\}$ (i.e., it is ‘glued’ to its next positive position, the negative part being thus ‘censored away’). If tails is thrown, then it is sent to 0 and the process is killed. In the event $Y^\natural$ is glued, it continues to evolve as an independent copy of X, flipping a new coin on first pass into $({-}\infty,0)$, and so on. Had we allowed $\mathfrak{q}=1$, then this case would correspond to the censored stable process of [Reference Kyprianou, Pardo and Watson23].

The characteristic exponent of $Y^\natural$, denoted by $\Psi^\natural$, may then be identified as

(4.4) \begin{align}\Psi^\natural(\theta)=\frac{\Gamma(\frac{\alpha}{2}-\textrm{i}\theta)\Gamma(\alpha-\textrm{i}\theta)}{\Gamma({-}\gamma+\frac{\alpha}{2}-\textrm{i}\theta)\Gamma(\gamma+\frac{\alpha}{2}-\textrm{i}\theta)}\times\frac{\Gamma(1-\frac{\alpha}{2}+\textrm{i}\theta)\Gamma(1+\textrm{i}\theta)}{\Gamma(1-\gamma-\frac{\alpha}{2}+\textrm{i}\theta)\Gamma(1+\gamma-\frac{\alpha}{2}+\textrm{i}\theta)}, \quad \theta\in \mathbb{R},\nonumber\\ \end{align}

where $\gamma\,:\!=\,\frac{1}{\pi}\arcsin(\sqrt{\mathfrak{q}}\sin(\pi\frac{\alpha}{2}))$; see the comments below the proof of Proposition 4.2. By noting that $\gamma<\frac{\alpha}{2}\land (1-\frac{\alpha}{2})$, we can check that the two factors either side of the sign $\times$ identify the Wiener–Hopf factors and that $Y^\natural$ belongs to the double hypergeometric class.

As indicated above, we can also introduce a real-valued version, $X^*$, of the ricocheted stable process. Indeed, we can define the process $X^*$ similarly to $Y^*$ but with some slight differences. On crossing the origin from $(0,\infty)$ to $({-}\infty,0)$, independently with probability $\mathfrak{p}$ it is ricocheted and with probability $1-\mathfrak{p}$, rather than being killed, it continues to evolve with the dynamics of a stable process. Additionally, as the state space of $X^*$ is $\mathbb{R}$, when passing from $({-}\infty, 0)$ to $(0,\infty)$, the same path adjustment is made on the independent flip of a coin, albeit with a different probability $\hat{\mathfrak{p}}$. Note, in the special case that $\mathfrak{p}=\hat{\mathfrak{p}}=0$, we have that $X^*$ is equal in law to nothing more than the underlying stable process X from which its paths are derived. The reader should also bear in mind that we do not assume any a priori relation between $\mathfrak{p}$ and $\hat{\mathfrak{p}}$: we stress this because we use the notation $\hat{\rho}$ for $1-\rho$, so the reader might be misled into thinking that $\hat{\mathfrak{p}}=1-\mathfrak{p}$, which is not being assumed.

It is worth spending a little bit of time to address the question as to why the processes $X^*$, $Y^\natural$, and $Y^*$ are indeed self-similar Markov processes (ssMp). As alluded to earlier, the fact that $Y^*$ is a ssMp comes from the reasoning of [Reference Budd4]. The explanation given there was via the use of the associated infinitesimal generators, which, in principle, needs a little care with the rigorous association of the generator to the appropriate semigroup of $Y^*$. We argue here that there is a relatively straightforward pathwise justification that comes directly out of the Lamperti transform for $Y^*$.

Proof. Our strategy will be simply to identify the pathwise definition of $Y^*$, $Y^\natural$, and $X^*$ directly with a Lamperti/Lamperti–Kiu decomposition. Once this has been done, the statement follows immediately from the bijection onto the class of pssMp/rssMp in the Lamperti/Lamperti–Kiu transform.

Let us start with the case of $Y^*$. From [Reference Caballero and Chaumont5] it is known that $\smash{(X_t\textbf{1}_{(t<\tau^-_0)}, t\geq 0)}$ is a pssMp; moreover, the Lévy process that underlies its Lamperti transform, say $\xi^\dagger$, has an explicit form for its characteristic exponent (indeed it belongs to the hypergeometric Lévy processes). As a Lévy process it falls into the category which are killed at an independent and exponentially distributed time with a strictly positive rate, say $q^\dagger$. Suppose we write $\Psi^\dagger$ for the characteristic exponent of $\xi^\dagger$; consider the process $\xi^0$ whose characteristic exponent is $\Psi^\dagger - q^\dagger$. That is to say, $\xi^0$ is the Lévy process whose dynamics are those of $\xi^\dagger$, albeit with the exponential killing removed.

Next, define a compound Poisson process $\xi^\circ$ with arrival rate $q^\dagger \mathfrak{p}$ and jump distribution which is equal in distribution to the random variable $\smash{\log (|X_{\tau^-_0}|/ X_{\tau^-_0-})}$. Note, straightforward scaling arguments (of both the stopping time $\tau^-_0$ and X) can be used to show that the distribution of $\log (|X_{\tau^-_0}|/ X_{\tau^-_0-})$ does not depend on the point of issue of X; see, for example, [Reference Kyprianou20, Chapter 13]. The characteristic exponent of this compound Poisson process is given by $q^\dagger \mathfrak{p}(1 - \mathbb{E}_1[(|X_{\tau^-_0}|/ X_{\tau^-_0-})^{\textrm{i}\theta}])$.

Finally, we build the Lévy process $\xi$ to have characteristic exponent given by

(4.5) \begin{equation}\Psi^*(\theta) =\Psi^\dagger(\theta) - q^\dagger + q^\dagger \mathfrak{p}(1 - \mathbb{E}_1[\big(|X_{\tau^-_0}|/ X_{\tau^-_0-}\big)^{\textrm{i}\theta}]) +q^\dagger (1-\mathfrak{p}).\end{equation}

In other words, the independent sum of $\xi^0$, the Lévy process $\xi^\dagger$ without killing, and the compound Poisson process described in the previous paragraph with the resulting stochastic process killed at an independent and exponentially distributed time with rate $q^\dagger (1-\mathfrak{p})$.

Now consider the Lamperti transform of $\xi$. By splitting the resulting path over the events corresponding to the arrival of points in the compound Poisson processes and the killing time, it is straightforward to see that the stochastic evolution of (3.1) matches that of $Y^*$ and hence they are equal in law. In other words, $Y^*$ is a positive self-similar Markov process.

In the above description, instead of killing at rate $q^\dagger (1-\mathfrak{p})$, we could also see this as the rate at which switching of an independent Markov chain occurs from $+1$ to $-1$. Accordingly, to build up a MAP which corresponds to the process $X^*$, we need to define

\begin{equation*}G_{1,-1}^*(\theta)\,:\!=\, \mathbb{E}_1[\big(|X_{\tau^-_0}|/ X_{\tau^-_0-}\big)^{\textrm{i}\theta}]=\frac{\Gamma(\alpha-\textrm{i}\theta)\Gamma(1+\textrm{i} \theta)}{\Gamma(\alpha)},\qquad \theta \in \mathbb{R}, \end{equation*}

where the second equality comes from [Reference Chaumont, Pantí and Rivero8, Corollary 11]. Note that, although we have seen the Fourier transform of $\log (|X_{\tau^-_0}|/ X_{\tau^-_0-})$ before, this is also the change in the radial exponent $\xi$ when $X^*$ passes from $(0,\infty)$ to $({-}\infty,0)$.

By replacing the role of $\rho$ by $\hat\rho$ in the definition of $\Psi^\dagger, q^\dagger, \mathfrak{p}$ (henceforth written $\hat{\Psi}^\dagger, \hat{q}^\dagger, \hat{\mathfrak{p}}$), we can similarly describe the part of the MAP that corresponds to $X^*$ until its first passage from $({-}\infty,0)$ to $(0,\infty)$ when issued from a negative value.

Thanks to the piecewise evolution of $X^*$, considered at each crossing of the origin, we can thus conclude that it is a rssMp with underlying MAP whose matrix exponent is given by

\begin{align*}&{\boldsymbol{\Psi}}^*(\theta) \\ &=- \left[\begin{array}{cc}\Psi^\dagger(\theta) - q^\dagger + q^\dagger \mathfrak{p}\big(1 - \mathbb{E}_1\big[\big(|X_{\tau^-_0}|/ X_{\tau^-_0-}\big)^{\textrm{i}\theta}\big]\big) & 0\\[8pt] 0 &\hat{\Psi}^\dagger(\theta) - \hat{q}^\dagger + \hat{q}^\dagger\hat{ \mathfrak{p}}\big(1 - \hat{\mathbb{E}}_1\big[\big(|X_{\tau^-_0}|/ X_{\tau^-_0-}\big)^{\textrm{i}\theta}\big]\big)\end{array}\right]\\[5pt] & \quad +\left[\begin{array}{cc}-q^\dagger (1-\mathfrak{p}) &q^\dagger (1-\mathfrak{p}) \mathbb{E}_1\big[\big(|X_{\tau^-_0}|/ X_{\tau^-_0-}\big)^{\textrm{i}\theta}\big] \\[12pt] \hat{q}^\dagger (1-\hat{\mathfrak{p}})\hat{\mathbb{E}}_1\big[\big(|X_{\tau^-_0}|/ X_{\tau^-_0-}\big)^{\textrm{i}\theta}\big]&-\hat{q}^\dagger (1-\hat{\mathfrak{p}}) \\[5pt] \end{array}\right]\\[8pt] &= \left[\begin{array}{cc}-\Psi^\dagger(\theta) + q^\dagger \mathfrak{p}\mathbb{E}_1\big[\big(|X_{\tau^-_0}|/ X_{\tau^-_0-}\big)^{\textrm{i}\theta}\big] & q^\dagger (1-\mathfrak{p}) \mathbb{E}_1\big[\big(|X_{\tau^-_0}|/ X_{\tau^-_0-}\big)^{\textrm{i}\theta}\big] \\[10pt] \hat{q}^\dagger (1-\hat{\mathfrak{p}})\hat{\mathbb{E}}_1\big[\big(|X_{\tau^-_0}|/ X_{\tau^-_0-}\big)^{\textrm{i}\theta}\big] &-\hat{\Psi}^\dagger(\theta) + \hat{q}^\dagger\hat{ \mathfrak{p}} \hat{\mathbb{E}}_1\big[\big(|X_{\tau^-_0}|/ X_{\tau^-_0-}\big)^{\textrm{i}\theta}\big]\end{array}\right] , \qquad \theta \in \mathbb{R} ,\end{align*}

where $\hat{\mathbb{P}}$ is the law of $-X$.

Finally, we consider the case of $Y^\natural$ and recall that $\rho=\frac{1}{2}$. The analysis is the same as in the case of $Y^*$ except that the compound Poisson process $\xi^\circ$ there gets replaced with a different compound Poisson process $\xi^\bullet$. From the path analysis of $Y^\natural$ the jumps of $\xi^\bullet$ have the distribution of $\log (X_{\sigma_0}/ X_{\tau^-_0-})$, while its rate is $q^\dagger\mathfrak{q}$. The characteristic function of $\log (X_{\sigma_0}/ X_{\tau^-_0-})$ follows from the analysis in [Reference Kyprianou, Pardo and Watson23, Proposition 4.2]:

\begin{equation*}\mathbb{E}_1\left[\left(\frac{X_{\sigma_0}}{X_{\tau^-_0-}}\right)^{\textrm{i} \theta}\right]=\frac{\Gamma(1-\frac{\alpha}{2}+\textrm{i}\theta)\Gamma(\frac{\alpha}{2}-\textrm{i}\theta)\Gamma(1+\textrm{i}\theta)\Gamma(\alpha-\textrm{i}\theta)}{\Gamma(\frac{\alpha}{2})\Gamma(1-\frac{\alpha}{2}) \Gamma(\alpha)},\qquad \theta\in\mathbb{R}.\end{equation*}

This completes the proof.

From the above proof we can recover the result of [Reference Budd4]. Indeed, from [Reference Caballero and Chaumont5, Corollary 1],

\begin{equation*}\Psi^\dagger(\theta)=\frac{\Gamma(\alpha-\textrm{i}\theta)\Gamma(1+\textrm{i}\theta)}{\Gamma(\alpha\hat{\rho}-\textrm{i}\theta)\Gamma(1-\alpha\hat{\rho}+\textrm{i}\theta)},\qquad \theta \in \mathbb{R},\end{equation*}

and in particular $q^\dagger =\Psi^\dagger(0)=\Gamma(\alpha)/(\Gamma(\alpha\hat{\rho})\Gamma(1-\alpha\hat{\rho}))$. It now follows from (4.5) that

\begin{align*}\Psi^*(\theta)& =\Psi^\dagger(\theta) - q^\dagger \mathfrak{p} \mathbb{E}_1[(|X_{\tau^-_0}|/ X_{\tau^-_0-})^{\textrm{i}\theta}]\\[5pt] &=\frac{\Gamma(\alpha-\textrm{i}\theta)\Gamma(1+\textrm{i}\theta)}{\Gamma(\alpha\hat{\rho}-\textrm{i}\theta)\Gamma(1-\alpha\hat{\rho}+\textrm{i}\theta)} - \frac{\Gamma(\alpha)}{\Gamma(\alpha\hat{\rho})\Gamma(1-\alpha\hat{\rho})}\mathfrak{p} \frac{\Gamma(\alpha-\textrm{i}\theta)\Gamma(1+\textrm{i}\theta)}{\Gamma(\alpha)} \\[5pt] &=\frac{\Gamma(\alpha-\textrm{i}\theta)\Gamma(1+\textrm{i}\theta)}{\pi}\left[\sin(\pi(\alpha\hat{\rho} -\textrm{i}\theta))-\mathfrak{p}\sin(\pi\alpha\hat{\rho})\right], \qquad \theta\in\mathbb{R},\end{align*}

which agrees with (4.2). Likewise, we can deduce the explicit form of $\Psi^\natural$ reported in (4.4).

In a similar spirit, we can proceed to compute the matrix exponent ${\boldsymbol{\Psi}}^*$ to obtain

\begin{align*}&{\boldsymbol{\Psi}}^*(\theta)\notag\\ &= \frac{\Gamma(\alpha-\textrm{i}\theta)\Gamma(1+\textrm{i}\theta)}{\pi}\left[\begin{array}{c@{\quad}c}\mathfrak{p}\,{\sin}(\pi\alpha\hat{\rho})-{\sin}(\pi(\alpha\hat{\rho} -\textrm{i}\theta)) & (1-\mathfrak{p})\,{\sin}(\pi\alpha\hat{\rho})\\[7pt] (1-\hat{\mathfrak{p}})\,{\sin}(\pi\alpha\rho)& \hat{\mathfrak{p}}\,{\sin}(\pi\alpha\rho) -\,{\sin}(\pi(\alpha\rho-\textrm{i}\theta))\end{array}\right]\end{align*}

for $\theta\in \mathbb{R}$. For $\mathfrak{p}=\hat{\mathfrak{p}}=0$ this reduces to the matrix exponent of a stable process killed on hitting 0, as discussed in [Reference Kyprianou21]. We conclude this discussion with the following corollary which identifies ${\boldsymbol{\Psi}}$ in a form that is similar to the double hypergeometric analogue.

Corollary 4.2. For $\theta\in\mathbb{R}$,

\begin{align*}&{\boldsymbol{\Psi}}^*(\theta) = \\ & \hspace*{-2pt}{-}2^\alpha\!\!\left[\begin{array}{c@{\quad}c}\frac{\Gamma \left(\frac{1+\textrm{i}\theta}{2}\right)\Gamma\left(\frac{2+\textrm{i}\theta}{2}\right)}{\Gamma\left( \frac{\sigma+b+\textrm{i}\theta}{2}\right)\Gamma\left(\frac{\sigma-b+\textrm{i}\theta+2}{2}\right)}\frac{\Gamma \left(\frac{\alpha-\textrm{i}\theta}{2}\right)\Gamma\left(\frac{1+\alpha-\textrm{i}\theta}{2}\right)}{\Gamma\left( \frac{b-\sigma-\textrm{i}\theta}{2}\right)\Gamma\left( \frac{2-\sigma-b-\textrm{i}\theta}{2}\right)} &-\frac{\Gamma \left(\frac{1+\textrm{i}\theta}{2}\right)\Gamma\left(\frac{2+\textrm{i}\theta}{2}\right)}{\Gamma\left( \frac{\sigma+b}{2}\right)\Gamma\left(\frac{\sigma-b+2}{2}\right)}\frac{\Gamma \left(\frac{\alpha-\textrm{i}\theta}{2}\right)\Gamma\left(\frac{1+\alpha-\textrm{i}\theta}{2}\right)}{\Gamma\left( \frac{b-\sigma}{2}\right)\Gamma\left( \frac{2-\sigma-b}{2}\right)}\\ &\\ -\frac{\Gamma \left(\frac{1+\textrm{i}\theta}{2}\right)\Gamma\left(\frac{2+\textrm{i}\theta}{2}\right)}{\Gamma\left( \frac{\hat{\sigma}+\hat{b}}{2}\right)\Gamma\left(\frac{\hat{\sigma}-\hat{b}+2}{2}\right)}\frac{\Gamma \left(\frac{\alpha-\textrm{i}\theta}{2}\right)\Gamma\left(\frac{1+\alpha-\textrm{i}\theta}{2}\right)}{\Gamma\left( \frac{\hat{b}-\hat{\sigma}}{2}\right)\Gamma\left( \frac{2-\hat{\sigma}-\hat{b}}{2}\right)}&\frac{\Gamma \left(\frac{1+\textrm{i}\theta}{2}\right)\Gamma\left(\frac{2+\textrm{i}\theta}{2}\right)}{\Gamma\left( \frac{\hat{\sigma}+\hat{b}+\textrm{i}\theta}{2}\right)\Gamma\left(\frac{\hat{\sigma}-\hat{b}+\textrm{i}\theta+2}{2}\right)}\frac{\Gamma \left(\frac{\alpha-\textrm{i}\theta}{2}\right)\Gamma\left(\frac{1+\alpha-\textrm{i}\theta}{2}\right)}{\Gamma\left( \frac{\hat{b}-\hat{\sigma}-\textrm{i}\theta}{2}\right)\Gamma\left( \frac{2-\hat{\sigma}-\hat{b}-\textrm{i}\theta}{2}\right)}\end{array}\right]\!,\end{align*}

where, as before, $\sigma=-\alpha\hat{\rho}+1/2$ and $b\,:\!=\,\arccos(\mathfrak{p}\cos(\pi\sigma))/{\pi}$, and $\hat\sigma$ and $\hat{b}$ have the obvious meaning.

We note that the condition $\{\mathfrak{p},\hat{\mathfrak{p}}\}\subset [0,1)$ ensures that J is irreducible. If $\mathfrak{p}=1$, then, albeit possibly only after the first crossing of the origin (in the case that $X^*$ is issued from a negative value), we are basically in the case of the pssMp $Y^*$. An analogous statement can be made for $\hat{\mathfrak{p}}=1$.

For real $\theta$ in some neighbourhood of 0, the explicit expression for ${\boldsymbol{\Psi}}^*({-}\textrm{i} \theta)$ can be used to determine its Perron–Frobenius eigenvalue $\chi(\theta)$ [Reference Asmussen1, Paragraph XI.2C]. From the aforesaid reference, it is known that $\chi(0)=0$ and that, when it exists, $\chi'(0)$ dictates the long-term behaviour of $(\xi,J)$, the underlying MAP of $X^*$, in the sense that a.s. $\lim_{t\to\infty}\xi_t/t=\chi'(0)$ [Reference Asmussen1, Corollary XI.2.8]. Moreover, $\xi$ drifts to $\infty$, oscillates, or drifts to $-\infty$, according to whether $\chi'(0)$ is valued $>0$, $=0$, or $<0$ [Reference Asmussen1, Proposition XI.2.10]. By the Lamperti–Kiu transform, this corresponds to $\vert X^*\vert$ drifting to $\infty$, oscillating between 0 and $\infty$, and hitting 0 continuously, respectively.

An explicit computation yields

\begin{equation*}\chi'(0)=\Gamma(\alpha)\frac{(1-\hat{\mathfrak{p}})\sin(\pi\alpha\rho)\cos(\pi\alpha\hat{\rho})+(1-\mathfrak{p})\sin(\pi\alpha\hat{\rho})\cos(\pi\alpha\rho)}{(1-\hat{\mathfrak{p}})\sin(\pi\alpha\rho)+(1-\mathfrak{p})\sin(\pi\alpha\hat{\rho})}.\end{equation*}

From this, we note that when $\mathfrak{p}=\hat{\mathfrak{p}}$ (in particular when both are equal to 0 and $X^*=(X_t\textbf{1}_{(t<\tau^-_0)}, t\geq 0)$), we obtain $\chi'(0)(\sin(\pi\alpha\rho)+\sin(\pi\alpha\hat{\rho}))=\Gamma(\alpha)\sin(\pi\alpha)$ (which is consistent with the polarity of 0 for X when $\alpha\in (0,1]$ and the fact that X hits 0 when $\alpha\in (1,2)$). Furthermore, whether or not $\mathfrak{p}=\hat{\mathfrak{p}}$, $\chi'(0)>0$ whenever $\{\alpha\rho,\alpha\hat{\rho}\}\subset (0,\frac{1}{2})$, in particular whenever $\alpha< \frac{1}{2}$, but in general the sign of $\chi'(0)$ will depend on $\alpha,\rho,\mathfrak{p},\hat{\mathfrak{p}}$ in a non-trivial ‘balancing’ way. For instance, if $\hat{\rho}\alpha\in (\frac{1}{2},1)$, $\hat{\mathfrak{p}}$ is sufficiently close to 0, while $\mathfrak{p}$ is sufficiently close to 1, then $X^*$ will hit 0 continuously, even though one may also have $\alpha< 1$, in which case $(X_t\textbf{1}_{(t<\tau^-_0)}, t\geq 0)$ will not be absorbed continuously at the origin. More generally, we have the following proposition.

Proposition 4.3. $X^*$ hits zero continuously if and only if

\begin{equation*}(1-\hat{\mathfrak{p}})\sin(\pi\alpha\rho)\cos(\pi\alpha\hat{\rho})+(1-\mathfrak{p})\sin(\pi\alpha\hat{\rho})\cos(\pi\alpha\rho)<0,\end{equation*}

equivalently $\sin(\pi\alpha)<\hat{\mathfrak{p}}\sin(\pi\alpha\rho)\cos(\pi\alpha\hat{\rho})+\mathfrak{p}\sin(\pi\alpha\hat{\rho})\cos(\pi\alpha\rho)$; for this condition to prevail it is necessary (but not sufficient) that $\alpha\hat{\rho}>\frac{1}{2}$ or $\alpha\rho>\frac{1}{2}$.

Proof. The denominator of $\chi'(0)$ is always strictly positive because $0<\alpha\rho<1$ and $0<\alpha\hat{\rho}<1$ (see, for instance, [Reference Kyprianou21, p. 3] for the identification of the admissible pairs $(\alpha,\rho)$ under our standing assumptions). Therefore, the condition for $X^*$ to hit 0 continuously, i.e., $\chi'(0)<0$, becomes the stated one, with its equivalent form following upon using the addition formula for the sine. For the final claim we note that $\cos(\pi\alpha\hat{\rho})$ and $\cos(\pi\alpha\rho)$ are both non-negative unless $\alpha\hat{\rho}>\frac{1}{2}$ or $\alpha\rho>\frac{1}{2}$.

5. Proofs of main results

5.1 Proof of Theorem 2.1

Proof. Let $\alpha<\beta$ and $\gamma<\delta$ be strictly positive real numbers, and assume that $\{\alpha-\beta,\gamma-\delta,\gamma-\alpha,\delta-\alpha, \gamma-\beta,\delta-\beta\}\cap \mathbb{Z}=\emptyset$, i.e., assume $\gamma+\mathbb{Z}_{\leq 0}$, $\delta+\mathbb{Z}_{\leq 0}$, $\alpha+\mathbb{Z}_{\leq 0}$, and $\beta+\mathbb{Z}_{\leq 0}=\emptyset$ are pairwise disjoint. Then we claim that $\{\gamma,\delta\}+ \mathbb{Z}_{\leq 0}$ interlaces with $\{\alpha,\beta\}+\mathbb{Z}_{\leq 0}$ if and only if either

1. (I) there is a $k\in \mathbb{N}_0$ with $\beta-k-1<\gamma<\alpha<\delta-k<\beta-k$, or

2. (II) there is a $k\in \mathbb{N}$ with $\delta-k<\alpha<\gamma<\beta-k<\delta-k+1$.

The conditions are clearly sufficient. For necessity, assume the interlacing property. Clearly we must have $\alpha<\delta<\beta$ and then either (i) $\gamma<\alpha<\delta<\beta$ or else (ii) $\alpha <\gamma<\delta<\beta$. Assume (i) first. Then $\beta-1<\delta$. If even $\beta-1<\alpha$, then necessarily $\beta-1<\gamma$, and we are done. Otherwise, $\gamma<\alpha<\beta-1<\delta<\beta$. But then we must have $\gamma<\alpha<\delta-1<\beta-1<\delta<\beta$. An inductive argument gives the required result. Now assume (ii). We must have $\alpha <\gamma<\beta-1<\delta<\beta$. If $\delta-1<\alpha$ then we are done. Otherwise, $\alpha<\delta-1$ and necessarily $\alpha <\gamma<\beta-2<\delta-1<\beta-1<\delta<\beta$. An inductive argument again gives the result. Clearly (I) and (II) are exclusive.

Assume further that the interlacing property holds. The above characterisation implies that $-\alpha-\beta<-\gamma-\delta<-\alpha-\beta+1 $, i.e., $\gamma+\delta<\alpha+\beta<\gamma+\delta+1$.

Next, via Euler’s infinite product formula for the gamma function, we may write, for $z\in \mathbb{C}$ where the expression is defined,

\begin{equation*}\Phi(z)\,:\!=\,\frac{\Gamma(z+\alpha)\Gamma(z+\beta)}{\Gamma(z+\gamma)\Gamma(z+\delta)}=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\gamma)\Gamma(\delta)}\prod_{n=0}^\infty \frac{\big(1-\frac{z}{-n-\gamma}\big)\big(1-\frac{z}{-n-\delta}\big)}{\big(1-\frac{z}{-n-\alpha}\big)\big(1-\frac{z}{-n-\beta}\big)}.\end{equation*}

Then $\Phi$ is a real meromorphic function, and it follows from [Reference Levin24, Theorem 27.2.1] and from the interlacing condition together with $\{\alpha,\beta,\gamma,\delta\}\subset (0,\infty)$ that it maps the open upper complex half-plane into itself, i.e., it is a Pick function, and hence a non-constant completely monotone Bernstein function. It follows furthermore, see [Reference Levin24, Problem 27.2.1], that $\Phi$ admits the representation, for some $\{\omega_1,\omega_2\}\subset \mathbb{Z}\cup \{-\infty,\infty\}$, $\omega_1\leq\omega_2$, and at least for z in the open upper complex half-plane,

(5.1) \begin{equation}\Phi(z)=az+b+\frac{d}{z}+{\sum}_{n=\omega_1}^{\omega_2} c_n\left[\frac{1}{a_n-z}-\frac{1}{a_n}\right],\end{equation}

where $\{a,-d\}\subset [0,\infty)$, $b\in \mathbb{R}$, all the $c_n$ are non-negative, the sequence $a_n$ (where $n\in\{\omega_1, \omega_1+1, \ldots, \omega_2\}$) consists of non-zero real numbers and is strictly increasing, and finally $\sum_{n=\omega_1}^{\omega_2}{c_n}/{a_n^2}<\infty$. Then the $a_n$ and 0 must exhaust the poles of $\Phi$ and the $c_n$ must be the respective negatives of the residua of $\Phi$ at these poles, which can easily be computed: for $k\in \mathbb{N}_0$,

\begin{align*}\textrm{Res}(\Phi,-\alpha-k) & = ({-}1)^k\frac{\Gamma(\beta-\alpha-k)}{k!\Gamma(\gamma-\alpha-k)\Gamma(\delta-\alpha-k)} , \\[5pt] \textrm{Res}(\Phi,-\beta-k) & = ({-}1)^k\frac{\Gamma(\alpha-\beta-k)}{k!\Gamma(\gamma-\beta-k)\Gamma(\delta-\beta-k)}.\end{align*}

The coefficient d equals 0 because $\Phi$ has no pole at 0 (as $\{\alpha,\beta\}\subset (0,\infty)$). By continuity of $\Phi$ and dominated convergence for the right-hand side, the equality (5.1) extends to $[0,\infty)$. Then, by dominated convergence and by Stirling’s formula for the gamma function, using $\alpha+\beta<\gamma+\delta+1$, we see that $a=\lim_{z\to \infty}{\Phi(z)}/{z}=0$ (limit over $(0,\infty)$).

On the other hand, from the definition of the hypergeometric function ${_2F_1}$, for $s\in (0,\infty)$,

\begin{align*}f(s) & = \sum_{k=0}^\infty ({-}1)^{k+1}\frac{\Gamma(\beta-\alpha-k)}{k!\Gamma(\gamma-\alpha-k)\Gamma(\delta-\alpha-k)}\textrm{e}^{-(\alpha+k)s} \\[5pt] & \quad + \sum_{k=0}^\infty ({-}1)^{k+1}\frac{\Gamma(\alpha-\beta-k)}{k!\Gamma(\gamma-\beta-k)\Gamma(\delta-\beta-k)}\textrm{e}^{-(\beta+k)s}.\end{align*}

A direct computation using Tonelli–Fubini (note that we know a priori from the preceding that $\sum_{n=\omega_1}^{\omega_2}{c_n}/{a_n^2}<\infty$) then reveals that, for $z\in \mathbb{C}$ with $\textrm{Re}(z)\geq 0$,

(5.2) \begin{align}H(z)&\,:\!=\,\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\gamma)\Gamma(\delta)}+\int_0^\infty (1-\textrm{e}^{-z s})f(s) \, \textrm{d}s\nonumber\\[5pt] &=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\gamma)\Gamma(\delta)}+\sum_{k=0}^\infty ({-}1)^{k+1}\frac{\Gamma(\beta-\alpha-k)}{k!\Gamma(\gamma-\alpha-k)\Gamma(\delta-\alpha-k)}\left(\frac{1}{\alpha+k}-\frac{1}{\alpha+k+z}\right)\nonumber\\[5pt] & \quad + \sum_{k=0}^\infty ({-}1)^{k+1}\frac{\Gamma(\alpha-\beta-k)}{k!\Gamma(\gamma-\beta-k)\Gamma(\delta-\beta-k)}\left(\frac{1}{\beta+k}-\frac{1}{\beta+k+z}\right).\end{align}

Comparing (5.1) and (5.2), we see that certainly $\Phi=H$ on $[0,\infty)$ and hence, by analyticity and continuity, on the whole of the closed right complex half-plane, which identifies the subordinator corresponding to $\Phi$ as being pure-jump with Lévy density f. Incidentally, because all of the $c_n$ must be non-negative, we also obtain that the two series appearing in the above representation of f are each of positive terms only.

The assumption $\{\alpha,\beta,\gamma,\delta\}\subset (0,\infty)$ ensures that $\Phi(0)>0$, i.e., the subordinator corresponding to $\Phi$ is killed. By monotone convergence $\Phi$ has infinite or finite activity according as $\int_0^\infty f(s) \, \textrm{d}s = \lim_{z\to \infty}\Phi(z)=\infty$ or $<\infty$, and Stirling’s formula for the gamma function together with $\gamma+\delta<\alpha+\beta$ guarantees that it is the former that is taking place here.

Further, as a complete Bernstein function, the potential measure associated to $\Phi$ has a density with respect to Lebesgue measure; cf. [Reference Schilling, Song and Vondraček27, Remark 11.6] and [Reference Song and Vondraček28, Lemma 2.3]. In order to check that (2.2) gives the potential density, write it out using the definition of ${_2F_1}$: for $x\in (0,\infty)$,

\begin{align*}u(x) & = \sum_{k=0}^\infty ({-}1)^{k}\frac{\Gamma(\delta-\gamma-k)}{k!\Gamma(\alpha-\gamma-k)\Gamma(\beta-\gamma-k)}\textrm{e}^{-(\gamma+k)x} \\[5pt] & \quad +\sum_{k=0}^\infty ({-}1)^{k}\frac{\Gamma(\gamma-\delta-k)}{k!\Gamma(\alpha-\delta-k)\Gamma(\beta-\delta-k)}\textrm{e}^{-(\delta+k)x}.\end{align*}

Note that the two series consist of positive terms owing to the characterisations (I) and (II) and the properties of the sign of the gamma function on the real line. By Tonelli’s theorem we then compute the Laplace transform $\hat{u}$ of (2.2),

\begin{align*}\hat{u}(z) & = \sum_{k=0}^\infty ({-}1)^{k}\frac{\Gamma(\delta-\gamma-k)}{k!\Gamma(\alpha-\gamma-k)\Gamma(\beta-\gamma-k)}\frac{1}{\gamma+k+z} \\[5pt] & \quad + \sum_{k=0}^\infty ({-}1)^{k}\frac{\Gamma(\gamma-\delta-k)}{k!\Gamma(\alpha-\delta-k)\Gamma(\beta-\delta-k)}\frac{1}{\delta+k+z}\end{align*}

($z\in \mathbb{C}$, $\textrm{Re}(z)\geq 0$), and check that it coincides with $1/\Phi$.

To this end, let $z\in \mathbb{C}$, $\textrm{Re}(z)\geq 0$; we verify that $\hat{u}(z)=\frac{1}{\Phi(z)}$. But $\Phi$ being a complete Bernstein function implies (see [Reference Schilling, Song and Vondraček27, Proposition 7.1]) that $\frac{\textrm{id}_{(0,\infty)}}{\Phi}$ is itself a complete Bernstein function. Then [Reference Schilling, Song and Vondraček27, Remark 6.4] guarantees that we can write

\begin{equation*}\frac{1}{\Phi(z)}=\frac{\Gamma(\gamma+z)\Gamma(\delta+z)}{\Gamma(\alpha+z)\Gamma(\beta+z)}=\frac{a}{z}+b+\int_{(0,\infty)}\frac{1}{z +t}\eta(\textrm{d} t),\qquad z\in (0,\infty),\end{equation*}

where $\{a,b\}\subset [0,\infty)$ and $\eta$ is a measure on $\mathcal{B}_{(0,\infty)}$ such that $\int (1+t)^{-1}\eta(\textrm{d} t)<\infty$. Clearly we must have $b=0$ because $\lim_{z\to\infty}{1}/{\Phi}(z)=0$. The measure $\eta$ is supported by $\{\delta,\gamma\}+\mathbb{N}_0$, which follows by appealing to, e.g., [Reference Schilling, Song and Vondraček27, Corollary 6.3]. Moreover, $a=0$ because $\lim_{z\downarrow 0}{1}/{\Phi(z)}<\infty$. By analytic extension, the masses of $\eta$ are identified by identifying the residues of $\Phi(z)$, which concludes the argument.

When, ceteris paribus, the interlacing condition fails, it follows from its representation as an infinite product given above, and from the characterisation of Pick maps of [Reference Levin24, Theorem 27.2.1], that the function $\Phi(z)$ is not Pick, hence also not a non-constant complete Bernstein function.

The final assertion of the theorem follows by recalling that complete Bernstein functions are closed under pointwise limits [Reference Schilling, Song and Vondraček27, Corollary 7.6(ii)].

Let us make some remarks concerning the above proof. First, to establish the complete Bernstein property of (1.3) under the ‘interlacing conditions’ we could have relied directly on the results of [Reference Kuznetsov, Kyprianou and Pardo16] (see Lemma 1 and the discussion following its proof therein). However, we wanted to establish the precise, not just sufficient, conditions under which the ‘non-constant complete Bernstein’ property obtains, which is why we have provided above a direct and detailed study of the Pick property of (1.3). Besides, this has the advantage of keeping our paper more self-contained.

Secondly, the expression (2.1) for the Lévy density of course does not just ‘fall from the sky’; rather, it is obtained by twice differentiating the inverse Laplace transform of

\begin{equation*}z\mapsto \frac{1}{z^2}\frac{\Gamma(\alpha+z)\Gamma(\beta+z)}{\Gamma(\gamma+z)\Gamma(\delta+z)}\end{equation*}

(cf. [Reference Schilling, Song and Vondraček27, Eq. (3.4)]), and the latter in turn is obtained (at least on a pro forma level) by the Heaviside residue expansion [Reference Henrici13, Theorem 10.7(c)]. A similar remark pertains to (2.2): we know [Reference Schilling, Song and Vondraček27, Eq. (5.20)] that the Laplace transform of U is $1/\Phi$, so that at least on a pro forma level we can Laplace invert via the Heaviside expansion.

Finally, we recover from the proof of Theorem 2.1 the non-obvious identities

\begin{align*}\frac{\Gamma(\alpha+z)\Gamma(\beta+z)}{\Gamma(\gamma+z)\Gamma(\delta+z)} & = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\gamma)\Gamma(\delta)}\\ & \quad +\sum_{k=0}^\infty ({-}1)^{k+1}\frac{\Gamma(\beta-\alpha-k)}{k!\Gamma(\gamma-\alpha-k)\Gamma(\delta-\alpha-k)}\left(\frac{1}{\alpha+k}-\frac{1}{\alpha+k+z}\right)\\ & \quad +\sum_{k=0}^\infty ({-}1)^{k+1}\frac{\Gamma(\alpha-\beta-k)}{k!\Gamma(\gamma-\beta-k)\Gamma(\delta-\beta-k)}\left(\frac{1}{\beta+k}-\frac{1}{\beta+k+z}\right),\\ &\qquad z\in \mathbb{C}\backslash (\{-\alpha,-\beta\}+\mathbb{Z}_{\leq 0}),\\ &\qquad \sum_{k=0}^\infty ({-}1)^{k}\frac{\Gamma(\delta-\gamma-k)}{k!\Gamma(\alpha-\gamma-k)\Gamma(\beta-\gamma-k)}\frac{1}{\gamma+k+z}\\ &\quad +\sum_{k=0}^\infty ({-}1)^{k}\frac{\Gamma(\gamma-\delta-k)}{k!\Gamma(\alpha-\delta-k)\Gamma(\beta-\delta-k)}\frac{1}{\delta+k+z}\\ & =\frac{\Gamma(\gamma+z)\Gamma(\delta+z)}{\Gamma(\alpha+z)\Gamma(\beta+z)},\qquad z\in \mathbb{C}\backslash (\{-\delta,-\gamma\}+\mathbb{Z}_{\leq 0}).\end{align*}

5.3. Proof of Proposition 2.2

As alluded to above, we will use the verification argument of [Reference Kuznetsov and Pardo19, Proposition 2]. The idea there is to verify that the right-hand side of (2.3) has certain analytical properties so that it matches the unique solution of the functional equation that the left-hand side of (2.3) must necessarily solve (cf. [Reference Maulik and Zwart25]).

Proof. The Laplace exponent $\psi_{c}(z)\,:\!=\,\log \textbf{E}[\exp(z\xi_1/{c})]$ is given by the map $z\mapsto \psi(z/{c})$, where $\psi$ is the Laplace exponent of $\xi$. The conditions on the parameters ensure that $\xi/{c}$ either drifts to $\infty$ or is killed (see Proposition 2.1(iii)), and that it satisfies the so-called Cramér condition, $\psi_{c}({-}\hat{\gamma}{c})=\psi({-}\hat{\gamma})=0$ and $\psi_{c}$ is finite on $({-}\hat{\alpha}{c},0)$, which contains $-\hat{\gamma}{c}$. Furthermore, letting $\mathcal{M}(s)$ be the right-hand side of (2.3), we see immediately from the properties of the double gamma function that $\mathcal{M}$ is analytic and free of zeros in the strip $\textrm{Re}(s)\in (0,1+\hat{\gamma}{c})$, and $\mathcal{M}(1)=1$ and $\mathcal{M}(s+1)=-s\mathcal{M}(s)/\psi_{c}({-}s)$ for $s\in (0,\hat{\gamma}{c})$.

It remains to check that $\lim_{\vert y\vert \to\infty}\vert \mathcal{M}(x+\textrm{i}y)\vert^{-1}\textrm{e}^{-2\pi \vert y\vert}=0$ uniformly in $x\in (0,\hat{\gamma}{c}+1)$. Lemma 1 in [Reference Kuznetsov and Pardo19] implies that, as $\vert y\vert\to\infty$,

\begin{equation*}\ln\left\vert \frac{G(p+\textrm{i}y;{c})}{G(q+\textrm{i}y;{c})}\right\vert=\frac{p-q}{{c}}\textrm{Re}[(q+\textrm{i}y)\ln(q+\textrm{i}y)]+o(\vert y\vert)=-\pi\frac{p-q}{2{c}}\vert y\vert+o(\vert y\vert)\end{equation*}

uniformly in bounded real p and q. Note that this was observed in [Reference Budd4, p. 44], albeit there without the uniformity, which is needed for our purposes. At the same time, by Stirling’s asymptotic formula for the gamma function [Reference Gradshteyn and Ryzhik11, Formula 8.327], as $\vert y\vert\to \infty$, we have $\ln\vert \Gamma(x+\textrm{i}y)\vert=\textrm{Re}[(x+\textrm{i}y)(\ln(x+\textrm{i}y)-1)]+o(\vert y\vert)=-\frac{\pi}{2}\vert y\vert+o(\vert y\vert)$ uniformly for x bounded in $ \mathbb{R}$.

Therefore, it follows that, uniformly for $x\in (0,\hat{\gamma}{c}+1)$, as $\vert y\vert\to \infty$, we have

\begin{equation*}\ln\vert \mathcal{M}(x+\textrm{i}y)\vert=-\frac{\pi}{2}\left(1+\frac{1}{2}(\gamma+\delta-\alpha-\beta+\hat{\alpha}+\hat{\beta}-\hat{\gamma}-\hat{\delta})\right)\vert y\vert+o(\vert y\vert).\end{equation*}

Because $\gamma+\delta\leq \alpha+\beta$ and $\hat{\alpha}+\hat{\beta}\leq \hat{\gamma}+\hat{\delta}+1$ (see Theorem 2.1), which gives us that $1+(\gamma+\delta-\alpha-\beta+\hat{\alpha}+\hat{\beta}-\hat{\gamma}-\hat{\delta})/2<4$, the proof is now complete.