Scaling Features of Two Special Markov Chains Involving Total Disasters

Abstract

Catastrophe Markov chain population models have received a lot of attention in the recent past. We herewith consider two special cases of such models involving total disasters, both in discrete and in continuous-time. Depending on the parameters range, the two models can show up a recurrence/transience transition and, in the critical case, a positive/null recurrence transition. The collapse transition probabilities are chosen in such a way that the models are exactly solvable and, in case of positive recurrence, intimately related to the extended Sibuya and Pareto–Zipf distributions whose divisibility and self-decomposability properties are shown relevant. The study includes: existence and shape of the invariant measure, time-reversal, return time to the origin, contact probability at the origin, extinction probability, height and length of the excursions, a renewal approach to the fraction of time spent in the catastrophic state, scale function, first time to collapse and first-passage times, divisibility properties.

Appendix

1.1 Discrete Infinite-Divisibility and Self-decomposability

For the sake of completeness, let us briefly introduce the notion of self-decomposability.

Definition 1

Let $X\ge 0$ be an integer-valued random variable. The probability generating function (pgf) $\phi \left( z\right) :=\mathbb {E}\left( z^{X}\right)$ is the one of a discrete self-decomposable (SD) variable X if for any $u\in \left( 0,1\right)$, there is a pgf $\phi _{u}\left( z\right)$ (depending on u) such that (see [22]),

$$\begin{aligned} \phi \left( z\right) =\phi \left( 1-u\left( 1-z\right) \right) \cdot \phi _{u}\left( z\right) . \end{aligned}$$

(15)

Define the u-thinned version of X,  say $u\circ X,$ as the random sum

$$\begin{aligned} u\circ X\overset{d}{=}\sum _{x=1}^{X}B_{x}\left( u\right) , \end{aligned}$$

(16)

with $\left( B_{x}\left( u\right) \right) _{x\ge 1}$ a sequence of iid Bernoulli variables such that $\mathbb {P}\left( B_{x}\left( u\right) =1\right) =u$, independent of X.

This binomial thinning operator, acting on discrete rvs, has been defined in [22]; it stands as the discrete version of the change of scale $X\rightarrow u\cdot X$ for continuous rvs X. If $\phi \left( z\right)$ is the pgf of the SD random variable X obeying Eq. (15), then X can be additively (self-)decomposed as

$$\begin{aligned} X\overset{d}{=}u\circ X^{\prime }+X_{u}. \end{aligned}$$

(17)

Here, X and $X^{\prime }$ have the same distribution and $u\circ X^{\prime }$ is independent of the remaining random variable $X_{u}$ with pgf, say $\phi _{u}\left( z\right) .$

Remark

The self-decomposability idea also (pre-)exists for continuous rvs on $\mathbb {R}_{+}$: $X>0$ is said to be SD if, for any $u\in \left( 0,1\right)$

$$\begin{aligned} X\overset{d}{=}u\cdot X^{\prime }+X_{u}. \end{aligned}$$

(18)

with X and $X^{\prime }$ having the same distribution and $u\cdot X^{\prime }$ being independent of the remaining random variable $X_{u}>0.$ If $\Phi \left( \lambda \right) =\mathbb {E}\left( e^{-\lambda X}\right)$, $\lambda \ge 0$, is to be the Laplace-Stieltjes transform (LST) of X, SD, then for any $u\in \left( 0,1\right)$, there is a LST $\Phi _{u}\left( \lambda \right)$ (depending on u) such that

$$\begin{aligned} \Phi \left( \lambda \right) =\Phi \left( \lambda u\right) \cdot \Phi _{u}\left( \lambda \right) . \end{aligned}$$

The two notions of self-decomposability are related as follows: let $Y>0$ be a continuous rv. Then Y is self-decomposable if and only if the discrete random variable supported by $\mathbb {N}_{0}$ defined by: $X=P\left( Y\right)$ (where $P\left( Y\right)$ is a Poisson rv with random intensity Y) is discrete self-decomposable [18, Corollary 1]. Indeed,

$$\begin{aligned} \phi _{X}\left( z\right) =\mathbb {E}\left( z^{X}\right) =\Phi _{Y}\left( 1-z\right) \end{aligned}$$

and, with $\phi _{X_{u}}\left( z\right) =\mathbb {E}\left( z^{X_{u}}\right) =\Phi _{u}\left( 1-z\right) =\mathbb {E}\left( e^{-\left( 1-z\right) Y_{u}}\right)$, the pgf of $X_{u}=P\left( Y_{u}\right)$

$$\begin{aligned} \phi _{X}\left( z\right) =\phi _{X}\left( 1-u\left( 1-z\right) \right) \cdot \phi _{X_{u}}\left( z\right) \Leftrightarrow \Phi _{Y}\left( \lambda \right) =\Phi _{Y}\left( \lambda u\right) \cdot \Phi _{Y_{u}}\left( \lambda \right) . \end{aligned}$$

$\square$

In such cases, the pmf of X is related to the density f of Y by:

$$\begin{aligned} \mathbb {P}\left( X=x\right) =\frac{1}{x!}\int _{0}^{\infty }y^{x}e^{-y}f\left( y\right) dy,\text { }x\in \mathbb {N}_{0}. \end{aligned}$$

This raises the question of which Y are SD? We first need to recall the notion of an HCM rv. Following [23, (5.15) p. 371], a HCM positive rv is one whose density f obeys that

$$\begin{aligned} \forall y>0:\text { the function }x\rightarrow f\left( xy\right) f\left( y/x\right) \text {, }x>0 \end{aligned}$$

is completely monotone on $\left( 2,\infty \right)$ as a function of $z=x+1/x$. Completely monotone functions h obey: $\left( -1\right) ^{k}h^{\left( k\right) }\left( x\right) \ge 0$ for all $k\ge 0$ in some range of x.

With $a>0$, consider now the rv $Y=G\left( a\right) ^{1/\beta }$, the $1/\beta -$power of $G\left( a\right) \overset{d}{\sim }$Gamma$\left( a,1\right)$. This rv is hyperbolically completely monotone (HCM) if and only if $\left| \beta \right| \le 1$ [23, ex. 12.8]. This is also true of the so-called Generalized Inverse Gaussian rvs [10], with density

$$\begin{aligned} f_{\lambda ,\delta _{1},\delta _{2}}\left( x\right) =\left( \frac{\delta _{1} }{\delta _{2}}\right) ^{\alpha /2}\frac{1}{2K_{\alpha }\left( \sqrt{\delta _{1}\delta _{2}}\right) }x^{\alpha -1}\exp \left\{ -\frac{1}{2}\left( \delta _{1}x+\delta _{2}/x\right) \right\} \text {, }x>0 \end{aligned}$$

(19)

in the parameter range: $\delta _{2}\ge 0$, $\delta _{1}>0$ if $\lambda >0$ , $\delta _{2}>0$, $\delta _{1}\ge 0$ if $\lambda <0$ and $\delta _{2}>0$, $\delta _{1}>0$ if $\lambda =0.$ Such densities include Gamma distributions ($\delta _{2}=0$) and inverse Gamma distributions ($\delta _{1}=0$) as particular cases. $K_{\alpha }$ is the modified Bessel function of the second kind.

HCM rvs constitute a subclass of Generalized-Gamma-Convolution (GGC) rvs [3, Proposition 2] and GGC rvs are SD [3, Theorem 1]. We refer to [23, Sect. 5] and [2], for the precise definition of GGC rvs. So, whenever Y is a GGC rv, it is SD and $X=P\left( Y\right)$, as a Poisson-mixture with respect to a SD distribution, is discrete-SD. Self-decomposable distributions are unimodal.

Coming back to discrete self-decomposability itself, the following representation result is also known to hold true, see [22]. Let $R\left( z\right)$ (with $r_{0}=R\left( 0\right) >0$) be the canonical function defined through

$$\begin{aligned} \phi \left( z\right) =\mathbb {E}\left( z^{X}\right) =e^{-\int _{z}^{1}R\left( z^{\prime }\right) dz^{\prime }}. \end{aligned}$$

(20)

The random variable X is discrete SD if and only if the function $h\left( z\right) :=1-\left( 1-z\right) R\left( z\right) /r_{0}$ defines a pgf such that $h\left( 0\right) =0$ (see [19, Lemma 2.13]). Consequently, X is discrete SD if and only if, for some $r_{0}>0$, its pgf can be written in the form

$$\begin{aligned} \phi \left( z\right) =e^{-r_{0}\int _{z}^{1}\frac{1-h\left( z^{\prime }\right) }{1-z^{\prime }}dz^{\prime }}. \end{aligned}$$

(21)

This means that the series coefficients $\left( r_{x}=\left[ z^{x}\right] R\left( z\right) ,\text { }x\in \mathbb {N}_{0}\right)$ of $R\left( z\right)$ constitute a non-negative, non-increasing sequence of x [22, Theorem 4.13 p. 271]. As a result, the associated probability system $\mathbb {P }\left( X=x\right) :=\pi _{x}$, $x\in \mathbb {N}_{0}$ of X, if SD, is unimodal, with mode at the origin if and only if $r_{0}=\frac{\pi _{1}}{\pi _{0}}\le 1$. The SD subclass of infinitely divisible distributions (ID) therefore consists of unimodal distributions, with mode possibly at the origin [22, Theorem 2.3].

Note that X is ID if and only if the sequence $\left( r_{x},x\in \mathbb {N} _{0}\right)$ is non-negative only, with [as can be checked from (20)] the sequences $\left( \pi _{x},\text { }r_{x},\text { }x\in \mathbb {N}_{0}\right)$ related by the convolution formula

$$\begin{aligned} \left( x+1\right) \pi _{x}=\sum _{y=0}^{x}\pi _{y}r_{x-y}\text {, }x\in \mathbb {N} _{0}. \end{aligned}$$

We recall that for rvs with integral support $\mathbb {N}_{0},$ the notion of an infinitely divisible rv coincides with the one of a compound Poisson rv.

Definition 2

A compound Poisson rv is one which is obtained as an independent Poisson sum of positive iid rvs which are called compounding rvs.

A rv with probability mass 0 at 0 cannot be ID.

With $r>0$, the pgf of ID rvs takes the form: $\phi \left( z\right) =e^{-r\left( 1-h\left( z\right) \right) }$ where $h\left( z\right)$ is the pgf of the compounding rvs, obeying $h\left( 0\right) =0$.

Remark

Let $Y_{\infty }\ge 1$ be a rv such that $Y_{\infty }-1$ is SD. In the study of the MCC we encountered the delicate problem of deciding whether or not the mixed rv $X_{\infty }\overset{d}{\sim }\pi _{0}\delta _{0}+\left( 1-\pi _{0}\right) Y_{\infty }$ was ID or SD. Consider the simpler case where $Y_{\infty }-1$ is Geometric$\left( p\right) ,$ which is SD. Then

$$\begin{aligned} \phi _{X_{\infty }}\left( z\right)= & {} \pi _{0}+\left( 1-\pi _{0}\right) \frac{pz}{1-qz}=\frac{\pi _{0}-\left( \pi _{0}-p\right) z}{1-qz} \\ \log \phi _{X_{\infty }}\left( z\right)= & {} \log \left( \pi _{0}-\left( \pi _{0}-p\right) z\right) -\log \left( 1-qz\right) \\ R\left( z\right)= & {} \log \phi _{X_{\infty }}\left( z\right) ^{\prime }=- \frac{\pi _{0}-p}{\pi _{0}-\left( \pi _{0}-p\right) z}+\frac{q}{1-qz} \\ r_{x}= & {} \left[ z^{x}\right] R\left( z\right) =q^{x+1}-\left( 1-\frac{p}{\pi _{0}}\right) ^{x+1} \end{aligned}$$

Observing $q\ge 1-\frac{p}{\pi _{0}}$, a necessary and sufficient condition for $X_{\infty }$ to be ID is $1-\frac{p}{\pi _{0}}\ge 0$ ($\pi _{0}\ge p$), leading to $r_{x}\ge 0$ for all $x\ge 0$. Under this condition, it will be SD if and only if $r_{x+1}\le r_{x}$, meaning

$$\begin{aligned} \left( \frac{1-\frac{p}{\pi _{0}}}{q}\right) ^{x}\le \pi _{0}\text { for all }x. \end{aligned}$$

This will always be the case if in addition $1-\frac{p}{\pi _{0}}\le q\pi _{0}$, equivalently if $\pi _{0}\le p/q.$ If $p\ge q$ ($p\ge 1/2$), $X_{\infty }$ is SD if $\pi _{0}\ge p.$ If $p<q$ ($p<1/2$), $X_{\infty }$ will be SD only if $p\le \pi _{0}\le p/q$. In the range $\pi _{0}\in \left( p/q,1\right)$ it is only ID. $\square$

Complete monotonicity: A rv X with support $\mathbb {N}$ is completely monotone if, for some probability measure $\pi$ on $\left[ 0,1\right]$, the following Hausdorff representation holds

$$\begin{aligned}&\overline{F}\left( x\right) :=\mathbb {P}\left( X>x\right) =\int _{0}^{1}u^{x}\pi \left( du\right) \text {, }x\in \left\{ 0,1,2,\ldots \right\} \\&\mathbb {P}\left( X=x\right) =\pi _{x}=\int _{0}^{1}u^{x}\left( 1-u\right) \pi \left( du\right) \text {, }x\in \left\{ 1,2,\ldots \right\} . \end{aligned}$$

If this is the case, for all $x\in \left\{ 0,1,2,\ldots \right\}$ (see [23, p. 77])

$$\begin{aligned} \left( -1\right) ^{k}\Delta ^{\left( k\right) }\overline{F}\left( x\right) \ge 0\text {, equivalently }\left( -1\right) ^{k}\Delta ^{\left( k\right) }\pi _{x}\ge 0 \end{aligned}$$

where $\Delta :$$\Delta h\left( x\right) =h\left( x+1\right) -h\left( x\right)$ is the right-shift operator and $\Delta ^{\left( k\right) }$ its $k-$th iterate. If this is the case, the rv $X-1,$ with support $\mathbb {N}_{0}$ , is completely monotone, log-convex and therefore infinitely divisible (see [23, Theorem 10.4 p. 77]).

1.2 On Simple Markov Realizations of ID and SD Distributions

The catastrophe Markov chains that have been studied in this draft showed up invariant measures that can be either ID or SD (such as $Y_{\infty }-1$ always or $X_{\infty }$ itself in some parameter range on $p_{0}$). Different Markov processes can have the same invariant equilibrium measure and here are natural ones whenever the latter is either ID or SD:

• The ID case: Consider a time-inhomogeneous Poisson process $P\left( R_{t}\right)$ with decaying rate function $re^{-t}$ and intensity $R_{t}=r\left( 1-e^{-t}\right)$, $r>0.$ Consider the compound-Poisson process (with independent increments):

  $$\begin{aligned} X_{t}=\sum _{k=1}^{P\left( R_{t}\right) }\Delta _{k} \end{aligned}$$

  where $\left( \Delta _{k};k\ge 1\right)$ is the iid sequence of the positive jumps occurring at the jump times of $P\left( R_{t}\right)$. If $h\left( z\right) =\mathbb {E}\left( z^{\Delta _{1}}\right)$,

  $$\begin{aligned} \phi _{t}\left( z\right) =\mathbb {E}\left( z^{X_{t}}\right) =e^{-R_{t}\left( 1-h\left( z\right) \right) }\underset{t\rightarrow \infty }{\rightarrow } e^{-r\left( 1-h\left( z\right) \right) } \end{aligned}$$

  which is the pgf of an ID rv. A mechanism responsible of the decay of the population, when balanced by incoming immigrants with sizes $\Delta ,$ will produce an ID limiting population size.

• The SD case (subcritical branching with immigration): Consider now a time-homogeneous compound Poisson process $P\left( rt\right) ,$$t\ge 0$, $P\left( 0\right) =0,$ so with pgf

  $$\begin{aligned} \mathbb {E}\left( z^{P\left( rt\right) }\right) =\exp \left\{ -rt\left( 1-h\left( z\right) \right) \right\} , \end{aligned}$$

  (22)

  where $h\left( z\right)$, with $h\left( 0\right) =0$, is the pgf of the jumps arriving at the jump times of $P\left( rt\right)$ having rate $r>0.$ Let now

  $$\begin{aligned} \varphi _{t}\left( z\right) =1-e^{-t}\left( 1-z\right) \text {,} \end{aligned}$$

  (23)

  be the pgf of a pure-death (rate-1) Greenwood branching process started with one particle at $t=0$, [9]. This expression of $\varphi _{t}\left( z\right)$ is easily seen to be the solution to $\overset{.}{ \varphi }_{t}\left( z\right) =f\left( \varphi _{t}\left( z\right) \right) =1-\varphi _{t}\left( z\right)$, $\varphi _{0}\left( z\right) =z$, as is usual for a pure-death continuous-time Bellman-Harris branching processes with affine branching mechanism $f\left( z\right) =r_{d}\left( 1-z\right)$ and fixing the death rate to be $r_{d}=1$, [11]. The distribution function of the lifetime of the initial particle is thus $1-e^{-t}$. Let $X_{t}$ with $X_{0}=0$ be a random process counting the current size of some population for which a random number of individuals (determined by $h\left( z\right)$) immigrate at the jump times of $P\left( rt\right) ,$ each of which being independently and immediately subject to the latter pure death Greenwood process. We have

  $$\begin{aligned} \phi _{t}\left( z\right) :=\mathbb {E}\left( z^{X_{t}}\right) =\exp -r\int _{0}^{t}\left( 1-h\left( \varphi _{t-s}\left( z\right) \right) \right) ds\text {, }\phi _{0}\left( z\right) =1, \end{aligned}$$

  (24)

  with $\phi _{t}\left( 0\right) =\mathbb {P}\left( X_{t}=0\right) =\exp -r\int _{0}^{t}\left( 1-h\left( 1-e^{-s}\right) \right) ds,$ the probability that the population is extinct at t. It holds

  $$\begin{aligned} \phi _{t}\left( z\right)= & {} e^{-r\int _{0}^{t}\left( 1-h\left( 1-e^{-s}\left( 1-z\right) \right) \right) ds}=e^{-r\int _{\varphi _{0}\left( z\right) =z}^{\varphi _{t}\left( z\right) }\frac{1-h\left( z^{\prime }\right) }{ 1-z^{\prime }}dz^{\prime }}\underset{t\rightarrow \infty }{\rightarrow }\phi _{\infty }\left( z\right) \nonumber \\= & {} e^{-r\int _{z}^{1}\frac{1-h\left( z^{\prime }\right) }{1-z^{\prime }}dz^{\prime }}. \end{aligned}$$

  (25)

  So, $X:=X_{\infty },$ as the limiting population size of this pure-death process with immigration, is a SD rv, [24]. As in the ID case, a mechanism responsible of the decay of the population is balanced by incoming immigrants.

Remark

If instead of a pure-death Greenwood branching process, the immigrants shrink, more generally, according to any subcritical branching process with branching mechanism f, as those whose pgf obeys $\overset{.}{\varphi }_{t}\left( z\right) =f\left( \varphi _{t}\left( z\right) \right)$, then, with $f^{\prime }\left( 1\right) <0,$$\varphi _{t}\left( z\right) \rightarrow 1$ as $t\rightarrow \infty$ and

$$\begin{aligned} \phi _{t}\left( z\right) =e^{-r\int _{0}^{t}\left( 1-h\left( \varphi _{s}\left( z\right) \right) \right) ds}=e^{-r\int _{\varphi _{0}\left( z\right) =z}^{\varphi _{t}\left( z\right) }\frac{1-h\left( z^{\prime }\right) }{f\left( z^{\prime }\right) }dz^{\prime }}\rightarrow \phi _{\infty }\left( z\right) =e^{-r\int _{z}^{1}\frac{1-h\left( z^{\prime }\right) }{f\left( z^{\prime }\right) }dz^{\prime }}. \end{aligned}$$

The obtained limiting pgf is the one of a self-decomposable rv induced by the subcritical semigroup $\varphi _{t}\left( z\right)$ generated by $f\left( z\right)$, [24]. Recall $f\left( z\right) =\varphi \left( z\right) -z$ where $\varphi \left( z\right)$ is the pgf of the branching number per capita in a Bellman–Harris process. $\square$