On asymptotic structure of the critical Galton-Watson Branching Processes with infinite variance and Immigration

TL;DR

This paper investigates the long-term behavior of critical Galton-Watson branching processes with infinite variance and immigration, focusing on their transition functions and convergence to invariant measures.

Contribution

It provides new insights into the asymptotic structure and limit properties of such processes with infinite variance and immigration.

Findings

Convergence of transition functions to invariant measures

Characterization of asymptotic behavior in infinite variance cases

Extension of classical results to processes with immigration

Abstract

We observe the Galton-Watson Branching Processes. Limit properties of transition functions and their convergence to invariant measures are investigated.

Equations

$$p_{ij}=\textrm{coefficient of}\;\;s^{j}\;\;\textrm{in}\;\;h(s)\bigl{(}f(s)\bigr{)}^{i},\quad\parbox{68.28644pt}{ {} $s\in[0,1)$,}$$

$${\mathcal{P}}_{n}^{(i)}(s):=\sum\limits_{j\in{\mathcal{S}}}{p_{ij}^{(n)}s^{j}}=\bigl{(}{f_{n}(s)}\bigr{)}^{i}\prod\limits_{k=0}^{n-1}{h\bigl{(}{f_{k}(s)}\bigr{)}}\quad\parbox{85.35826pt}{{for any} {} $i\in\mathcal{S}$,}$$

$$\displaystyle{\textsf{E}}\bigl{[}{\left.{X_{n}}\right|X_{0}=i}\bigr{]}=\left\{\begin{array}[]{l}\displaystyle{a}n+i\hfill,\quad\parbox{79.6678pt}{{when} {} $m=1$,}\\ \\ \displaystyle\left({{\displaystyle{a}\over\displaystyle{m-1}}+i}\right)m^{n}-{\displaystyle{a}\over\displaystyle{m-1}}\;\hfill,\quad\parbox{79.6678pt}{{when} {} $m\neq 1$,}\end{array}\right.$$

$$f(s)=s+(1-s{)}^{1+\nu }\mathcal{L}\left(\frac{1}{1-s}\right),$$

$${Q_{n}^{\nu}}\cdot\mathcal{L}\left({{{1}\over{{Q_{n}}}}}\right)\sim{{1}\over{\nu n}}\quad\parbox{68.28644pt}{{as} {} $n\to\infty$.}$$

$${\mathcal{U}}_n(s):=\frac{f_n(s)-f_n(0)}{f_n(0)-f_{n-1}(0)}⟶U(s)$$

$$U\left(f(s)\right)=U(s)+1,$$

$${\mathcal{U}}_{n}(s)\sim{U}_{n}(s):=\left[{1-{{R_{n}(s)}\over{Q_{n}}}}\right]{\nu{n}}\quad\parbox{68.28644pt}{{as} {} $n\to\infty$.}$$

$$R_n(s)=\frac{\mathcal{N}\left(n\right)}{{\left(\nu n\right)}^{1/\nu }}\cdot \left[1-\frac{U_n(s)}{\nu n}\right],$$

$${\mathcal{N}}(n)\cdot\mathcal{L}^{{1\mathord{\left/{\vphantom{1\nu}}\right.\kern-1.2pt}\nu}}\left({{{(\nu n)^{{1\mathord{\left/{\vphantom{1\nu}}\right.\kern-1.2pt}\nu}}}\over{{\mathcal{N}}(n)}}}\right)\longrightarrow 1\quad\parbox{68.28644pt}{{as} {} $n\to\infty$,}$$

$${\mathcal{M}}_{n}(s):=1-{{\Lambda\bigl{(}{R_{n}(s)}\bigr{)}}\over{\Lambda\left({Q_{n}}\right)}}.$$

$${\mathcal{M}}_n(s)$$

$$n\cdot{\mathcal{M}}_{n}(s)\longrightarrow U(s)\quad\parbox{68.28644pt}{{as} {} $n\to\infty$,}$$

$$\mathcal{V}(s)=\frac{1}{\nu \Lambda \left(1-s\right)}.$$

$$U(s)=\mathcal{V}(s)-\mathcal{V}(0).$$

$${\mathcal{V}}\bigl{(}{f_{n+1}(s)}\bigr{)}-{\mathcal{V}}\bigl{(}{f_{n}(s)}\bigr{)}\longrightarrow 1\quad\parbox{68.28644pt}{{as} {} $n\to\infty$.}$$

$${\mathcal{V}}\bigl{(}{f_{n}(s)}\bigr{)}-{\mathcal{V}}(s)=n\cdot\bigl{(}{1+o(1)}\bigr{)}\quad\parbox{68.28644pt}{{as} {} $n\to\infty$.}$$

$$\Lambda\bigl{(}{R_{n}(s)}\bigr{)}={{\Lambda\left({1-s}\right)}\over{\Lambda\left({1-s}\right)\nu n+1}}\bigl{(}{1+o(1)}\bigr{)}\quad\parbox{68.28644pt}{{as} {} $n\to\infty$.}$$

$$p_{00}^{(n)}\sim K\exp\left\{{\int\limits_{1}^{e^{n}}{{{\ln h\left({1-\varphi(y)}\right)}\over y}\,dy}}\right\}\quad\parbox{68.28644pt}{{as} {} $n\to\infty$,}$$

$$\sum\limits_{m=0}^{\infty}{\Bigl{[}{\left({1-h(f_{m}(0)}\right)\left({1-f^{\prime}(f_{m}(0)}\right)}\Bigr{]}}<\infty,$$

$$p_{00}^{(n)}\sim K_{1}\exp\left\{{\int\limits_{0}^{f_{n}(0)}{{{\ln h(y)}\over{f(y)-y}}\,dy}}\right\}\quad\parbox{68.28644pt}{{as} {} $n\to\infty$.}$$

$$1-h(s)=(1-s{)}^{\delta }\ell \left(\frac{1}{1-s}\right),$$

$$\frac{\mathcal{L}\left(\lambda x\right)}{\mathcal{L}(x)}=1+\alpha (x)$$

$$\alpha(x)={o}\left({{{\mathcal{L}\left(x\right)}\over{x^{\nu}}}}\right)\quad\parbox{68.28644pt}{{as} {} $x\to\infty$.}$$

$$\frac{\ell \left(\lambda x\right)}{\ell (x)}=1+\beta (x)$$

$$\beta(x)={o}\left({{{{\ell}\left(x\right)}\over{x^{\delta}}}}\right)\quad\parbox{68.28644pt}{{as} {} $x\to\infty$.}$$

$${\mathcal{P}}_n(s)={\mathcal{P}}_n^{(0)}(s).$$

$${\mathcal{P}}_{n}(s)\sim K(s)\exp\left\{-{\int\limits_{s}^{f_{n}(s)}{{{1-h(y)}\over{f(y)-y}}\Bigl{[}{1+\delta(1-y)}\Bigr{]}dy}}\right\}$$

$${\mathcal{P}}_{n}(s)\sim K(s)\exp\left\{-{\int\limits_{s}^{f_{n}(s)}{{{1-h(y)}\over{f(y)-y}}\Bigl{[}{1+{o}\bigl{(}{{\Lambda}\left({1-y}\right)}\bigr{)}}\Bigr{]}dy}}\right\}\quad\parbox{68.28644pt}{{as} {} $n\to\infty$.}$$

$$p_{00}^{(n)}\sim A\exp\left\{-{{{\mathcal{N}}^{\,\nu}(n)}\over{\delta-\nu}}\cdot{\ell}\left({{{(\nu n)^{{1\mathord{\left/{\vphantom{1\nu}}\right.\kern-1.2pt}\nu}}}\over{{\mathcal{N}}(n)}}}\right)\right\}\quad\parbox{68.28644pt}{{as} {} $n\to\infty$,}$$

Full text

1 Introduction

Let $\left\{{X_{n}},n\in\mathbb{N}_{0}\right\}$ be the Galton-Watson Branching Process allowing Immigration (GWPI), where $\mathbb{N}_{0}=\left\{0\right\}\cup\mathbb{N}$ and $\mathbb{N}=\left\{{1,2,\ldots}\right\}$. This is a homogeneous Markov chain with state space $\mathcal{S}\subset\mathbb{N}_{0}$ and whose transition probabilities are

[TABLE]

where $h(s)=\sum\nolimits_{j\in\mathcal{S}}{h_{j}s^{j}}$ and $f(s)=\sum\nolimits_{j\in\mathcal{S}}{p_{j}s^{j}}$ are probability generating functions (PGF’s). The variable $X_{n}$ is interpreted as the population size in GWPI at the moment $n$. An evolution of the process will occurs by following scheme. An initial state is empty that is $X_{0}=0$ and the process starts owing to immigrants. Each individual at time $n$ produces $j$ progeny with probability $p_{j}$ independently of each other so that $p_{0}>0$. Simultaneously in the population $i$ immigrants arrive with probability $h_{i}$ in each moment $n\in\mathbb{N}$. These individuals undergo further transformation obeying the reproduction law $\left\{{p_{j}}\right\}$ and $n$-step transition probabilities $p_{ij}^{(n)}:={\textsf{P}}\left\{{\left.{X_{n+k}=j}\right|X_{k}=i}\right\}$ for any $k\in\mathbb{N}$ are given by

[TABLE]

where $f_{n}(s)$ is $n$-fold iteration of PGF $f(s)$; see for example [6]. Thus the transition probabilities $\left\{{p_{ij}^{(n)}}\right\}$ are completely defined by the probabilities $\left\{{p_{j}}\right\}$ and $\left\{{h_{j}}\right\}$.

Classification of states of the chain $\left\{{X_{n}}\right\}$ is one of fundamental problems in theory of GWPI. Direct differentiation of (1) gives

[TABLE]

where $m=f^{\prime}(1-)$ is mean per-capita offspring number and $a=h^{\prime}(1-)$. The received formula for {\textsf{E}}\bigl{[}{\left.{X_{n}}\right|X_{0}=i}\bigr{]} shows that classification of states of GWPI depends on a value of the parameter $m$. Process $\left\{{X_{n}}\right\}$ is classified as sub-critical, critical and supercritical if $m<1$, $m=1$ and $m>1$ accordingly.

The above described population process was considered first by Heathcote [3] in 1965. Further long-term properties of $\mathcal{S}$ and a problem of existence and uniqueness of invariant measures of GWPI were investigated by Seneta [12], Pakes [8], [9] and by many other authors. Therein some moment conditions for PGF $f(s)$ and $h(s)$ was required to be satisfied. In aforementioned works of Seneta the ergodic properties of $\left\{{X_{n}}\right\}$ were investigated. He has proved that when $m\leq 1$ the process $\left\{{X_{n}}\right\}$ has an invariant measure $\left\{{\mu_{k},\;k\in{\mathcal{S}}}\right\}$ which is unique up to multiplicative constant. Pakes [9] have shown that in supercritical case ${\mathcal{S}}$ is transient. In the critical case ${\mathcal{S}}$ can be transient, null-recurrent or ergodic. In this case, if in addition to assume that $2b:=f^{\prime\prime}(1-)<\infty$, properties of ${\mathcal{S}}$ depend on value of parameter $\lambda={a\mathord{\left/{\vphantom{a{b}}}\right.\kern-1.2pt}b}$: if $\lambda>1$ or $\lambda<1$, then ${\mathcal{S}}$ is transient or null-recurrent accordingly. In the case when $\lambda=1$, Pakes [8] studied necessary and sufficient conditions for a null-recurrence property. Limiting distribution law for critical process $\left\{{X_{n}}\right\}$ was found first by Seneta [11]. He has proved that the normalized process ${{X_{n}}\mathord{\left/{\vphantom{{X_{n}}{\left({b}{n}\right)}}}\right.\kern-1.2pt}{\left({b}{n}\right)}}$ has limiting Gamma distribution with density function ${\Gamma^{-1}(\lambda)}x^{\lambda-1}e^{-x}$ provided that $0<\lambda<\infty$, where $x>0$ and $\Gamma(*)$ is Euler’s Gamma function. This result has been established also by Pakes [8] without reference to Seneta. Afterwards Pakes [6], [7], has obtained principally new results for all cases $m<\infty$ and $b=\infty$.

Throughout the paper we keep on the critical case only and $b=\infty$. Our reasoning will bound up with elements of slow variation theory in sense of Karamata; see [10]. Remind that real-valued, positive and measurable function $L(x)$ is said to be slowly varying (SV) at infinity if ${{L(\lambda x)}\mathord{\left/{\vphantom{{L(\lambda x)}{L(x)}}}\right.\kern-1.2pt}{L(x)}}\to 1$ as $x\to\infty$ for each $\lambda>0$. We refer the reader to [1], [2] and [10] for more information.

In second section we study invariant measures of the simple Galton-Watson (GW) Process. In third section the invariant properties of GWPI will be investigated.

2 Invariant measures of GW Process

Let $\left\{{Z_{n},n\in\mathbb{N}_{0}}\right\}$ be the simple GW Branching Process without immigration given by offspring PGF $f(s)$. Discussing this case we will assume that the offspring PGF $f(s)$ has the following representation:

[TABLE]

where $0<\nu\leq 1$ and $\mathcal{L}(x)$ is SV at infinity. By the criticality of the process the condition $\left[{f_{\nu}}\right]$ implies that $b=\infty$. This includes the case $b<\infty$ when $\nu=1$ and $\mathcal{L}(t)\rightarrow{b}$ as $t\to\infty$.

Consider PGF ${f_{n}(s)}:=\textsf{E}\left[s^{Z_{n}}\left|{Z_{0}=1}\right.\right]$ and write ${R_{n}(s)}:=1-f_{n}(s)$. Evidently ${Q_{n}}:=R_{n}(0)$ is the survival probability of the process. By arguments of Slack [13] one can be shown that if the condition $\left[{f_{\nu}}\right]$ holds then

[TABLE]

Slack [13] also has shown that

[TABLE]

for $s\in[0,1)$, where the limit function $U(s)$ satisfies the Abel equation

[TABLE]

so that $U(s)$ is PGF of invariant measure for the GW process $\left\{{Z_{n}}\right\}$. Combining $\left[{f_{\nu}}\right]$, (3) and (4) and considering properties of the process $\left\{{Z_{n}}\right\}$ we have

[TABLE]

So we proved the following lemma.

Lemma 2.1**.**

If the condition $\left[{f_{\nu}}\right]$ holds then

[TABLE]

where the function ${\mathcal{N}}(x)$ is SV at infinity and

[TABLE]

and the function $U_{n}(s)$ enjoys following properties:

• •

$U_{n}(s)\longrightarrow{U}(s)$* as $n\to\infty$ so that the equation (5) holds;*

• •

$\lim_{s\uparrow 1}U_{n}(s)=\nu n$* for each fixed $n\in\mathbb{N}$;*

• •

$U_{n}(0)=0$* for each fixed $n\in\mathbb{N}$.*

Evidently that this lemma is generalization of (3) and herein it established by more simple proof rather than as shown in [4].

Further writing $\Lambda(y)=y^{\nu}{\mathcal{L}}\left({{1\mathord{\left/{\vphantom{1{y}}}\right.\kern-1.2pt}y}}\right)$ we consider the function

[TABLE]

It follows from (6) and from the properties of SV-function that

[TABLE]

where \rho_{n}(s)={\mathcal{O}}\bigl{(}{{1\mathord{\left/{\vphantom{{1}{n}}}\right.\kern-1.2pt}n}}\bigr{)} uniformly for all $s\in[0,1)$.

Thus we obtain the following assertion.

Lemma 2.2**.**

If the condition $\left[{f_{\nu}}\right]$ holds then

[TABLE]

where $U(s)$ is PGF of invariant measure of GW Process.

In the following Lemma we find out an explicit form of PGF of $U(s)$. Write

[TABLE]

Lemma 2.3**.**

If the condition $\left[{f_{\nu}}\right]$ holds then

[TABLE]

Proof 2.4**.**

In pursuance of reasoning from [2, p. 401] we obtain the following relation:

[TABLE]

Thence summing by $n$ we find

[TABLE]

Keeping our designation we easily will transform last equality to a form of

[TABLE]

Combining (8), (9) and (11) we reach (10).

3 Invariant measures of GWPI

Consider GWPI. Pakes [7] has proved the following theorem.

Theorem P1 [7]. If $m=1$ then

[TABLE]

where $\varphi(y)$ is decreasing SV-function. If

[TABLE]

*then *

[TABLE]

Herein $K$ and $K_{1}$ are some constants.

Since this point we everywhere will consider the case that immigration PGF $h(s)$ has the following form:

[TABLE]

where $0<\delta\leq 1$ and $\ell(x)$ is SV at infinity.

Our results appear provided that conditions $\left[{f_{\nu}}\right]$ and $\left[{h_{\delta}}\right]$ hold and $\delta>\nu$. As it has been shown in [7] that in this case $\mathcal{S}$ is ergodic. Namely we improve statements of Theorem P1. Herewith we put forward an additional requirement concerning $\mathcal{L}(x)$ and ${\ell}(x)$. So since $\mathcal{L}(x)$ is SV we can write

[TABLE]

for each $\lambda>0$, where $\alpha(x)\to 0$ as $x\to\infty$. Henceforth we suppose that some positive function $g(x)$ is given so that $g(x)\to 0$ and \alpha(x)={o}\bigl{(}g(x)\bigr{)} as $x\to\infty$. In this case $\mathcal{L}(x)$ is called SV with remainder $\alpha(x)$; see [2, p. 185, condition SR3]. Wherever we exploit the condition $\left[\mathcal{L}_{\alpha}\right]$ we will suppose that

[TABLE]

And also by perforce we suppose the condition

[TABLE]

for each $\lambda>0$, where

[TABLE]

Since $f_{n}(s)\uparrow 1$ for all $s\in[0,1)$ in virtue of (1) it sufficiently to observe the case $i=0$ as $n\to\infty$. Write

[TABLE]

The following theorem is generalization of the Theorem P1.

Theorem 3.1**.**

Let conditions $\left[{f_{\nu}}\right]$, $\left[{h_{\delta}}\right]$ hold. If $\delta>\nu$ then

[TABLE]

as $n\to\infty$, where $K(s)$ is a bounded function for $s\in[0,1)$ and $\delta(x)\to 0$ as $x\downarrow 0$. If in addition, the conditions $\left[\mathcal{L}_{\alpha}\right]$ and (11) are satisfied then

[TABLE]

Corollary 3.2**.**

Let conditions $\left[{f_{\nu}}\right]$, $\left[{h_{\delta}}\right]$ hold. If $\delta>\nu$ then

[TABLE]

where $A$ is a positive constant and ${\mathcal{N}}(x)$ is SV at infinity defined in (7).

We make sure that at the conditions of second part of Theorem 3.1 PGF ${\mathcal{P}}_{n}(s)$ converges to a limit ${\pi}(s)$ which we denote by the power series representation

[TABLE]

In our conditions we can establish a speed rate of this convergence.

Theorem 3.3**.**

Let conditions $\left[{f_{\nu}}\right]$, $\left[{h_{\delta}}\right]$ hold and $\delta>\nu$. Then ${\mathcal{P}}_{n}(s)$ converges to ${\pi}(s)$ which generates the invariant measures $\left\{{\pi}_{j}\right\}$ for GWPI. The convergence is uniform over compact subsets of the open unit disc. If in addition, the conditions $\left[\mathcal{L}_{\alpha}\right]$, (11) and $\left[{\ell}_{\beta}\right]$ are fulfilled then

[TABLE]

where ${\mathcal{N}}_{\delta}(x)={\mathcal{N}}^{\delta}(x){\ell(x)}$, the function ${\mathcal{N}}(x)$ is defined in (7) and

[TABLE]

as $n\to\infty$ and $\nu_{n}(s)=\nu{n}+{\Lambda}^{-1}\left({1-s}\right)$.

The following result is direct consequence of Theorem 3.3.

Corollary 3.4**.**

If conditions of Theorem 3.3 hold then

[TABLE]

where ${\mathcal{N}_{\delta}(n)}$ is SV at infinity and

[TABLE]

Remark 3.5**.**

The analogous result as in Theorem 3.3 has been proved in [5] provided that $\delta=1$ and $f^{\prime\prime\prime}(1-)<\infty$.