Bivariate Bernstein-gamma functions and moments of exponential functionals of subordinators

TL;DR

This paper introduces bivariate Bernstein-gamma functions, establishes asymptotic bounds, and applies these results to derive explicit formulas for the moments of exponential functionals of subordinators, advancing understanding of Lévy process functionals.

Contribution

It extends Bernstein-gamma functions to the bivariate case, derives asymptotic bounds, and provides a new explicit convolution formula for moments of exponential functionals of subordinators.

Findings

Derived Stirling-type asymptotic bounds for bivariate Bernstein-gamma functions.

Established an explicit infinite convolution formula for the Mellin transform of exponential functionals.

Connected the results to the study of Lévy process functionals on finite time horizons.

Abstract

In this paper, we extend recent work on the functions that we call Bernstein-gamma to the class of bivariate Bernstein-gamma functions. In the more general bivariate setting, we determine Stirling-type asymptotic bounds which generalise, improve upon and streamline those found for the univariate Bernstein-gamma functions. Then, we demonstrate the importance and power of these results through an application to exponential functionals of L\'evy processes. In more detail, for a subordinator (a non-decreasing L\'evy process) $(X_s)_{s\geq 0}$, we study its \textit{exponential functional}, $\int_0^t e^{-X_s}ds $, evaluated at a finite, deterministic time $t>0$. Our main result here is an explicit infinite convolution formula for the Mellin transform (complex moments) of the exponential functional up to time $t$ which under very minor restrictions is shown to be equivalent to an infinite series. We believe this work can be regarded as a stepping stone towards a more in-depth study of general exponential functionals of L\'evy processes on a finite time horizon.

Equations

$$W_{\varphi }(z+1)=\varphi (z)W_{\varphi }(z),\mathtt{Re}(z)>0;W_{\varphi }(1)=1,$$

$$W_{\kappa }\left(\zeta ,z+1\right)=\kappa \left(\zeta ,z\right)W_{\kappa }\left(\zeta ,z\right),\mathtt{Re}(\zeta )\ge 0,\mathtt{Re}(z)>0;W_{\kappa }\left(\zeta ,1\right)=1.$$

$${\mathcal{M}}_{I_{\varphi }(t)}(z+1)=\frac{\Gamma (z+1)}{W_{\varphi }(z+1)}-\sum _{k=1}^{\infty }\frac{{\prod }_{i=1}^k[\varphi (z+i)-\varphi (k)]}{{\prod }_{j=1}^{k-1}[\varphi (j)-\varphi (k)]}\frac{e^{-\varphi (k)t}}{\varphi (k)}\frac{\Gamma (z+1)}{W_{{\varphi }_{(k)}}(z+1)},$$

$$\begin{array}{rl}& \mathbb{E}\left[{\left({\int }_0^{e_q}{e}^{-{\xi }_s}ds\right)}^z\right]={\kappa }_{q,-}(0)\frac{\Gamma (z+1)}{W_{{\kappa }_{q,+}}(z+1)}{W}_{{\kappa }_{q,-}}(-z),\mathtt{Re}\left(z\right)\in \left(-1,0\right],\end{array}$$

$$\begin{array}{rl}& \frac{1}{q}\mathbb{E}\left[{\left({\int }_0^{e_q}{e}^{-{\xi }_s}ds\right)}^z\right]={\int }_0^{\infty }{e}^{-qt}\mathbb{E}\left[{\left({\int }_0^t{e}^{-{\xi }_s}ds\right)}^z\right]dt\end{array}$$

$$\varphi (z)$$

$$=\varphi (0)+\text{d}z+z{\int }_0^{\infty }{e}^{-zy}\overline{\Pi }(y)dy,$$

$$\begin{array}{rl}& \mathbb{E}\left[e^{-z{X}_t}\right]=e^{-\varphi (z)t}\text{ for z∈C[0,∞) and t≥0.}\end{array}$$

$$X_t=\{\begin{array}{l}{X}_t,t<e_{\varphi (0)}\\ \infty ,t\ge e_{\varphi (0)}.\end{array}$$

$$W_{\varphi }(z+1)=\varphi (z)W_{\varphi }(z),\text{ for z∈C(0,∞) with }{W}_{\varphi }(1)=1.$$

$$I_{\varphi }(t):={\int }_0^t{e}^{-X_s}ds={\int }_0^{\min \left\{t,e_{\varphi (0)}\right\}}{e}^{-X_s}ds,t\in [0,\infty ],$$

$$\kappa \left(\zeta ,z\right)=\kappa (0,0)+{\mathtt{d}}_1\zeta +{\mathtt{d}}_{2}z+{\int }_0^{\infty }{\int }_0^{\infty }\left(1-e^{-\zeta x_1-z{x}_2}\right)\mu (d{x}_1,d{x}_2),$$

$${\int }_0^{\infty }{\int }_0^{\infty }\min \left\{x_1,1\right\}\min \left\{x_2,1\right\}\mu (d{x}_1,d{x}_2)<\infty .$$

$$W_{\kappa }\left(\zeta ,z+1\right)=\kappa \left(\zeta ,z\right)W_{\kappa }\left(\zeta ,z\right),z\in {\mathbb{C}}_{\left(0,\infty \right)},\zeta \in {\mathbb{C}}_{[0,\infty )};$$

$$\begin{array}{rl}\kappa \left(q,z\right)& =\kappa \left(q,0\right)+{\mathtt{d}}_{2}z+{\int }_0^{\infty }\left(1-e^{-z{x}_2}\right)\left({\int }_0^{\infty }{e}^{-q{x}_1}\mu \left(d{x}_1,d{x}_2\right)\right)\\ & =:{\varphi }_q(0)+{\mathtt{d}}_{2}z+{\int }_0^{\infty }\left(1-e^{-zx}\right){\mu }_q(dx)=:{\varphi }_q(z),\end{array}$$

$$\frac{e^{-{\gamma }_{\kappa }(\zeta )z}}{\kappa \left(\zeta ,z\right)}\prod _{k=1}^{\infty }\frac{\kappa \left(\zeta ,k\right)}{\kappa \left(\zeta ,k+z\right)}{e}^{\frac{{\kappa }_z^{\prime }(\zeta ,k)}{\kappa \left(\zeta ,k\right)}z},$$

$${\gamma }_{\kappa }(\zeta )=\underset{n\to \infty }{\lim }\left(\sum _{k=1}^n\frac{{\kappa }_z^{\prime }(\zeta ,k)}{\kappa (\zeta ,k)}-{\log }_0\left(\kappa (\zeta ,n)\right)\right),$$

$$W_{\kappa }(\zeta ,z)=\frac{e^{-{\gamma }_{\kappa }(\zeta )z}}{\kappa \left(\zeta ,z\right)}\prod _{k=1}^{\infty }\frac{\kappa \left(\zeta ,k\right)}{\kappa \left(\zeta ,k+z\right)}{e}^{\frac{{\kappa }_z^{\prime }(\zeta ,k)}{\kappa \left(\zeta ,k\right)}z},$$

$$W_{\kappa }(\zeta ,z)=\frac{1}{\kappa \left(\zeta ,z\right)}\underset{n\to \infty }{\lim }{e}^{z{\log }_0\kappa \left(\zeta ,n\right)}\prod _{k=1}^n\frac{\kappa \left(\zeta ,k\right)}{\kappa \left(\zeta ,k+z\right)}.$$

$$L_{\kappa }\left(\zeta ,z\right):={\int }_{1\to 1+z}{\log }_0\left(\kappa \left(\zeta ,\chi \right)\right)d\chi ,$$

$$L_{\kappa }\left(\zeta ,z\right)={\int }_{1\to 1+z}{\log }_0\left(\kappa \left(\zeta ,\chi \right)\right)d\chi =z{\int }_0^1{\log }_0\left(\kappa \left(\zeta ,1+zv\right)\right)dv.$$

$$\mathrm{P}(u):=\left(u-\lfloor u\rfloor \right)\left(1-\left(u-\lfloor u\rfloor \right)\right).$$

$$E_{\kappa }(\zeta ,z)=\frac{1}{2}{\int }_1^{\infty }\mathrm{P}(u){\left({\log }_0\left(\frac{\kappa \left(\zeta ,u+z\right)}{\kappa \left(\zeta ,u\right)}\right)\right)}^{\prime \prime }du.$$

$$W_{\kappa }\left(\zeta ,z\right)=\frac{{\kappa }^{\frac{1}{2}}\left(\zeta ,1\right)}{\kappa \left(\zeta ,z\right){\kappa }^{\frac{1}{2}}\left(\zeta ,1+z\right)}{e}^{L_{\kappa }\left(\zeta ,z\right)}{e}^{-E_{\kappa }\left(\zeta ,z\right)},$$

$$\underset{\kappa \in {\mathcal{B}}^2}{\sup }\underset{\left(\zeta ,z\right)\in {\mathbb{C}}_{\left[0,\infty \right)}\times {\mathbb{C}}_{\left(0,\infty \right)}}{\sup }\left|E_{\kappa }\left(\zeta ,z\right)\right|\le 2,$$

$$\underset{\mathtt{Re}\left(z\right)\to \infty }{\lim }{E}_{\kappa }\left(\zeta ,z\right)=\frac{1}{2}{\int }_1^{\infty }\mathrm{P}(u)\left({\left(\frac{{\kappa }_z^{\prime }\left(\zeta ,u\right)}{\kappa \left(\zeta ,u\right)}\right)}^2-\frac{{\kappa }_z^{\prime \prime }(\zeta ,u)}{\kappa \left(\zeta ,u\right)}\right)du:=T_{\kappa }\left(\zeta \right),$$

$$\underset{u\to \infty }{\lim }\frac{\varphi (u+a)}{\varphi (u)}=1.$$

$$\underset{u\to \infty }{\lim }\frac{\varphi (u+z)}{\varphi (u)}=1.$$

$$\underset{u\to \infty }{\lim }\frac{\Gamma (u+1)(u+1{)}^z}{\Gamma (u+1+z)}=1.$$

$$\underset{u\to \infty }{\lim }\frac{W_{\varphi }(u+1){\varphi }^z(u+1)}{W_{\varphi }(u+1+z)}=1.$$

$$\begin{split}\mathbb{E}[I_{\phi}^{z}(t)]=1\ast\frac{zt^{z-1}}{e^{\phi(1)t}}\ast\overset{\infty}{\underset{k=1}{\raisebox{-2.32501pt}{\scalebox{2.5}{$\ast$}}}}\Bigg{[}\delta_{0}(dt)&+\sum_{m=1}^{\infty}\frac{(z)^{(m)}(-[\phi(k+1)-\phi(k)])^{m}}{(m!)^{2}}\frac{mt^{m-1}}{e^{\phi(k)t}}\\ &+[\phi(z+k)-\phi(k)]\sum_{m=0}^{\infty}\frac{(z)^{(m)}(-[\phi(k+1)-\phi(k)])^{m}}{(m!)^{2}}\frac{t^{m}}{e^{\phi(k)t}}\Bigg{]},\end{split}$$

Full text

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main\sethead[0][][0 \sectiontitle] \hdrtitle0 \sethead[0][ADAM BARKER, MLADEN SAVOV][] EXPONENTIAL FUNCTIONALS OF SUBORDINATORS0 \setfoot[][][]

BIVARIATE BERNSTEIN-GAMMA FUNCTIONS AND MOMENTS OF EXPONENTIAL FUNCTIONALS OF SUBORDINATORS

A. Barker   and M. Savov Department of Mathematics and Statistics, University of Reading, Reading, United Kingdom   E-mail: Adam.Barker@pgr.reading.ac.ukInstitute of Mathematics and Informatics, Bulgarian Academy of Sciences, ”Akad. Georgi Bonchev” bl. 8, Sofia 1113, Bulgaria   E-mail: mladensavov@math.bas.bg

Abstract

In this paper, we extend recent work on the functions that we call Bernstein-gamma to the class of bivariate Bernstein-gamma functions. In the more general bivariate setting, we determine Stirling-type asymptotic bounds which generalise, improve upon and streamline those found for the univariate Bernstein-gamma functions. Then, we demonstrate the importance and power of these results through an application to exponential functionals of Lévy processes. In more detail, for a subordinator (a non-decreasing Lévy process) $(X_{s})_{s\geq 0}$, we study its exponential functional, $\int_{0}^{t}e^{-X_{s}}ds$, evaluated at a finite, deterministic time $t>0$. Our main result here is an explicit infinite convolution formula for the Mellin transform (complex moments) of the exponential functional up to time $t$ which under very minor restrictions is shown to be equivalent to an infinite series. We believe this work can be regarded as a stepping stone towards a more in-depth study of general exponential functionals of Lévy processes on a finite time horizon.

Keywords:

Lévy processes; Complex analysis; Special functions; Financial mathematics

1 Introduction, Background and Motivation

Each Bernstein function, $\phi$, has been shown to have a unique associated Bernstein-gamma function, $W_{\phi}$, defined by the recurrent equation

[TABLE]

see [33, Section 6] or [32]. In this work we study a suitable generalisation of the class of Bernstein-gamma functions. Each bivariate Bernstein function, $\kappa$ (the Lévy ​​-Khintchine exponent of a possibly-killed bivariate subordinator), is shown to have a unique bivariate Bernstein-gamma function, $W_{\kappa}$, defined by the recurrent equation

[TABLE]

Our main analytical result on bivariate Bernstein-gamma functions, Theorem 2.9, provides a general Stirling asymptotic representation for $W_{\kappa}$. It is an improvement upon [32, Theorem 3.3], which only gives a Stirling representation for the absolute value in the univariate case. Also, building upon [33, Theorem 6.1], Theorem 2.8 gives a Weierstrass product representation for $W_{\kappa}$. The univariate Bernstein-gamma functions are intimately linked to Markovian self-similarity and other important quantities in probability and spectral theory, see [1, 21, 32, 33, 34] and the discussion below. In this work we will demonstrate that bivariate Bernstein-gamma functions also play a role in probability theory via the study of exponential functionals of Lévy processes up to a deterministic horizon, although we expect further applications to appear. In more detail, we apply our results on bivariate Bernstein-gamma functions to exponential functionals of Lévy processes, as follows. For a subordinator $(X_{s})_{s\geq 0}$, we study its exponential functional, defined as $I_{\phi}(t):=\int_{0}^{t}e^{-X_{s}}ds$, $t\in[0,\infty)$. Our main results concern information on the Mellin tranform of $I_{\phi}(t)$, that is $\mathcal{M}_{I_{\phi}(t)}(z+1):=\mathbb{E}[I_{\phi}(t)^{z}]$, for $\mathtt{Re}(z)>0$. Here we only highlight the following representation:

[TABLE]

which holds under a minor regularity condition and where $\phi_{(k)}(w):=\phi(w+k)-\phi(k)$ is a Bernstein function and $W_{\phi_{(k)}}$ is its corresponding univariate Bernstein-gamma function, see Definitions 2.1, 2.2. We emphasize that $\frac{\Gamma(z+1)}{W_{\phi_{(k)}}(z+1)},k\geq 1,$ are the Mellin transforms of exponential functionals of subordinators on infinite horizon.

The question of finding precise information on the distribution of the exponential functional of a Lévy process up to a finite, deterministic time was posed in the 1990’s, see e.g. [11, Remark 3.2], yet very few works have since been able to cover this case. For the Brownian motion case an extensive study of the law of the exponential functional has been carried out in [22]. Recently, interesting results concerning moments of exponential functionals of processes with independent increments have been discussed in [35] where in particular $\mathbb{E}\left[I^{n}_{\phi}(t)\right],n\geq 1,$ have been computed when $\phi(0)=0$, see Theorem 2.17 below. In this work we provide an expression for any complex moments with positive real part. This work can be considered a stepping stone for a more in-depth study of exponential functionals of more general Lévy processes up to a deterministic horizon, and the reason why the Mellin transform is a suitable starting point in such an endeavour can be explained as follows. Consider a Lévy process $\xi$ which is killed at independent exponentially distrubuted random time $e_{q},q>0$. Then, in a sequence of papers [23, 30, 31, 32], it has been shown gradually that in general

[TABLE]

where $\kappa_{q,\pm}$ are bivariate Bernstein functions corresponding to the Wiener-Hopf factors of $\xi$, see for example [4, Chapter VI] for an introduction to Wiener-Hopf factorization. On the other hand

[TABLE]

and one can try to understand $\mathbb{E}\left[\left(\int_{0}^{t}e^{-\xi_{s}}ds\right)^{z}\right]$ through standard Laplace inversion. The point where subordinators always appear is the expression $\frac{\Gamma(z+1)}{W_{\kappa_{q,+}}(z+1)}$, which corresponds to an exponential functional of a killed subordinator. Therefore, it seems likely that our results, such as $\left(\ref{introformula}\right)$, may have implications well beyond subordinators.

The study of exponential functionals of Lévy processes has received much attention in recent years. Advancements in the general theory can be found in [2, 3, 6, 8, 23, 26, 27, 30, 31, 32, 35, 37]. These quantities have been used in various areas of probability theory, such as branching processes and processes in random environments, see [25, 20, 28], spectral theory of non-self-adjoint semigroups, see [33, 34], positive self-similar Markov processes, see [7, 19], financial mathematics, see [15] and [9, Section 6.3]. Exponential functionals up to random exponential horizon have also appeared in the study of Asian options, see [14, 17, 29]. For Asian options, which are valued according to an integral of the form $\int_{0}^{t}e^{-X_{u}}du$, where $X_{u}$ denotes an asset price at time $u$, one needs to consider exponential functionals up to deterministic horizon, but the latter have proved to be extremely hard to deal with, and that is why researchers have focused on their Laplace transform. From this perspective our studies, which deal with obtaining knowledge of the exponential functional up to a finite, deterministic time, can be relevant to pricing of Asian options. We refer to [9, 12, 38] for further details on applications of exponential functionals.

The remainder of the paper is structured as follows: Section 2.1 introduces notation and key quantities; Section 2.2 provides the statements of the main results on Bernstein-gamma functions and bivariate Bernstein-gamma functions; Section 2.3 contains the statements of the main results on exponential functionals, including the formula $\left(\ref{introformula}\right)$ for the Mellin transform; Sections 3 and 4 contain the proofs of the main results; Section 5 collects functional properties and results on bivariate Bernstein functions, which can be of independent interest; Section 6 contains proofs of the remaining lemmas.

2 Main Results

2.1 Preliminary Definitions and Notation

We start by defining some complex-analytical quantities. We use $\mathbb{C}$ to denote the complex plane. For any $z\in\mathbb{C}$, we write $z=\mathtt{Re}(z)+i\mathtt{Im}(z)$ and we set $z=|z|e^{iargz}$ with the branch of the argument function defined via the convention $arg:\mathbb{C}\mapsto\left(-\pi,\pi\right]$. For any $z\in\mathbb{C}$, set $z=|z|e^{i\arg z}$ with the branch of the argument function defined via the convention $\arg:\mathbb{C}\mapsto\left(-\pi,\pi\right]$. We put $\log_{0}:\mathbb{C}\setminus\left(-\infty,0\right]\mapsto\mathbb{C}$ for the main branch of the complex logarithm whereby $\log_{0}(z)=\ln|z|+i\arg z$. For any $-\infty\leq a<b\leq\infty$, we denote by $\mathbb{C}_{\left(a,b\right)}=\left\{z\in\mathbb{C}:\,a<\mathtt{Re}(z)<b\right\}$ and for any $a\in\left(-\infty,\infty\right)$ we set $\mathbb{C}_{a}=\left\{z\in\mathbb{C}:\,\mathtt{Re}(z)=a\right\}$. The notation $\mathbb{C}_{\left[a,b\right)}=\left\{z\in\mathbb{C}:\,a\leq\mathtt{Re}(z)<b\right\}$ and all possible variations thereof denote strips whose boundary lines are included or not in the respective subset of the complex plane. We use $\mathtt{A}_{\left(a,b\right)}$ for the set of holomorphic functions on $\mathbb{C}_{\left(a,b\right)}$, whereas if $-\infty<a$ then $\mathtt{A}_{\left[a,b\right)}$ stands for the holomorphic functions on $\mathbb{C}_{\left(a,b\right)}$ that can be extended continuously to $\mathbb{C}_{a}$. Similarly, we have the spaces $\mathtt{A}_{\left[a,b\right]}$ and $\mathtt{A}_{\left(a,b\right]}$. We employ $\mathbb{C}^{2}$ for the two dimensional complex numbers with $\mathbb{C}^{2}_{\left(a,b\right)}$ standing for $\mathbb{C}_{\left(a,b\right)}\times\mathbb{C}_{\left(a,b\right)}$ and $\mathtt{A}^{2}_{\left(a,b\right)}$ standing for the class of bivariate holomorphic functions on $\mathbb{C}^{2}_{\left(a,b\right)}$.

Now, let us state key definitions for Bernstein-gamma functions, Lévy processes, and exponential functionals.

Definition 2.1**.**

A function $\phi$ is a Bernstein function, that is $\phi\in\mathcal{B}$, if for all $z\in\mathbb{C}_{[0,\infty)}$,

[TABLE]

where $\phi(0),\textup{d}\in[0,\infty)$, $\Pi$ denotes a measure on $[0,\infty)$ satisfying $\int_{0}^{\infty}\min\left\{y,1\right\}\Pi(dy)<\infty$, and $\overline{\Pi}(x):=\int_{x}^{\infty}\Pi(dy),$ for $x\geq 0$. Note that in any case $\phi\in\mathtt{A}_{\left[0,\infty\right)}$. For further background on Bernstein functions, we refer to the book [36] or to the paper [32, Section 3].

With each $\phi\in\mathcal{B}$ there is an associated, possibly-killed subordinator (non-decreasing Lévy process) $X=(X_{t})_{t\geq 0}$, whose Lévy ​​-Khintchine exponent is defined by the relation

[TABLE]

For a subordinator with Lévy ​​-Khintchine exponent $\phi$ as in $\left(\ref{eq:Bern}\right)$, $\textup{d}\geq 0$ is the linear drift, and $\Pi$ is the Lévy measure, which determines the size and intensity of its jumps. If $\phi(0)>0$, then we say that the subordinator $X$ is killed at rate $\phi(0)$, and it follows that for an independent exponential random variable $e_{\phi(0)}$ with rate parameter $\phi(0)$,

[TABLE]

If our original, unkilled subordinator $X$ has Laplace exponent $\phi$, then the process $X$ killed at rate $q>0$ has Laplace exponent $\phi_{q}(\lambda)=q+\phi(\lambda)$.

We recall that if $Y$ is an almost surely positive random variable then $\mathcal{M}_{Y}(z):=\mathbb{E}\left[Y^{z-1}\right]$ is by definition its Mellin transform which is always well-defined at least for $z\in\mathbb{C}_{1}=1+i\mathbb{R}$. Now we define Bernstein-gamma functions.

Definition 2.2**.**

For each $\phi\in\mathcal{B}$, its associated Bernstein-gamma function $W_{\phi}$ is defined, for $z\in\mathbb{C}_{\left(0,\infty\right)}$, as the solution in the space of Mellin transforms of positive random variables of the recurrent equation

[TABLE]

The existence of $W_{\phi}$ for any $\phi\in\mathcal{B}$ is proven in [32, Section 4], where an extensive study of its complex-analytical properties has been carried out. Finally, we define the exponential functional of a subordinator:

Definition 2.3**.**

For a subordinator with Laplace exponent $\phi\in\mathcal{B}$, its exponential functional is

[TABLE]

where the terminal value, for $t=\infty$, can also be denoted by $I_{\phi}:=I_{\phi}(\infty)=\int_{0}^{\infty}e^{-X_{s}}ds$. Note that if $\phi(0)=0$ then $e_{\phi(0)}=\infty$ almost surely.

2.2 Bivariate   Bernstein-Gamma   Functions   and   their   Stirling   Type  Approximation

To extend the theory of Bernstein-gamma functions to the bivariate setting, we first define $\mathcal{B}^{2}$, the class of bivariate Bernstein functions, which generalises the class $\mathcal{B}$ to the bivariate case.

Definition 2.4**.**

We say that a function $\kappa$ is a bivariate Bernstein function if for all $\zeta,z\in\mathbb{C}_{[0,\infty)}$,

[TABLE]

where $\kappa(0,0),{\mathtt{d}}_{1},{\mathtt{d}}_{2}\in[0,\infty)$ and $\mu$ is a measure on $(0,\infty)\times(0,\infty)$ such that

[TABLE]

Note that according to Lemma 5.1 we have that $\kappa\in\mathtt{A}^{2}_{\left[0,\infty\right)}$.

Observe that $\kappa\in\mathcal{B}^{2}$ if and only if $\kappa$ is the bivariate Laplace exponent of a possibly killed bivariate subordinator, see [13, p.27] for further details. For some important properties of the class $\mathcal{B}^{2}$, see Proposition 5.2, which collects key results on the class $\mathcal{B}^{2}$. These are natural extensions of known properties of the univariate class $\mathcal{B}$ but seem not to have appeared in the literature. Now, let us define the class of bivariate Bernstein-gamma functions.

Definition 2.5**.**

We say that $W_{\kappa}$ is a bivariate Bernstein-gamma function if

[TABLE]

for each $\zeta\in\mathbb{C}_{\left[0,\infty\right)}$, $W_{\kappa}\left(\zeta,1\right)=1$; $W_{\kappa}\in\mathtt{A}^{2}_{\left(0,\infty\right)}$ and for any $q\in\left[0,\infty\right)$ the function $W_{\kappa}(q,\cdot)$ is the Mellin transform of a positive random variable.

Remark 2.6**.**

Taking $\zeta=q\in[0,\infty)$ in the formula in (2.5), we can write $\kappa\left(q,z\right)$ as

[TABLE]

where, crucially, $\phi_{q}\in\mathcal{B}$. Then, using (2.4), note that $W_{\kappa}\left(q,z\right)\equiv W_{\phi_{q}}(z)$. This gives a starting point from which we can begin to understand bivariate Bernstein-gamma functions through known univariate results, then we can extend results from $W_{\kappa}\left(q,z\right)$, $q\in[0,\infty)$, to $W_{\kappa}\left(\zeta,z\right)$, $\zeta\in\mathbb{C}_{\left[0,\infty\right)}$.

Remark 2.7**.**

Note the the reduction from the bivariate to the univariate case is simply done by taking $\kappa(q,z)=q+\phi(z),\phi\in\mathcal{B}$. This corresponds simply to the killing of one-dimensional subordinator.

It is proven in [33, Section 6] that $W_{\phi}$ admits an absolutely convergent Weierstrass product representation. In the following Theorem 2.8, we extend this infinite product representation of $W_{\phi}$ to $W_{\kappa}$.

Theorem 2.8**.**

If $\kappa\in\mathcal{B}^{2}$, then $W_{\kappa}$ as in Definition 2.5 exists and is unique, and the following product representation, defined for $\left(\zeta,z\right)\in\mathbb{C}_{\left[0,\infty\right)}\times\mathbb{C}_{\left(0,\infty\right)}$ by

[TABLE]

[TABLE]

satisfies $\left(\ref{eq:BivBernGam}\right)$. In particular, it follows that for $\left(\zeta,z\right)\in\mathbb{C}_{\left[0,\infty\right)}\times\mathbb{C}_{\left(0,\infty\right)}$

[TABLE]

so that the product in $\left(\ref{eq:Bern-GammaExt}\right)$ is indeed a product representation of $W_{\kappa}$. Moreover, $\gamma_{\kappa}\in\mathtt{A}_{\left[0,\infty\right)}$, and for $\left(\zeta,z\right)\in\mathbb{C}_{\left[0,\infty\right)}\times\mathbb{C}_{\left(0,\infty\right)}$, we can express $W_{\kappa}(\zeta,z)$ as

[TABLE]

The proof of this theorem is provided in Section 3. We proceed with the derivation of the Stirling type approximation for $W_{\kappa}$. For this purpose we need some more notation. Firstly, we introduce the function $L_{\kappa}$, which contains the main asymptotic contribution in $W_{\kappa}$, and is defined, for $\left(\zeta,z\right)\in\mathbb{C}_{\left[0,\infty\right)}\times\mathbb{C}_{\left(0,\infty\right)}$, as

[TABLE]

where the integral denotes the path integral along a contour starting from $1$ and ending at $1+z$ which lies in the domain of analyticity of $\log_{0}\left(\kappa\left(\zeta,\cdot\right)\right)$. If $\mathtt{Re}(z)>-1$ then there is a straight line connecting $1$ to $1+z$ in the domain of analyticity of $\log_{0}\left(\kappa\left(\zeta,\cdot\right)\right)$ and we have

[TABLE]

Now, for $\left(\zeta,z\right)\in\mathbb{C}_{\left[0,\infty\right)}\times\mathbb{C}_{\left(0,\infty\right)}$, $\mathtt{Re}\left(\kappa\left(\zeta,z\right)\right)>0$ by (5.8), so $L_{\kappa}$ is well-defined on $\mathbb{C}_{\left[0,\infty\right)}\times\mathbb{C}_{\left(0,\infty\right)}$. We denote the floor function $\lfloor u\rfloor=\max\left\{n\in\mathbb{N}:\,n\leq u\right\}$, and define

[TABLE]

The function $E_{\kappa}$ corresponds to the error term in our Stirling approximation, and is defined, for $\left(\zeta,z\right)\in\mathbb{C}_{\left[0,\infty\right)}\times\mathbb{C}_{\left(0,\infty\right)}$, as

[TABLE]

Now we are ready to state Theorem 2.9, the Stirling asymptotic representation for $W_{\kappa}$. For the absolute value of the univariate case a similar, but less wieldy, asymptotic representation has been derived in [32, Theorem 4.2].

Theorem 2.9**.**

Let $\kappa\in\mathcal{B}^{2}$. Then, for $\left(\zeta,z\right)\in\mathbb{C}_{\left[0,\infty\right)}\times\mathbb{C}_{\left(0,\infty\right)}$, we have that

[TABLE]

[TABLE]

[TABLE]

*and for each fixed $z\in\mathbb{C}_{\left(0,\infty\right)}$, we have that $\lim\limits_{\mathtt{Re}(\zeta)\to\infty}E_{\kappa}\left(\mathtt{Re}\left(\zeta\right),z\right)=0$. *

The proof of this theorem is provided in Section 3. Now we state two key lemmas. Lemma 2.10, is a complex generalisation of the result [32, Prop. 3.1 (8)], that for all $\phi\in\mathcal{B}$, $a\in\mathbb{R}$, and $u\in(0,\infty)$, uniformly among $a$ in compact intervals in $\mathbb{R}$,

[TABLE]

Lemma 2.10**.**

For each $\phi\in\mathcal{B}$, $z\in\mathbb{C}$, uniformly among $z$ in compact subsets of the complex plane $\mathbb{C}$,

[TABLE]

The next lemma is a generalisation, from the standard gamma function to the class of Bernstein-gamma functions, of the following result. For each $z\in\mathbb{C}$,

[TABLE]

Lemma 2.11**.**

For each $\phi\in\mathcal{B}$, $z\in\mathbb{C}_{\left(0,\infty\right)}$, uniformly among $z$ in compact subsets of the complex half-plane $\mathbb{C}_{\left[0,\infty\right)}$,

[TABLE]

The proof of Lemma 2.11 builds upon Lemma 2.10. Both proofs are contained in Section 6.

2.3 Applications to Exponential Functionals up to a Finite Time

The first of our key results on exponential functionals up to a finite time is an infinite convolution formula for the Mellin transform:

Theorem 2.12**.**

For each possibly killed subordinator with Laplace exponent $\phi\in\mathcal{B}$, for $t\in(0,\infty)$, and for $z\in\mathbb{C}_{(0,\infty)}$, the Mellin transform, $\mathcal{M}_{I_{\phi}(t)}(z+1)=\mathbb{E}[I_{\phi}^{z}(t)]$, satisfies

[TABLE]

*where the symbol $\ast$ denotes convolution $(f\ast g)(t):=\int_{0}^{t}f(s)g(t-s)ds$, the symbol

$\ast$

means infinite convolution, $\delta_{0}(dt)$ is the Dirac measure at the point [math], and $(z)^{(m)}:=z(z+1)\cdots(z+m-1)$ is the rising factorial function.*

While this result is interesting on its own, it makes the computations of the moments hard, and for this purpose we shall express our formula as an infinite sum rather than an infinite convolution. This requires a slight regularity condition on the underlying Bernstein function:

Definition 2.13** (Regularity Condition).**

We impose that the derivative of our subordinator’s Laplace exponent satisfies $\beta(\phi^{\prime})>-1$, where $\beta(\phi^{\prime})$ denotes the lower Matuszewska index of $\phi^{\prime}$, defined as the infimum of $\beta\in\mathbb{R}$ for which there exists $C>0$ such that for each $\Lambda>1$, $\phi^{\prime}(\lambda x)/\phi^{\prime}(x)\geq(1+o(1))C\lambda^{\beta}$, uniformly in $\lambda\in[1,\Lambda]$, as $x\to\infty$. See [10, p68] for more details but we highlight that cases like $\phi\in\mathcal{B}$ with $d>0$ and/or $\bar{\Pi}(x)\sim x^{-\alpha},x\to 0,\alpha\in\left(0,1\right)$ satisfy this condition but are a small sample of cases that fall under this definition.

Theorem 2.14**.**

For each subordinator whose Laplace exponent $\phi$ satisfies the condition in Definition 2.13, for all $t\in(0,\infty]$, and for all $z\in\mathbb{C}_{(0,\infty)}$,

[TABLE]

where $\phi_{(k)}(w):=\phi(w+k)-\phi(k)$ is a Bernstein function, and $W_{\phi_{(k)}}$ is its corresponding Bernstein-gamma function.

Remark 2.15**.**

*Our constraint excludes cases in which $\phi$ is slowly varying (see [10, p6]), but it should be noted that even for e.g. $\phi(x)=\ln(1+x)$, Theorem 2.14 still holds in a region of the form $\mathtt{Re}(z)>c_{t,\phi}$, with the region depending on $t$ and $\phi$. We are unable to obtain any partial results in only the most pathological cases, e.g. $\phi(x)=\ln(\ln(e+x))$, for which the rate of growth of $\phi(x)$ to $\infty$, as $x\to\infty$, is too slow. We emphasise that the Matuszewska indices of the derivatives of the Bernstein functions in this remark are of value precisely $-1$. *

Remark 2.16**.**

Note that since $\int_{0}^{e_{q}}e^{-X_{s}}ds$ has negative moments of order betwen $(-1,0)$, see [Theorem 2.4][32], then clearly $I_{\phi}(t)$ has the same moments for any $t>0$. Their evaluation is excluded in the statement of this theorem as there is a technical difficulty to obtain a similar expression. However, from (4) below, one can attack the problem from the fact that

[TABLE]

where $\varphi_{q}(z)=\phi(z)+q$ and therefore $\mathbb{E}[I_{\phi}^{z}(t)]$ is the convolution in $t$ of $\mathbb{E}[I_{\phi}^{z-1}(t)]$ with $ze^{-\phi(z)t}$ or

[TABLE]

This equation can be analysed by suitable differentiation and rearrangement.

One can quite easily obtain, using elementary methods, an explicit formula for the positive integer moments of the exponential functional of a subordinator, as it appears in the recent work [35, Corollary 1].

Theorem 2.17** (Salminen, Vostrikova 2018).**

For all $\phi\in\mathcal{B}$ with $\phi(0)=0$, and for all $n\in\mathbb{N}$,

[TABLE]

Using our methodology, based on Laplace inversion, we can deduce the following formula for integer moments, which extends [35, Corollary 1] to integer moments of killed subordinators.

Corollary 2.18**.**

For each subordinator with Laplace exponent $\phi\in\mathcal{B}$, for $t\in(0,\infty)$ and $n\in\mathbb{N}$,

[TABLE]

*where $\phi^{*}(j)=\phi(j),j\geq 1,$ and $\phi^{*}(0)=0$ and if $\phi(0)=0$ then $\phi^{*}=\phi$. *

Remark 2.19**.**

Note that as $t\to\infty$ the second relation of (2.24) yields the well known formula

[TABLE]

but it also offers an asymptotic expansion of the speed of convergence in $t$, the first term of which is exponential of value $e^{-\phi(1)t}$.

We proceed with the proofs of our results.

3 Proofs for Bivariate Bernstein-Gamma Functions

Before providing the proofs for Section 2.2, we state some key results on the class of Bernstein functions $\mathcal{B}$, which can be found in [33, Section 4].

Proposition 3.1**.**

For all $\phi\in\mathcal{B}$ and $z\in\mathbb{C}_{\left(0,\infty\right)}$, we can express the derivative of $\phi$ as

[TABLE] 2. 2.

Each $\phi\in\mathcal{B}$ is non-decreasing on $[0,\infty)$, and $\phi^{\prime}$ is completely monotone, positive, and non-increasing on $[0,\infty)$. Hence, $\phi$ is strictly log-concave on $[0,\infty)$, so for all $u\in[0,\infty)$,

[TABLE]

We are ready to start with the proof of the main results.

Proof of Theorem 2.8.

Since for $\zeta=q\geq 0$ we have that $W_{\kappa}\left(q,z\right)=W_{\phi_{q}}(z)$ with $\gamma_{\kappa}(q)$ computed as in (2.9), see Remark 2.6, then

[TABLE]

We extend analytically $\overline{W_{\kappa}}\left(\cdot,z\right)$ to $\mathbb{C}_{\left[0,\infty\right)}$. First, we extend $\gamma_{\kappa}(\cdot)$. Fix $\zeta\in\mathbb{C}_{\left(0,\infty\right)}$ and an open ball $B_{\zeta}$ centred at $\zeta$ with the closed ball $\overline{B_{\zeta}}$ satisfying $\overline{B_{\zeta}}\subset\mathbb{C}_{\left(0,\infty\right)}$. From item 5 of Proposition 5.2, that is $\mathtt{Re}(\kappa(\zeta,k))>0,k\geq 1$, and $\kappa\in\mathtt{A}^{2}_{\left(0,\infty\right)}$, see Lemma 5.1, we deduce that for any $n\in\mathbb{N}^{+}:=\left\{1,2,\cdots\right\}$

[TABLE]

Moreover, we observe that

[TABLE]

where in the first inequality we have used first order Taylor’s expansion of $\frac{\kappa^{\prime}_{z}\left(\zeta,x+k\right)}{\kappa(\zeta,x+k)}$ about $x=0$ and immediate bounds, in the second we have applied (5.6) and (5.7), and the third follows from the well-known $\sum_{k=1}^{\infty}k^{-2}=\pi^{2}/6$. Henceforth,

[TABLE]

Therefore, $f_{n}$ are uniformly bounded holomorphic functions on $\overline{B_{\zeta}}$ and from the dominated convergence theorem which is applicable thanks to (3.5) we deduce that

[TABLE]

From classical result of complex analysis we conclude that $f\in\mathtt{A}_{B_{\zeta}}$ and since $\zeta\in\mathbb{C}_{\left(0,\infty\right)}$ we get that $f\in\mathtt{A}_{\left(0,\infty\right)}$. Since by the very definition of $\gamma_{\kappa}$, see (2.9), $\gamma_{\kappa}=\lim\limits_{n\to\infty}f_{n}=f$ we conclude that $f=\gamma_{\kappa}\in\mathtt{A}_{\left(0,\infty\right)}$. The claim that $\gamma_{\kappa}\in\mathtt{A}_{\left[0,\infty\right)}$ is then affirmed by the fact that $\kappa\left(\zeta,k\right),\kappa^{\prime}_{z}\left(\zeta,x+k\right),\,k\geq 1,x\in\left[0,1\right],$ extend continuously to $\zeta\in i\mathbb{R}$ and that the uniform bound (3.5) is valid for $\zeta\in i\mathbb{R}$. In view of $\gamma_{\kappa}\in\mathtt{A}_{\left[0,\infty\right)}$ to analytically extend in $\zeta$ the right-hand side of the infinite product in (2.8), it suffices from Hartog’s theorem, see [18, Section 2.4], to fix $z\in\mathbb{C}_{\left(0,\infty\right)}$ and show using Montel’s theorem that the infinite product, which we record in (3.6) below, converges absolutely

[TABLE]

For any $k\geq 1$, we get using the Taylor’s expansion

[TABLE]

where $a_{k}\in\left[0,1\right]$. Thus,

[TABLE]

From (5.6) we have that on any ball $\overline{B_{\zeta}}\subset\mathbb{C}_{\left(0,\infty\right)}$ centred at $\zeta$ and any $k\geq 1$

[TABLE]

From (5.9), (5.3), (5.7) and the fact that $\left|\phi_{\mathtt{Re}(\zeta)}^{\prime\prime}\right|=-\phi_{\mathtt{Re}(\zeta)}^{\prime\prime}$ decreases on $[0,\infty)$ we also observe that for any $k\geq 1$

[TABLE]

Therefore, using $\log_{0}\left(1+w\right)=w+\mathrm{O}\left(w^{2}\right)$, as $|w|\sim 0$, we get for all $k$ large enough

[TABLE]

and hence from (3.6)

[TABLE]

This shows that $\overline{W_{\kappa}}\in\mathtt{A}^{2}_{\left(0,\infty\right)}$ and it equates the infinite product in (2.8). To deduce that $W_{\kappa}=\overline{W_{\kappa}}$ and therefore $W_{\kappa}$ is represented as in (2.8) we proceed to demonstrate first that (2.11) holds. For this it suffices to use that the infinite product in (2.8) is absolutely convergent and the limit that deifines $\gamma_{\kappa}$, see (2.9). Indeed, we simply write

[TABLE]

Then

[TABLE]

provided $\lim\limits_{n\to\infty}\frac{\kappa\left(\zeta,n+1\right)}{\kappa\left(\zeta,n\right)}=1$. This is an elementary consequence of

[TABLE]

inequality (5.10) and item 8 of Lemma 5.2. Clearly, $\overline{W_{\kappa}}\left(\zeta,1\right)=1$. Now, $\overline{W_{\kappa}}$ satisfies all conditions of Definition 2.5 and therefore as it coincides with $W_{\kappa}$ for $\zeta=q\in\left[0,\infty\right)$ we deduce they coincide on $\mathbb{C}_{\left[0,\infty\right)}\times\mathbb{C}_{\left(0,\infty\right)}$. The uniqueness of $W_{\kappa}$ follows from the fact that the positive random variable associated with the Mellin transform $W_{\kappa}(q,z)$ is unique, for each $q\geq 0$, see e.g. [32, Section 4] and the discussion in [33, Section 6]. This concludes the proof of the theorem. ∎

Next, we prove the Stirling asymptotic representation for a bivariate Bernstein-gamma function.

Proof of Theorem 2.9.

Let us denote $f(x):=\log_{0}\left(\kappa\left(\zeta,x+z\right)/\kappa\left(\zeta,x\right)\right)$ and $S_{n}:=\sum_{k=1}^{n}f(k)$. Then applying $\left(\ref{eq:Bern-GammaExtLim}\right)$, for each $\left(\zeta,z\right)\in\mathbb{C}_{\left[0,\infty\right)}\times\mathbb{C}_{\left(0,\infty\right)}$, we can express $W_{\kappa}(\zeta,z)$ as

[TABLE]

Then applying [24, Section 8.2, (2.01),(2.03)] with $m=1$ in their notation, we can write $S_{n}$ as

[TABLE]

where recalling that $\mathrm{P}(u)=\left(u-\lfloor u\rfloor\right)\left(1-\left(u-\lfloor u\rfloor\right)\right)$, we set

[TABLE]

Let us first show that

[TABLE]

and estimate uniformly $\left|E_{\kappa}\right|$. For this purpose we estimate using (5.10) and (5.11) that

[TABLE]

where in the last inequality we have invoked the fact that $\mathtt{Re}\left(z\right)\geq 0$. Therefore with the help of the last inequality and

[TABLE]

we obtain that

[TABLE]

and hence (3.16). The bound (2.16) also follows from (3.17). The limit (2.17) is easily deduced from (3.17) wherein the first two terms in the upper bound vanish as $\mathtt{Re}\left(z\right)\to\infty$. Let us show that for any fixed $z$, $\lim\limits_{\mathtt{Re}(\zeta)\to\infty}E_{\kappa}\left(\mathtt{Re}(\zeta),z\right)=0$. From (2.5) we easily get that for any $z\in\mathbb{C}_{\left(0,\infty\right)}$

[TABLE]

whereas from (5.2) we get since $\lim\limits_{\mathtt{Re}(\zeta)\to\infty}\mu_{\mathtt{Re}(\zeta)}(dy)\stackrel{{\scriptstyle w}}{{=}}0dy$ and

[TABLE]

that for any $z\in\mathbb{C}_{\left(0,\infty\right)}$

[TABLE]

By the same reasoning $\lim\limits_{\mathtt{Re}(\zeta)\to\infty}\kappa^{\prime\prime}_{z}(\mathtt{Re}(\zeta),z)=0$. This together with (3.19), (3.20) and the dominated convergence theorem (applicable due to (3.17)) yield that

[TABLE]

Therefore, since $\left|\kappa(\infty,z)\right|=\infty$ whenever $d_{2}>0$ it follows that $\lim\limits_{\mathtt{Re}(\zeta)\to\infty}E_{\kappa}\left(\mathtt{Re}(\zeta),z\right)=0$. Next, we investigate the term $f\left(n\right)$ in (3.14). We write

[TABLE]

Clearly, from (5.9), Proposition 3.1 2 and $\mathtt{Re}\left(z\right)\geq 0$

[TABLE]

Thus, from (5.3) and (5.6) we arrive at

[TABLE]

Thus, if $|z|=\mathrm{o}\left(n\right)$

[TABLE]

and, as $n\to\infty,z=\mathrm{o}\left(n\right)$, we get that

[TABLE]

Henceforth we obtain from (3.14) that

[TABLE]

Next, we estimate $I_{n}=\int_{1}^{n}f(x)dx$, as $n\to\infty$. From (5.8) and since $\left(\zeta,z,x\right)\in\mathbb{C}_{\left[0,\infty\right)}\times\mathbb{C}_{\left(0,\infty\right)}\times\left[1,\infty\right)$ both $\mathtt{Re}\left(\kappa\left(\zeta,x\right)\right)>0,\mathtt{Re}\left(\kappa\left(\zeta,z+x\right)\right)>0$. Therefore, $\arg\left(\frac{\kappa\left(\zeta,x+z\right)}{\kappa\left(\zeta,x\right)}\right)\in\left(-\pi,\pi\right)$ and thus

[TABLE]

Thus, fixing the parallelogram in $\mathbb{C}$ with vertices $1,1+z,z+n,n$, we see from the Cauchy integral theorem applied to the function $\log_{0}\left(\kappa\left(\zeta,\cdot\right)\right)$ which is holomorphic in an open neighbourhood of the parallelogram that

[TABLE]

From the latter and (3.24) we get that

[TABLE]

and hence (3.13) can be re-expressed as follows

[TABLE]

However, as (3.25) holds true we get using (3.22) when $|z|=\mathrm{o}\left(n\right)$ that

[TABLE]

and hence

[TABLE]

Finally, combining together the results (3.16), (3.22), (3.28) and (3.29), we deduce, as required for (2.15), that

[TABLE]

This completes the proof of the theorem. ∎

4 Proofs for Exponential Functionals of Lévy Processes

Our starting point in determining the Mellin transform of the exponential functional up to a finite time is the following equation, proven in [33, Proposition 6.1.2]. For all $\phi\in\mathcal{B}$ and $z\in\mathbb{C}_{(-1,\infty)}$,

[TABLE]

This allows us to understand $I_{\phi}(t)$, $t<\infty$, in terms of the exponential functionals $I_{\varphi_{q}}(\infty)$ of the subordinators with Lévy ​​-Khintchine exponent $\varphi_{q}(w):=\phi(w)+q$, for $q\geq 0$, through the following argument:

[TABLE]

The formula in Theorem 2.12 comes from inverting this Laplace transform. First, we will express the Bernstein-gamma function $W_{\varphi_{q}}(z+1)$ as a product of simple Laplace transforms in the variable $q$, which facilitates our proof of Theorem 2.12.

Lemma 4.1**.**

For each subordinator with Laplace exponent $\varphi_{q}$, for all $z\in\mathbb{C}_{(-1,\infty)}$,

[TABLE]

We prove this lemma in Section 6. Next, let us state some important facts about convolutions and Laplace transforms. We define these as $(f\ast g)(t):=\int_{0}^{t}f(s)g(t-s)ds$, and \mathcal{L}\big{|}_{q}\{f\}:=\int_{0}^{\infty}e^{-qs}f(s)ds, respectively. Then we have

[TABLE]

[TABLE]

Moreover, for $f(t):=t^{z-1},g(t):=mt^{m-1}$,

[TABLE]

where we recall that $(z)^{(m)}=z\left(z+1\right)\cdots(z+m-1)$ is the rising factorial function. Similarly, with $f(t):=t^{z-1},g(t):=t^{m}$,

[TABLE]

Also, we can expand brackets for convolutions of exponentials, in the sense that

[TABLE]

Now we are ready to prove Theorem 2.12.

Proof of Theorem 2.12.

First, observe that since $\varphi^{\prime}_{q}$ is non-increasing, applying $\left(\ref{eq:phi'_phi}\right)$, we have

[TABLE]

Then using the generalised binomial series expansion for $|w|<1$

[TABLE]

we can rewrite the product in $\left(\ref{prodformula}\right)$ as

[TABLE]

Now, we can write each individual term in $\left(\ref{sumprod}\right)$ as a well-known Laplace transform, which will then allow us to invert the Laplace transform to yield the expression in $\left(\ref{convformula}\right)$ as an infinite convolution. One easily verifies that the terms in $\left(\ref{sumprod}\right)$ are the following Laplace transforms:

[TABLE]

and when $m=0$, the term in $\left(\ref{conv4}\right)$ is simply the Laplace transform of the unit point mass $\delta_{0}(dt)$. Substituting $\left(\ref{conv1}\right),\left(\ref{conv2}\right)$, $\left(\ref{conv3}\right),$ and $\left(\ref{conv4}\right)$ into the equation $\left(\ref{sumprod}\right),$ we arrive at the following formula:

[TABLE]

[TABLE]

Now, by $\left(\ref{laplacesum}\right)$ and $\left(\ref{expout}\right)$, noting $\Gamma(m)=(m-1)!$ and $\delta_{0}(\cdot)\ast f\equiv f$ for any function $f$, we have:

[TABLE]

and then the desired result $\left(\ref{convformula}\right)$ follows immediately by a simple Laplace inversion. ∎

Lemma 4.2, which we prove in Section 6, builds upon Theorem 2.12 by evaluating the first $n$ terms in the infinite convolution. In the proof of Theorem 2.14, we will take limits as $n\to\infty$.

Lemma 4.2**.**

For all $\phi\in\mathcal{B}$, $z\in\mathbb{C}_{(0,\infty)}$, and $n\in\mathbb{N}$, the following truncated convolution satisfies

[TABLE]

[TABLE]

[TABLE]

where $\gamma(z,u):=\int_{0}^{u}e^{-x}x^{z-1}dx$ denotes the lower incomplete gamma function.

The proof of Theorem 2.14 also requires the following lemmas, which are also proven later in Section 6.

Lemma 4.3**.**

For each $\phi\in\mathcal{B}$ and $c>0$, there exists a constant $C_{\phi,c}>0$ such that for all $n\geq 1$,

[TABLE]

Lemma 4.4**.**

For $\phi\in\mathcal{B}$ satisfying the condition in Definition 2.13 and for each $z\in\mathbb{C}_{(0,\infty)}$,

[TABLE]

Lemma 4.5**.**

For all $\phi\in\mathcal{B}$ and $z\in\mathbb{C}_{(0,\infty)}$, there is a constant $c_{\phi,z}>0$ such that for all $k\geq 1$,

[TABLE]

The structure of the proof of Theorem 2.14 is as follows: First, we will find the termwise limit, as $n\to\infty$, of each summand in $\left(\ref{nthtermeqn}\right)$. This gives a formula for the infinite convolution in $\left(\ref{convformula}\right)$, without the convolution with $1$, which means this limit is an expression for $\frac{d}{dt}\mathbb{E}[I_{\phi}^{z}(t)]$. To show that the termwise limits correspond to the limit of the whole expression in $\left(\ref{nthtermeqn}\right)$, we employ a dominated convergence argument. We finish the proof of Theorem 2.14 by integrating each term in our expression for $\frac{d}{dt}\mathbb{E}[I_{\phi}^{z}(t)]$, which requires a second dominated convergence argument.

Proof of Theorem 2.14.

Denoting $\phi_{(k)}(w):=\phi(w+k)-\phi(k)$, we first verify that $\phi_{(k)}$ is itself a Bernstein function, with unchanged drift and rescaled Lévy measure of the from $e^{-kx}\Pi(dx)$. Indeed, this follows by noting that $\left(\ref{eq:Bern}\right)$ gives

[TABLE]

Now, rewriting in terms of $\phi_{(k)}$, the convolution in $\left(\ref{nthtermeqn}\right)$ satisfies

[TABLE]

For the termwise limit as $n\to\infty$, first observe that under the conditions of Definition 2.13, relation (6.16) below implies that $\phi(\infty)=\infty$ and hence $\lim_{n\to\infty}zt^{z-1}/e^{\phi(n+1)t}=0$, and second that $\lim_{n\to\infty}\gamma(z,[\phi(n+1)-\phi(k)]t)=\Gamma(z)$. Repeatedly using that $W_{\phi_{(k)}}(w+1)=\phi_{(k)}(w)W_{\phi_{(k)}}(w)$ from $\left(\ref{eq:Bern-Gamma}\right)$, noting $W_{\phi_{(k)}}(1)=1$, and applying Lemma 2.11, it follows that as $n\to\infty$,

[TABLE]

This gives us the limit, as $n\to\infty$, of each summand in $\left(\ref{nthtermeqn}\right)$. Using the dominated convergence theorem, we will now show that we can exchange the order of limits and summation, which yields the following relation

[TABLE]

In order to apply the dominated convergence theorem, we must show that the sum in $\left(\ref{nthtermeqn}\right)$ is dominated by an absolutely convergent sum. From $\left(\ref{Wtoprod}\right)$, we can rewrite the sum in $\left(\ref{nthtermeqn}\right)$ as

[TABLE]

Then if we can show that there exist $n_{0}>0$ and $C>0$ such that for all $n\geq n_{0}$, and for all $k\geq 1$,

[TABLE]

and moreover if

[TABLE]

then the dominated convergence theorem applies, so $\lim_{n\to\infty}\sum_{k=1}^{\infty}F_{k}(n)=\sum_{k=1}^{\infty}\lim_{n\to\infty}F_{k}(n)$, as required for $\left(\ref{derivativeformula}\right)$. The absolute convergence of the sum in $\left(\ref{absconv}\right)$ is proven in Lemma 4.4, so now let us prove $\left(\ref{desiredbound}\right)$. Firstly, one can easily verify that for all $n,k\in\mathbb{N}$, with $z=a+ib$,

[TABLE]

The remaining term in the expression for $F_{k}(n)$ from $\left(\ref{F_k}\right)$ which depends on $n$ is

[TABLE]

Applying Theorem 2.9, we can rewrite $\left(\ref{remainingWterm}\right)$ as

[TABLE]

We will bound the absolute value of the terms in $\left(\ref{phi(k)}\right)$ separately. Firstly, using the result [32, Prop. 3.1.9] that $|\phi(a+ib)|\geq\phi(a)$ and the fact that $\phi$ is non-decreasing on $\mathbb{R}_{+}$, it follows that

[TABLE]

To bound the $E_{\phi_{(k)}}$ terms in $\left(\ref{phi(k)}\right)$, observe that by $\left(\ref{eq:Ek}\right)$, $\sup_{n\geq 1}\sup_{1\leq k\leq n}|E_{\phi_{(k)}}(z+n+1-k)|\leq 2,$ and $\sup_{n\geq 1}\sup_{1\leq k\leq n}|E_{\phi_{(k)}}(n+1-k)|\leq 2,$ so it follows immediately that for all $k\leq n$,

[TABLE]

Next, consider the $L_{\phi_{(k)}}$ terms in $\left(\ref{phi(k)}\right)$, which are defined in $\left(\ref{eq:A}\right)$. Part of the integrals cancel, so

[TABLE]

[TABLE]

where we substitute $N:=n+2$. Using the fact that $\log_{0}(w)=\ln(|w|)+i\arg(w)$, it follows that

[TABLE]

We consider the integrals in $\left(\ref{Lints}\right)$ along the contours $\gamma_{1},\gamma_{2}$, which are straight lines connecting $N-k$ to $N-k+a$ and $N-k+a$ to $N-k+a+ib$, respectively. Observe that along $\gamma_{1}$, $\arg(\phi_{(k)}(x))=0$, and note $|\gamma_{2}|=b$. Then for all $1\leq k\leq n$, the second integral in $\left(\ref{Lints}\right)$ satisfies

[TABLE]

Next, we consider the first integral in $\left(\ref{Lints}\right)$ over $\gamma_{1}$. Recalling $N=n+2$, this can be written as

[TABLE]

Comparing this with the $1/\phi^{a}_{(k)}(n+1-k)=e^{-a\ln(\phi_{(k)}(n+1-k))}$ term from $\left(\ref{phi(k)}\right)$, we get

[TABLE]

Observe that $(\phi(x+n+2)-y)/(\phi(n+1)-y)$ and $\phi(y)$ are non-decreasing in $y$. So for all $k\leq n$,

[TABLE]

Now, since $\phi^{\prime}$ is non-increasing on $\mathbb{R}_{+}$, applying Lemma 4.3, it follows that

[TABLE]

Finally, consider the first integral in $\left(\ref{Lints}\right)$ over the contour $\gamma_{2}$. To bound $\left(\ref{phi(k)}\right)$, we compare it to $1/\phi_{(k)}(n+1-k)^{ib}$. Then since a real number raised to an imaginary power has absolute value 1, we arrive at

[TABLE]

Combining together the bounds $\left(\ref{phieasybound}\right)$, $\left(\ref{Eeasybound}\right)$, $\left(\ref{Lgamma1bound}\right)$, and $\left(\ref{Lgamma2bound}\right)$, we conclude that there exists a constant $C(z,\phi)$ such that uniformly among $n\in\mathbb{N}$ and $1\leq k\leq n$, $|\left(\ref{phi(k)}\right)|\leq C(z,\phi)$, from which $\left(\ref{desiredbound}\right)$ follows, and therefore we have shown that $\left(\ref{derivativeformula}\right)$ holds. Now, applying $\left(\ref{derivativeformula}\right)$, and recalling from $\left(\ref{bgreln}\right)$ that $\int_{0}^{\infty}\frac{d}{ds}\mathbb{E}[I_{\phi}^{z}(s)]ds=\mathbb{E}[I_{\phi}^{z}(\infty)]=\Gamma(z+1)/W_{\phi}(z+1)$, we can express $\mathbb{E}[I_{\phi}^{z}(t)]$ as

[TABLE]

Now, we wish to exchange the order of integration and summation in $\left(\ref{intsumm}\right)$. If we can show that

[TABLE]

then by Fubini’s theorem, we can exchange the order of integration and summation, yielding the result

[TABLE]

Substituting this into $\left(\ref{intsumm}\right)$, we conclude, as required, that

[TABLE]

or relation (2.22) holds true. To see that $\left(\ref{intsumabsconv}\right)$ is finite, following the same argument as in $\left(\ref{asabove}\right)$, noting that $\phi$ is non-decreasing,

[TABLE]

and then it follows by Lemma 4.4 that $\left(\ref{intsumabsconv}\right)$ is a finite quantity and so the proof of Theorem 2.14 is complete.

∎

Before we prove Corollary 2.18, we state the following fact about Vandermonde matrices, see e.g. [16, p37]. For $a_{1},\dots,a_{n}\in\mathbb{C}$, the determinant of the $n\times n$ Vandermonde matrix has the form

[TABLE]

We derive some elementary consequence of this representation, which perhaps is located somewhere in the literature.

Proposition 4.6**.**

For any $n+1$ different complex numbers $a_{0},a_{1},\cdots,a_{n}$ it holds that

[TABLE]

Proof of Proposition 4.6.

To derive (4.36) we multiply through by $\underset{0\leq h<l\leq n}{\prod}[a_{h}-a_{l}]$, and we see that $\left(\ref{sum=0}\right)$ holds if and only if

[TABLE]

Applying the formula $\left(\ref{vandermondedeterminant}\right)$ for Vandermonde determinants, one can verify that the sum in $\left(\ref{vdet}\right)$ equals the following determinant, evaluated using the cofactor expansion along its first column:

[TABLE]

This determinant is 0 as the first two columns are identical, so $\left(\ref{vdet}\right)=0$. This proves (4.36). ∎

Proof of Corollary 2.18.

We specialise (4) for the case $z=n$ and use the recurrent relation (2.4) to get

[TABLE]

From here we immediately get that for any $t>0$

[TABLE]

Clearly, for $n=1$,

[TABLE]

which gives the first identity of (2.24) for $n=1$. We proceed by induction assuming that the first identity of (2.24) holds for $n=N$ and all $t>0$. We consider $\mathbb{E}\left[I^{N+1}_{\phi}(t)\right]$. From above and the inductive hypothesis we have that

[TABLE]

Now, since $\phi^{*}(j),0\leq j\leq N$, are different numbers, from (4.36) we conclude that

[TABLE]

Hence, substituting above

[TABLE]

which verifies the inductive hypothesis and thus the first identity of (2.24) is proven. For the second we use again that (4.39) holds true to deduce that for any $n$

[TABLE]

and upon substitution, (4.39) follows. ∎

The proof of Lemma 4.2 relies upon the following lemma, which we shall prove after Lemma 4.2.

Lemma 4.7**.**

For each $\phi\in\mathcal{B}$, and for all $n\geq 1$,

[TABLE]

Proof of Lemma 4.2.

We use a proof by induction. For the base case, $n=1$, we need to verify

[TABLE]

[TABLE]

Applying the properties of convolutions $\left(\ref{m-1}\right)$, $\left(\ref{m}\right)$, and $\left(\ref{expout}\right)$, we can write

[TABLE]

as required for the base case. For the inductive step, assume that the formula in $\left(\ref{nthtermeqn}\right)$ holds for the $(n-1)$th convolution. Noting that

[TABLE]

this means that

[TABLE]

We are going to evaluate the convolution of $\left(\ref{indhyp}\right)$ with the next term in $\left(\ref{convformula}\right)$, that is, the expression

[TABLE]

For brevity, we label the above convolution as

[TABLE]

First, observe that $\left((1A)+\sum_{k=1}^{n-1}(1k)\right)\ast\delta_{0}(dt)=(1A)+\sum_{k=1}^{n-1}(1k)$. We will see that this contribution cancels with some of the other terms arising from $\left(\ref{convs}\right)$, which we evaluate individually.

Evaluating the $(1A)\ast(HG1)$ term

Applying the properties $\left(\ref{expout}\right)$ and $\left(\ref{m-1}\right)$, we can write

[TABLE]

Evaluating the $(1A)\ast(HG2)$ term

Applying the properties $\left(\ref{expout}\right)$ and $\left(\ref{m}\right)$, we can write

[TABLE]

Evaluating each $(1k)\ast(HG1)$ term

To evaluate $(1k)\ast(HG1)$, we first consider the quantity

[TABLE]

Observe that

[TABLE]

Then by $\left(\ref{expout}\right)$ and $\left(\ref{m-1}\right)$,

[TABLE]

Now, multiplying $\left(\ref{simple1}\right)$ by suitable constants as in $\left(\ref{convs}\right)$, we can express $(1k)\ast(HG1)$ as

[TABLE]

Evaluating each $(1k)\ast(HG2)$ term

To evaluate $(1k)\ast(HG2)$, we first consider the quantity

[TABLE]

Recall that $e^{-\phi(k)t}\int_{0}^{t}e^{-[\phi(n)-\phi(k)]s}s^{z-1}ds=e^{-\phi(k)t}\ast t^{z-1}e^{-\phi(n)t}$. Then by $\left(\ref{expout}\right)$ and $\left(\ref{m}\right)$,

[TABLE]

Now, multiplying $\left(\ref{simple2}\right)$ by suitable constants as in $\left(\ref{convs}\right)$, we can express $(1k)\ast(HG2)$ as

[TABLE]

Combining $\left(\ref{1A*HG1}\right)$, $\left(\ref{1A*HG2}\right)$ , $\left(\ref{1k*HG1}\right)$, and $\left(\ref{1k*HG2}\right)$ gives an unwieldly formula for $\left(\ref{convs}\right)$. Observe that \big{[}(1A)+\sum_{k=1}^{n-1}(1k)\big{]}\ast\delta_{0}(dt)=(1A)+\sum_{k=1}^{n-1}(1k) cancels respectively with the second terms from $\left(\ref{1A*HG1}\right)$ and $\left(\ref{1k*HG1}\right)$. Then

[TABLE]

Multiplying $\left(\ref{tidy2}\right)$ by $1=\frac{\phi(n)-\phi(k)}{\phi(n)-\phi(k)}$, we can combine the terms $\left(\ref{tidy2}\right)$ and $\left(\ref{tidy3}\right)$, then recalling the definition $\gamma(z,u):=\int_{0}^{u}x^{z-1}e^{-x}dx$, we apply a simple change of variables to get

[TABLE]

Therefore we see that the terms in $\left(\ref{tidy2}\right)+\left(\ref{tidy3}\right)$ correspond to the desired sum between $1$ and $n-1$ as in equation $\left(\ref{nthtermeqn}\right)$, and now all that remains for the proof of Lemma 4.2 is to verify that

[TABLE]

To show this, first observe that integral which appears in $\left(\ref{tidy1}\right)$ and $\left(\ref{tidy4}\right)$ can be written as

[TABLE]

from which it follows that $\left(\ref{almostdone}\right)$ holds if and only if

[TABLE]

then rearranging and dividing through by $[\phi(z+n)-\phi(n)]$, we see that $\left(\ref{almostdone}\right)$ holds if and only if

[TABLE]

Multiplying both sides by $\underset{1\leq h<l\leq n}{\prod}[\phi(l)-\phi(h)]$, one can verify that $\left(\ref{almostdone}\right)$ holds if and only if

[TABLE]

But this equation holds by Lemma 4.7, and so the proof of Lemma 4.2 is complete. ∎

Proof of Lemma 4.7.

Applying $\left(\ref{vandermondedeterminant}\right)$, then using elementary row and column operations, we get

[TABLE]

Next, we add a scalar multiple of the first row to each of the other rows, yielding

[TABLE]

Evaluating this using the first row’s cofactor expansion, we get two terms, the first of which is

[TABLE]

since the last column is a linear combination of the other columns, and the remaining term is

[TABLE]

∎

5 Auxiliary results on bivariate Bernstein functions

Following (2.7), for each $\zeta\in\mathbb{C}_{\left[0,\infty\right)}$ we can write

[TABLE]

where $\bar{\mu}_{\zeta}(y)=\int_{y}^{\infty}\mu_{\zeta}(dx),y>0$. We then have the following elementary claim which we provide without proof.

Lemma 5.1**.**

Let $\kappa\in\mathcal{B}^{2}$. Then $\kappa\in\mathtt{A}^{2}_{\left[0,\infty\right)}$. Moreover, for any $\zeta\in\mathbb{C}_{\left[0,\infty\right)}$ we have that $\kappa\left(\zeta,z\right)=\phi_{\zeta}(z)$ and $\phi_{\zeta}\in\mathtt{A}_{\left[0,\infty\right)}$, see (5.1). The measure $\mu_{\zeta}$ is a complex measure on $\left(0,\infty\right)$. If $\zeta\in\mathbb{R}$ then $\phi_{\zeta}\in\mathcal{B}$ if and only if $\int_{0}^{\infty}\min\left\{1,x\right\}\mu_{\zeta}(dx)<\infty$, which is always the case if $\zeta\geq 0$.

The next result collects results which may be regarded as simple extensions of results available for $\phi\in\mathcal{B}$ which can be found in [33, Section 4] or [32, Section 3].

Proposition 5.2**.**

Let $\kappa\in\mathcal{B}^{2}$. Then each of the following items holds:

For all $\left(\zeta,z\right)\in\mathbb{C}^{2}_{\left(0,\infty\right)}$,

[TABLE] 2. 2.

For each $\zeta\in\left[0,\infty\right)$, $\phi_{\zeta}$ is non-decreasing on $[0,\infty)$, and $\phi_{\zeta}^{\prime}$ is completely monotone, positive, and non-increasing on $[0,\infty)$. In particular, $\phi_{\zeta}$ is strictly log-concave on $[0,\infty)$. 3. 3.

For all $\left(\zeta,z\right)\in\mathbb{C}^{2}_{\left[0,\infty\right)}\setminus\left\{\left(0,0\right)\right\}$, we have that

[TABLE]

and moreover, the inequality $\left(\ref{eq:ratioBounds}\right)$ is valid for $\zeta=z=0$ when $\kappa\left(0,0\right)>0$. 4. 4.

For all $\left(\zeta,z\right)\in\mathbb{C}_{\left[0,\infty\right)}\times\mathbb{C}_{(0,\infty)}$, we have

[TABLE]

and

[TABLE]

In particular, taking $z=\mathtt{Re}(z)=u\in[0,\infty)$ and $\zeta\in\mathbb{C}_{\left(0,\infty\right)}$, we have

[TABLE]

[TABLE] 5. 5.

For all $\left(\zeta,z\right)\in\mathbb{C}^{2}_{\left[0,\infty\right)}\setminus\left\{\left(0,0\right)\right\}$, we have that

[TABLE]

and if in addition $\mathtt{Re}(\zeta)>0$ denoting $B_{\zeta}$ as a ball centred at $\zeta$ with $\overline{B_{\zeta}}\subset\mathbb{C}_{\left(0,\infty\right)}$, we have that $\inf_{\chi\in\overline{B_{\zeta}}}\mathtt{Re}\left(\kappa\left(\chi,z\right)\right)>0$. 6. 6.

For all $\left(\zeta,z\right)\in\mathbb{C}_{\left[0,\infty\right)}\times\mathbb{C}_{(0,\infty)}$ and for all $n\geq 1$

[TABLE] 7. 7.

For all $\left(\zeta,z\right)\in\mathbb{C}_{\left[0,\infty\right)}\times\mathbb{C}_{(0,\infty)}$, we have that

[TABLE]

[TABLE] 8. 8.

For $\zeta\in\mathbb{C}_{\left[0,\infty\right)}$ and any $A>0$ we have that

[TABLE]

Proof of Proposition 5.2.

Item 1 follows immediately by a simple rearrangement using (5.1), and item 2 follows from the fact that $\phi_{\zeta}\in\mathcal{B}$ for $\zeta\in\left[0,\infty\right)$, using item 2 of Proposition 3.1. Now, let us prove item 3. Let $\left(\left(\xi_{t},\eta_{t}\right)\right)_{t\geq 0}$ denote the (possibly killed) bivariate subordinator associated to $\kappa$. Then for an independent exponential random variable $e_{q}$ with parameter $q>0$, we have

[TABLE]

so that

[TABLE]

for all $q>0$, from which it follows, taking limits as $q\to 0$, that (5.3) holds, and similarly (5.3) is valied, for $\mathtt{Re}\left(\zeta\right)=\mathtt{Re}\left(z\right)=0$, if $\kappa\left(0,0\right)>0$. Let us prove item 4. We estimate the last term in (5.2) to get that

[TABLE]

where the last inequality follows from the fact that $\phi_{\mathtt{Re}(\zeta)}\in\mathcal{B}$, $\phi_{\mathtt{Re}(\zeta)}^{\prime}\geq 0$ on $[0,\infty)$ and the left-hand side of inequality (3.2). Relation (5.13) together with (5.2) show the very first inequality in (5.4) whereas the second follows from an application of (5.3) and the third from the monotonicity of $\phi_{\mathtt{Re}(\zeta)}$ on $[0,\infty)$. We proceed to establish relation (5.5). For this purpose we note from (5.2) that

[TABLE]

where for the last inequality we have invoked (5.13). It remains to bound the very last integral. However, if $\phi\in\mathcal{B}$ then $\phi^{\prime}$ is completely monotone, see item 2 of Proposition 3.1, and hence $\phi^{\prime\prime}\leq 0$ on $[0,\infty)$. Differentiating the last expression of (3.1) and utilizing once again (5.13) we thus arrive at

[TABLE]

and hence

[TABLE]

Therefore, collecting the terms we obtain that

[TABLE]

and employing (5.3) we get that

[TABLE]

This ends the proof of (5.5). Relation (5.7) follows by a simple substitution $z=u$ in (5.5), whereas (5.6) is derived in the following manner. The first inequality is deduced by not splitting the term $\frac{\phi_{\zeta}(z)-\phi_{\zeta}(0)}{z}$ in the last identity of (5.2) and substituting in the first inequality of (5.4) with $z=\mathtt{Re}\left(z\right)=u$. To obtain the second inequality we observe from (5.3) that for any $\left(\zeta,u\right)\in\mathbb{C}_{\left[0,\infty\right)}\times\left(0,\infty\right)$

[TABLE]

Next, we note from (5.1) that

[TABLE]

Therefore, (5.14) is further estimated from as (5.3)

[TABLE]

which proves (5.6). We proceed with item 5. Let $\left(\xi,\eta\right):=\left(\left(\xi_{t},\eta_{t}\right)\right)_{t\geq 0}$ be the possibly killed bivariate subordinator associated to $\kappa$. Then, for all $\left(\zeta,z\right)\in\mathbb{C}^{2}_{\left[0,\infty\right)}\setminus\left\{\left(0,0\right)\right\}$,

[TABLE]

and (5.8) follows. The last claim follows from the inequality above, the fact that $\inf_{\chi\in\overline{B_{\zeta}}}\inf\mathtt{Re}\left(\zeta\right)>0$ since the closed ball $\overline{B_{\zeta}}\subset\mathbb{C}_{\left(0,\infty\right)}$ and $\mathbb{P}\left(\xi_{1}>0\right)>0$ almost surely. Next we prove item 6. Differentiating (5.1) $n\geq 1$ times with respect to $z$, then taking absolute values, it follows that

[TABLE]

which establishes (5.9) and item 6. For item 7, we apply first (5.9) with $n=1,2$, and then (5.3) to yield

[TABLE]

Applying (5.6) when $n=1$ in $\left(\ref{n=1,2}\right)$, we deduce that $\left(\ref{eq:kappa'_kappa_1}\right)$ holds. Similarly, applying $\left(\ref{eq:kappa''_kappa}\right)$ when $n=2$ in $\left(\ref{n=1,2}\right)$, it follows that $\left(\ref{eq:kappa''_kappa_1}\right)$ holds, as required for item 7, and the proof is complete. It remains to consider item 8. We observe that

[TABLE]

However,

[TABLE]

where in the second inequality we have used (5.9) with $n=1$, in the first identity we have used that $\kappa^{\prime}_{z}(\mathtt{Re}\left(\zeta\right),\cdot)$ is non-increasing, see item 2 of Proposition 3.1, for the last inequality we have invoked (5.3) and the evaluation of the last limit follows from (5.6). ∎

6 Proofs of Auxiliary Lemmas

Proof of Lemma 2.10.

Consider the fact, see [36, Prop 3.6], that each Bernstein function $\phi\in\mathcal{B}$ preserves angular sectors, i.e. for each $w\in\mathbb{C}$, $\left|\arg(\phi(w))\right|\leq|\arg(w)|,$ where we use the convention that $\arg(w)\in(-\pi,\pi]$. Then, write $z=a+ib$ and without loss of generality assume that $a\in\left[c,d\right],b\in\left[c_{1},d_{1}\right]$, where $-\infty<c<d<\infty$ and $-\infty<c_{1}<d_{1}<\infty$. It follows that for all $u$ large enough that $u>-a$,

[TABLE]

Observe that $\lim_{u\to\infty}\left|\arctan\left(b/(u+a)\right)\right|=0$, uniformly for the specified compact range of $a,b$. Therefore, for each $\phi\in\mathcal{B}$, as $u\to\infty$,

[TABLE]

Writing $\phi$ as in $\left(\ref{eq:Bern'}\right)$, since $b$ is bounded, we have $|u+z|\sim u+a$, as $u\to\infty$, and by $\left(\ref{phirealresult}\right)$, as $u\to\infty$,

[TABLE]

Combining $\left(\ref{ree}\right)$, $\left(\ref{upperb}\right)$, and $\left(\ref{lowerb}\right)$, the proof of Lemma 2.10 is complete. ∎

Proof of Lemma 2.11.

Write $z=a+ib$ and without loss of generality assume that $a\in\left[c,d\right],b\in\left[c_{1},d_{1}\right]$, where $0\leq c<d<\infty$ and $-\infty<c_{1}<d_{1}<\infty$. For $u>-a$ (so that $W_{\phi}(u+z+1)$ is well-defined), applying Lemma 2.10 alongside $\left(\ref{eq:Stirling}\right)$ and $\left(\ref{eq:limEk}\right)$ from Theorem 2.9, it follows that as $u\to\infty$,

[TABLE]

Now, recalling the definition of the $L_{\phi}$ error terms in $\left(\ref{eq:A}\right)$, part of the integrals cancel, yielding

[TABLE]

Writing $N=u+2$ for brevity, and noting that $\log_{0}(\phi(w))=\ln(|\phi(w)|)+i\arg(\phi(w))$, we have

[TABLE]

We will integrate along the contours $\gamma_{1}$ and $\gamma_{2}$, which are straight lines connecting $N$ to $N+a$, and $N+a$ to $N+a+ib=N+z$, respectively. Note that along $\gamma_{1}$, $\arg(\phi(w))=\arg(w)=0$, so we need only consider $\gamma_{2}$ for the second integral in $\left(\ref{integrls}\right)$. Now, $|\gamma_{2}|=b$, so

[TABLE]

since $\phi$ preserves angular sectors [36, Prop 3.6]. Now, $\sup_{w\in\gamma_{2}}|\arg(w)|=\arctan(b/(N+a))$, which converges to [math] as $N=u+2\to\infty$ and $b$ belongs to a compact interval, and we conclude that

[TABLE]

Now let us consider the first integral in $\left(\ref{integrls}\right)$ over $\gamma_{1}$ and $\gamma_{2}$ separately. We will compare the $\gamma_{1}$ contribution with $a\ln(\phi(u+1))$, and the $\gamma_{2}$ part with $ib\ln(\phi(u+1))$. For the $\gamma_{1}$ part,

[TABLE]

[TABLE]

By Lemma 2.10, $\underset{n\to\infty}{\lim}\phi(u+2+x)/\phi(u+1)=1$, uniformly in $x\in[0,a]$, so $\lim_{u\to\infty}\left(\ref{int1gamma1}\right)=0$, and

[TABLE]

For the contour $\gamma_{2}$, one can verify by applying the same argument as for $\gamma_{1}$, that

[TABLE]

Finally, substituting the individual limits $\left(\ref{l2}\right),\left(\ref{l3}\right)$, and $\left(\ref{l4}\right)$ into $\left(\ref{l0}\right)$, we conclude that

[TABLE]

∎

Proof of Lemma 4.1.

Firstly, by $\varphi_{q}(w)=q+\phi(w)\in\mathcal{B}$ and Lemma 2.11, we can rewrite

[TABLE]

Recalling the relations $W_{\varphi_{q}}(w+1)=\varphi_{q}(w)W_{\varphi_{q}}(w)$ and $W_{\varphi_{q}}(1)=1$ from $\left(\ref{eq:Bern-Gamma}\right)$, it follows that

[TABLE]

Now, observing that we can rewrite $\varphi_{q}(n+1)$ in the form

[TABLE]

we conclude, as required for Lemma 4.1, that

[TABLE]

∎

Proof of Lemma 4.3.

Fix $y\geq 1$ such that $\int_{0}^{y}x\Pi(dx)>0$. Applying simple inequalities, we get

[TABLE]

which is a finite constant, independent of $n$, as required.

∎

Proof of Lemma 4.4.

To show that the sum in $\left(\ref{absconvlemmaeqn}\right)$ is absolutely convergent, we consider terms separately. First, consider the $1/W_{\phi_{(k)}}(z)$ term. Applying $\left(\ref{eq:Stirling}\right)$ and $\left(\ref{eq:Ek}\right)$ from Theorem 2.9,

[TABLE]

First we bound $|e^{-L_{\phi_{(k)}}(z+1)}|$. Recalling $\log_{0}(w)=\ln(|w|)+i\arg(w)$, by $\left(\ref{eq:A}\right)$, with $z=a+ib$,

[TABLE]

But $\int_{2+a\mapsto 2+a+ib}\ln(|\phi_{(k)}(w)|)dw$ is pure imaginary, and $\int_{1\mapsto 2+a}\arg(\phi_{(k)}(w))dw=0$, so

[TABLE]

As $\phi_{(k)}$ preserves angular sectors[36, Prop 3.6], \big{|}\int_{2+a\mapsto 2+a+ib}\arg(\phi_{(k)}(w))dw\big{|}\leq b\pi/2, and so

[TABLE]

Now, $\phi_{(k)}$ is non-increasing, and $\phi^{\prime}_{(k)}$ is non-decreasing, so applying Lemma 4.3, we get

[TABLE]

where $C_{\phi}$ depends only on $\phi$. Hence there is a constant $K>0$ so that for all $k\geq 1$,

[TABLE]

from which it follows that, with a slightly different constant $K_{1}$, for all $k\geq 1$,

[TABLE]

Now, since (5.9) of Proposition 5.2 6 implies that $\left|\phi^{\prime}(\zeta)\right|\leq\phi^{\prime}(\mathtt{Re}\left(\zeta\right))),\zeta\in\mathbb{C}_{\left(0,\infty\right)}$ and $\phi^{\prime}$ is non-increasing on $\left(0,\infty\right)$, observe that

[TABLE]

and similarly $|\phi^{\frac{1}{2}}_{(k)}(z+2)|\leq|z+2|^{\frac{1}{2}}\phi^{\prime}(k)^{\frac{1}{2}}$. Then, substituting above, we have that

[TABLE]

and plugging this into $\left(\ref{absconvlemmaeqn}\right)$, it follows that for another constant $K_{2}>0$,

[TABLE]

Now, applying the bounds in Lemma 4.5, we see that for another constants $K_{3},K_{4}$,

[TABLE]

Now, recall that in Definition 2.13, we have imposed that $\beta(\phi^{\prime})>-1$, where $\beta(\cdot)$ denotes the lower Matuszewska index, see [10, p68] for the formal definition. One can verify that if $\beta(\phi^{\prime})>-1$, then the function $f(x):=x\phi^{\prime}(x)$ has lower Matuszewska index $\beta(f)>0$. Then applying [10, Prop 2.6.1(b)] to the function $f$, it follows that there exists a constant $c>0$ such that for all $k\geq 1$,

[TABLE]

Now, consider the lower index of the function $f$, defined by $\underline{\textup{ind}}(f):=\liminf_{x\to\infty}\ln(f(x))/\ln(x)$, or equivalently, $\underline{\textup{ind}}(f):=\sup\{\rho>0:\lim_{x\to\infty}x^{-\rho}f(x)=\infty\},$ see e.g. [5, p39]. From the result [10, Prop 2.2.5], $\underline{\textup{ind}}(f)\geq\beta(f)$, and hence $\underline{\textup{ind}}(f)\geq\beta(f)>0.$ Moreover, by $\left(\ref{eq:phi'_phi}\right)$, one can deduce that $\underline{\textup{ind}}(\phi)\geq\underline{\textup{ind}}(f)>0$, and in particular, there exist $c,\rho>0$ such that for all $k\geq 1$,

[TABLE]

Now, if $a>1/2$, then since $\phi$ is non-decreasing, by $\left(\ref{matuszbound}\right)$ and $\left(\ref{blumbound}\right)$, for another constant $K_{5}$,

[TABLE]

and then one can easily verify that this sum is finite, as required. On the other hand, if $a\leq 1/2$, then since $\phi^{\prime}$ is non-increasing,

[TABLE]

and again one can easily verify that this sum is finite, and the proof of Lemma 4.4 is complete.

∎

Proof of Lemma 4.5.

Writing $z=a+ib$, $a,b\in\mathbb{R}$, we first consider the product for $1\leq j\leq k-C_{z}$, where $C_{z}>a$ is to be determined later. We are going to change variables in the numerator so that the new variable ranges between [math] and $k-j$. The change of variables is motivated by the observation that

[TABLE]

so that after this change of variables, we can simply compare terms within each integral.

Consider first $u\in[0,1]$ chosen so that $w=(z+j)(1-u)+ku$. Then $(z+j)(1-u)+ku$ ranges between $z+j$ and $k$, as desired. Now, $\tau$ is simply defined as $\tau:=(k-j)u$, so that the ranges of integration match.

Now, changing variables to $\tau$ chosen such that $w=z+j+(1-z/(k-j))\tau$, we can rewrite $|\phi(k)-\phi(z+j)|$ as

[TABLE]

[TABLE]

[TABLE]

where we have used for the last inequality relation (5.9) of Proposition 5.2 6. We can now compare terms. Since $\tau\leq k-j,$ observe that $a-a\tau/(k-j)\geq 0$, and hence $a+j+(1-a/(k-j))\tau\geq j+\tau$, from which it follows, since $\phi^{\prime}$ is non-increasing on $\left(0,\infty\right),$ that

[TABLE]

Now we want to show that $|(1-z/(k-j))|\leq 1$. We consider its square for convenience:

[TABLE]

This does not exceed one if and only if

[TABLE]

[TABLE]

[TABLE]

So let us choose $C_{z}\geq\frac{|z|^{2}}{2\mathtt{Re}(z)}$. Then it follows that for all $1\leq j\leq k-C_{z}$,

[TABLE]

and hence the product simplifies substantially:

[TABLE]

where $\vee$ stands for the maximum function, we have used $|\phi^{\prime}(w)|\leq\phi^{\prime}(\mathtt{Re}(w))$, see (5.9), and the monotonicity of $\phi^{\prime}$. Now, $|z+j-k|\leq|z|+k-j\leq|z|+C_{z}$, because $j\geq k-C_{z}$, so this is

[TABLE]

Finally, observe that if $\phi^{\prime}(k+a-C_{z})\leq\phi^{\prime}(k)$ then the remaining product is $\leq 1$, and on the other hand, if $\phi^{\prime}(k+a-C_{z})\geq\phi^{\prime}(k)$, then uniformly among all $k$ large enough, we have from Lemma 4.3

[TABLE]

where $C>0$ is independent of $k$, and the proof of Lemma 4.5 is complete.

∎