Diffusion approximation of critical controlled multi-type branching processes

Branching processes can be well-applied to describe evolutionary systems, where elements reproduce according to certain probability laws. These processes are commonly used in population dynamics, and, in this context, systems are referred to as populations and elements as individuals or cells. In this framework, multi-type branching processes, that are appropriate for modelling the evolution of populations in which different types of individuals coexist, have been successfully applied. For instance, these processes are used for modelling polymerase chain reaction (see, e.g., Sagitov and Ståhlberg [30]), for modelling cell proliferation kinetics (see, e.g., González et al. [15], Kimmel and Axelrod [26, Chapter 5] or Yanev [37]) or for studying extinction of outbreak of diseases (see, e.g., Mwasunda et al. [27]). We will focus our attention on the class of controlled multi-type branching processes (CMBPs). This class was first introduced in González et al. [14]. The key feature of CMBPs is that the number of parents of each type at a given generation is determined by a random control mechanism that depends on the number of individuals of different types in the previous generation.

CMBPs form a wide family of branching processes that include, as particular cases, the well-known classical branching processes, namely, multi-type branching processes without or with immigration (MBPs or MBPIs), or two-sex branching processes (see details in Example 2.1). Although, for all these processes, individuals give rise to offspring independently of each others, in general, a CMBP no longer satisfies the additive property (see, e.g., Athreya and Ney [4, Chapter 1, page 3]). This is an important property that is satisfied, for instance, by MBPs and MBPIs. The lack of the additive property for a general CMBP makes it difficult to study this kind of processes and it comes from the following fact. Provided that we know the number of different types of individuals in a generation, in case of a general CMBP, the number of different types of individuals in the next generation is a random sum of some independent random variables, while, in case of a MBP or a MBPI, it is a non-random sum (in the sense that the number of summands is deterministic).

This paper aims to obtain a Feller-type diffusion approximation for some critical CMBPs (details of the model are given in Section 2) and to study the asymptotic behaviour of the relative frequencies of distinct types of individuals. The problems under consideration have a dual motivation. On the one hand, it has interest in itself from a theoretical point of view. Our theoretical results (see Theorem 3.3 and its corollaries) may allow us to study relevant applied problems as well. For example, our result on the asymptotic behaviour of relative frequencies may find applications in cell kinetics. On the other hand, the obtained scaling limit theorem can enable us to carry out further research on statistical inference. For example, one may complete the study on weighted conditional least square estimators for some critical CMBPs, started in González et al. [11], or one might start to investigate more general quasi-likelihood estimators. This intended research could contribute to the modeling of real data using CMBPs.

From a mathematical point of view, achieving functional limit theorems for critical branching processes has attracted the interest of many researchers since 1951, when Feller [10] was able to provide the first formulation for Galton–Watson processes. He proved that the sequence of appropriately scaled critical Galton–Watson processes converges in distribution to a non-negative diffusion process without drift (for a detailed proof based on infinitesimal generators, see also Ethier and Kurtz [9, Theorem 9.1.3]). The extension to branching processes with immigration started with the pioneering paper of Wei and Winnicki [33], where the limit process is a squared Bessel process that can be characterized as the pathwise unique strong solution of a certain stochastic differential equation (SDE). Focusing on controlled branching models, a further extension to the critical, single-type case was first carried out by Sriram et al. [32], and, more recently, by González et al. [13]. Furthermore, for some critical CMBPs, under quite involved technical conditions, González et al. [16, Corollary 4.1] proved a conditional weak limit theorem for the one-dimensional distributions provided that the explosion set has a positive probability. However, according to our knowledge, functional limit theorems for critical CMBPs are not available in the literature. Our present paper fills this gap, it is a natural extension of the result for critical single-type controlled branching processes (CBPs) in González et al. [13] to the multi-type case.

Assuming that the expectations of the control distributions satisfy a linear relationship with the population size additively perturbed by a function also depending on the population size (see (2.10)), we introduce a classification for such CMBPs based on the spectral radius of a matrix related to the offspring mean matrix and to the control distributions. Under some additional hypotheses, we prove that a suitably scaled and normalized critical CMBP converges weakly toward a squared Bessel process supported by a ray determined by an eigenvector of the aforementioned matrix, see Theorem 3.3. As corollaries, we are able to rediscover the known results on Feller-type diffusion approximations for critical, primitive MPBIs (see Corollaries 3.8 and 3.9). What is even more interesting is that we can apply our main theorem to get Feller-type diffusion approximations for some two-sex branching processes. We emphasize that no such results are available in the literature. Very recently, Bansaye et al. [5] have also proved a scaling limit theorem for a class of two-sex branching processes that combine classical asexual Galton–Watson processes and two-sex Galton–Watson branching processes introduced by Daley [8]. For a comparison of our results and theirs, see Remark 3.12. Finally, a result on the asymptotic behaviour of the relative frequencies for critical CMBPs is also derived from Theorem 3.3 (see Corollary 3.13). This kind of result has potential applications, for instance, in the field of cell kinetics, where it is more usual to measure relative frequencies instead of the absolute cell counts.

The proof of our main result (Theorem 3.3) follows the proof scheme of Theorem 3.1 in Ispány and Pap [21] for MBPIs, which is based on a weak convergence result for random step processes due to Ispány and Pap [20, Corollary 2.2] (see also Theorem A.4). This latter result has been applied in other papers to prove several scaling limit theorems, see, e.g., Ispány and Pap [21] and Ráth [28]. The lack of the additive property for a general CMBP makes the proof of our Theorem 3.3 more involved. Next, we outline the course of the proof, and we also point out the new ingredients in it. We start with determining the conditional moments of the branching process (see Proposition 2.3), which are essential to find out the asymptotic behaviour of some moments of the branching process (see Lemma A.3). The heart of the proof of Theorem 3.3 is an application of Theorem A.4 for a sequence of martingale differences formed from the CMBP in question (see (4.3) in Step 1). Then, in case of MBPIs, the Feller-type diffusion approximation follows straightforwardly from a continuous mapping theorem (see Theorem A.5), as Ispány and Pap [21] showed. However, this is not the case for general CMBPs, an extra additional work (see Step 3) is required. We also emphasize that the additive perturbation of the expectations of the control distributions mentioned above (see also (2.10)) results a new difficulty in the proofs compared to MBPIs, and it is addressed in the proofs of Steps 1 and 3.

The paper is structured as follows. In Section 2, CMBPs are defined, and, under the linearity assumption (2.11) on the conditional expectations, we introduce their classification by distinguishing subcritical, critical and supercritical CMBPs. By giving examples, we also point out that different types of classical branching processes can be viewed as particular cases of our model, thus illustrating its wide scope of applicability. In Section 3, we collect all the hypotheses that are assumed and we present all of our results obtained. Section 4 is devoted to the proof of Theorem 3.3, which is structured in four steps for an easy reading. We close the paper with an Appendix which contains some auxiliary results such as the asymptotic behaviour of the first and second moments of CMBPs and the second and fourth moments of a corresponding martingale difference (see Lemma A.3).

Let $(\Omega,\mathcal{F},\mathrm{P})$ be a fixed probability space on which all the random variables will be defined, and let $\mathbb{N}$, $\mathbb{Z}_{+}$, $\mathbb{R}$, $\mathbb{R}_{+}$ and $\mathbb{R}_{++}$ be the set of positive integers, non-negative integers, real numbers, non-negative real numbers, and positive real numbers, respectively. For all $\boldsymbol{x},\boldsymbol{y}\in\mathbb{R}^{p},$ let us denote by $\boldsymbol{x}\preceq\boldsymbol{y}$ if each coordinate of $\boldsymbol{x}$ is less than or equal to the corresponding coordinate of $\boldsymbol{y}$. For ${\boldsymbol{z}}=(z_{1},\ldots,z_{p})^{\top}\in\mathbb{R}^{p}$, let $|\boldsymbol{z}|\colonequals(|z_{1}|,\ldots,|z_{p}|)^{\top}\in\mathbb{R}_{+}^{p}$, and ${\boldsymbol{z}}^{+}\colonequals(z_{1}^{+},\ldots,z_{p}^{+})^{\top}\in\mathbb{R}_{+}^{p}$, where $x^{+}$ stands for the positive part of $x\in\mathbb{R}$ and ^⊤ for the transpose. The natural basis in $\mathbb{R}^{p}$ is denoted by $\boldsymbol{e}_{1},\ldots,\boldsymbol{e}_{p}$. The null vector in $\mathbb{R}^{p}$ is denoted by $\boldsymbol{0}_{p}$. The Borel sigma-algebra on $\mathbb{R}^{p}$ is denoted by $\mathcal{B}(\mathbb{R}^{p})$. The trace of a matrix $\mathsf{A}\in\mathbb{R}^{p\times p}$ is denoted by $\operatorname{tr}\left(\mathsf{A}\right)$. The $p\times p$ identity matrix is denoted by $\mathsf{I}_{p}$. For a matrix $\mathsf{A}\in\mathbb{R}^{l\times p}$, let $\operatorname{Null}(\mathsf{A})\colonequals\{\boldsymbol{x}\in\mathbb{R}^{p}:\mathsf{A}\boldsymbol{x}=\boldsymbol{0}_{l}\}$. Along the paper, we will not distinguish between the notations of the norm of a vector in $\mathbb{R}^{p}$ and that of a matrix in $\mathbb{R}^{p\times p}$. For ${\boldsymbol{z}}\in\mathbb{R}^{p}$, let $\|{\boldsymbol{z}}\|$ denote the Euclidean norm of ${\boldsymbol{z}}$, and for a matrix $\mathsf{A}\in\mathbb{R}^{p\times p}$, let $\|\mathsf{A}\|\colonequals\max\{\|\mathsf{A}{\boldsymbol{z}}\|:\|{\boldsymbol{z}}\|=1,\,{\boldsymbol{z}}\in\mathbb{R}^{p}\}$. For a positive semi-definite matrix $\mathsf{A}\in\mathbb{R}^{p\times p}$, let $\sqrt{\mathsf{A}}$ denote the unique symmetric positive semi-definite square root of $\mathsf{A}$. For a function $\boldsymbol{h}:\mathbb{R}^{p}\to\mathbb{R}^{p}$, by the notation $\boldsymbol{h}({\boldsymbol{z}})=\operatorname{O}(f({\boldsymbol{z}}))$ as $\|{\boldsymbol{z}}\|\to\infty$, where $f:\mathbb{R}^{p}\to\mathbb{R}_{+}$, we mean that there exist $C>0$ and $R>0$ such that $\|\boldsymbol{h}({\boldsymbol{z}})\|\leq Cf({\boldsymbol{z}})$ for all ${\boldsymbol{z}}\in\mathbb{R}^{p}$ with $\|{\boldsymbol{z}}\|>R$. Further, if for all $\epsilon>0$ there exists $R>0$ such that $\|\boldsymbol{h}({\boldsymbol{z}})\|\leq\epsilon f({\boldsymbol{z}})$ for all ${\boldsymbol{z}}\in\mathbb{R}^{p}$ with $\|{\boldsymbol{z}}\|>R$, then we write $\boldsymbol{h}({\boldsymbol{z}})=\operatorname{o}(f({\boldsymbol{z}}))$ as $\|{\boldsymbol{z}}\|\to\infty$. Convergence in probability is denoted by $\stackrel{{\scriptstyle\mathrm{P}}}{{\longrightarrow}}$. Some further notations for weak convergence of stochastic processes with càdlàg sample paths are recalled in Section 3.

For a fixed $p\in\mathbb{N}$ and a $\mathbb{Z}_{+}^{p}$–valued random vector $\boldsymbol{Z}_{0}$, let us consider a controlled $p$–type branching process $(\boldsymbol{Z}_{k})_{k\in\mathbb{Z}_{+}}$, defined recursively as

where $\boldsymbol{Z}_{k}$, $\boldsymbol{\phi}_{k}(\boldsymbol{z})$, $\boldsymbol{z}\in\mathbb{Z}_{+}^{p}$, and $\boldsymbol{X}_{k,j,i}$ are $\mathbb{Z}_{+}^{p}$–valued random vectors:

The intuitive interpretation of the process $(\boldsymbol{Z}_{k})_{k\in\mathbb{Z}_{+}}$ is as follows:

• •

  $Z_{k,i}$ is the number of $i$–type individuals in the $k$–th generation,

• •

  $\phi_{k,i}(\boldsymbol{Z}_{k})$ is the number of $i$–type progenitors in the $k$–th generation,

• •

  $X_{k,j,i,l}$ is the number of $l$–type offsprings of the $j$–th $i$–type progenitor in the $k$–th generation.

Assume that $\{\boldsymbol{Z}_{0},\boldsymbol{\phi}_{k}(\boldsymbol{z}),\boldsymbol{X}_{k,j,i}:k\in\mathbb{Z}_{+},\,j\in\mathbb{N},\,\boldsymbol{z}\in\mathbb{Z}_{+}^{p},\,i\in\{1,\ldots,p\}\}$ are independent, $\{\boldsymbol{\phi}_{k}(\boldsymbol{z}):k\in\mathbb{Z}_{+}\}$ are identically distributed for each $\boldsymbol{z}\in\mathbb{Z}_{+}^{p}$ and $\{\boldsymbol{X}_{k,j,i}:k\in\mathbb{Z}_{+},\,j\in\mathbb{N}\}$ are also identically distributed for each $i\in\{1,\ldots,p\}$. The distributions of $\boldsymbol{\phi}_{0}(\boldsymbol{z})$, ${\boldsymbol{z}}\in\mathbb{Z}_{+}^{p}$, are called the control distributions.

Moreover, for clarity of the readers, let us write out (2.1) for controlled two-type branching processes. The number of $i$–type individuals in the initial generation is $Z_{0,i}$, $i=1,2$, and the recursive definition in (2.1) takes the form

Part (iv) of Example 2.1 serves as a good illustration of a two-type case.

It is easy to check that the process defined in (2.1) is a $\mathbb{Z}_{+}^{p}$-valued Markov chain. This model is very general, several other popular branching processes can be seen as particular cases of the CMBPs, see Example 2.1.

Let us introduce notations for some moments. In all what follows, we suppose that $\operatorname{E}\!\left[\|\boldsymbol{X}_{0,1,i}\|^{4}\right]<\infty$ for $i\in\{1,\ldots,p\}$ and $\operatorname{E}\!\left[\|\boldsymbol{\phi}_{0}(\boldsymbol{z})\|^{4}\right]<\infty$ for $\boldsymbol{z}\in\mathbb{Z}_{+}^{p}$, and we denote

where $i,l\in\{1,\ldots,p\}$ and ${\boldsymbol{z}}\in\mathbb{Z}_{+}^{p}$.

Note that the moments defined in (2.5) and (2.6) are of course well-defined under weaker assumptions, namely, under the existence of first and second order moments of the offspring and control distributions, respectively. We also remark that $\boldsymbol{m}_{i}$, $\mathsf{\Sigma}_{i}$ and $\zeta_{i,l}$, $i,l\in\{1,\ldots,p\}$, do not depend on the control distributions, while $\boldsymbol{\varepsilon}(\boldsymbol{z})$, $\mathsf{\Gamma}(\boldsymbol{z})$, and $\kappa_{i}(\boldsymbol{z})$, ${\boldsymbol{z}}\in\mathbb{Z}_{+}^{p}$, $i\in\{1,\ldots,p\}$, do depend.

Let us consider the canonical filtration of the process $\mathcal{F}_{k}\colonequals\sigma(\boldsymbol{Z}_{0},\ldots,\boldsymbol{Z}_{k}),$ $k\in\mathbb{Z}_{+},$ introduce the matrix $\mathsf{m}\colonequals(\boldsymbol{m}_{1},\ldots,\boldsymbol{m}_{p})\in\mathbb{R}_{+}^{p\times p},$ and the operator $\odot:\mathbb{R}^{p}\times(\mathbb{R}^{p\times p})^{p}\to\mathbb{R}^{p\times p}$, $\boldsymbol{z}\odot\boldsymbol{\mathsf{A}}\colonequals\sum_{i=1}^{p}z_{i}\mathsf{A}_{i}$ for $\boldsymbol{z}=(z_{1},\ldots,z_{p})^{\top}\in\mathbb{R}^{p}$ and $\boldsymbol{\mathsf{A}}=(\mathsf{A}_{1},\ldots,\mathsf{A}_{p})\in(\mathbb{R}^{p\times p})^{p}$.

The following proposition is a multi-type counterpart of Proposition 3.5 in González et al. [12].

From now on, we assume that there exist a matrix $\mathsf{\Lambda}\in\mathbb{R}^{p\times p}$ and a function $\boldsymbol{h}:\mathbb{Z}_{+}^{p}\to\mathbb{R}^{p}$ with $\|\boldsymbol{h}(\boldsymbol{z})\|=\operatorname{o}(\|{\boldsymbol{z}}\|)$ as $\|\boldsymbol{z}\|\to\infty$ such that

Roughly speaking, the assumption (2.10) means that the average quantity of ancestors can be expressed as a linear map of the number of individuals affected by a minor perturbation, which becomes insignificant (negligible) in comparison with the population size when this latter is large enough. It is worth noting that migration of parents, both emigration and immigration, is permitted under (2.10) (see, for instance, part (iii) of Example 2.1 assuming that $\|\operatorname{E}\!\left[\boldsymbol{M}_{0}({\boldsymbol{z}})\right]\|=\operatorname{o}(\|{\boldsymbol{z}}\|)$ as $\|{\boldsymbol{z}}\|\to\infty$). In the single-type case, i.e., in case of $p=1$, a corresponding condition has already been assumed (see, e.g., the condition (i) in Theorems 1 and 2 in González et al. [17]). In the multi-type case, the assumption (2.10) was also considered in González and del Puerto [11, condition (11.2)] and in González et al. [16, condition (4.2)], but only with a diagonal matrix $\mathsf{\Lambda}$ having non-negative entries.

Using (2.8) and (2.10), we get

We can introduce a classification for a subclass of CMBPs satisfying (2.11) based on the asymptotic behaviour of $\operatorname{E}\!\left[\boldsymbol{Z}_{k}\right]$ as $k\to\infty$. Taking expectations of both sides of (2.11), by a recursive argument, one can derive a formula for $\operatorname{E}\!\left[\boldsymbol{Z}_{k}\right]$, $k\in\mathbb{N}$ (see (A.5) in Appendix A). From this, one can see that in the description of the asymptotic behaviour of $\operatorname{E}\!\left[\boldsymbol{Z}_{k}\right]$ as $k\to\infty$, the powers of the matrix $\tilde{\mathsf{m}}\colonequals\mathsf{m}\mathsf{\Lambda}$ play a crucial role. We assume that $\tilde{\mathsf{m}}\in\mathbb{R}_{+}^{p\times p}$ and it has only one eigenvalue of maximum modulus $\tilde{\rho}$ with algebraic and geometric multiplicities equal one. By the spectral theory of matrices, the asymptotic behaviour of $(\mathsf{m}\mathsf{\Lambda})^{k}$ as $k\to\infty$ is determined by $\tilde{\rho}$, the other eigenvalues do not come into play. Motivated by this, by definition, we say that such a CMBP is subcritical if $\tilde{\rho}<1$, critical if $\tilde{\rho}=1$, and supercritical if $\tilde{\rho}>1$. It is important to highlight that whenever the matrix $\tilde{\mathsf{m}}$ is primitive (i.e., there exists $n\in\mathbb{N}$ such that all the entries of $\tilde{\mathsf{m}}^{n}$ are positive), the Perron–Frobenius theorem (see also Lemma A.1) guarantees the unique existence of $\tilde{\rho}$. Moreover, the usual classification of multi-type branching processes without and with immigration coincides with the one proposed here (see, e.g., Athreya and Ney [4, Chapter V, page 186] and Kaplan [25, page 948]).

Let $({\boldsymbol{Z}}_{k})_{k\in\mathbb{Z}_{+}}$ be a controlled $p$–type branching process given in (2.1). Let us introduce the following hypotheses.

Next, we give an example for a controlled $p$–type branching process for which the Hypotheses 1–5 hold.

Let $\mathbf{C}(\mathbb{R}_{+},\mathbb{R}^{p})$ be the space of $\mathbb{R}^{p}$–valued continuous functions on $\mathbb{R}_{+}$ and let us denote by $\mathbf{D}(\mathbb{R}_{+},\mathbb{R}^{p})$ the space of $\mathbb{R}^{p}$–valued càdlàg functions on $\mathbb{R}_{+}$ and by $\mathcal{D}_{\infty}(\mathbb{R}_{+},\mathbb{R}^{p})$ its Borel $\sigma$–algebra corresponding to the metric defined in equation (1.26) in Jacod and Shiryaev [22, Chapter VI]. For $\mathbb{R}^{p}$–valued stochastic processes with càdlàg paths $(\boldsymbol{\mathcal{Y}}_{t}^{(n)})_{t\in\mathbb{R}_{+}}$, $n\in\mathbb{N}$, and $(\boldsymbol{\mathcal{Y}}_{t})_{t\in\mathbb{R}_{+}}$, if the distribution of $(\boldsymbol{\mathcal{Y}}_{t}^{(n)})_{t\in\mathbb{R}_{+}}$ on the space $\left(\mathbf{D}(\mathbb{R}_{+},\mathbb{R}^{p}),\mathcal{D}_{\infty}(\mathbb{R}_{+},\mathbb{R}^{p})\right)$ converges weakly as $n\to\infty$ to the distribution of $(\boldsymbol{\mathcal{Y}}_{t})_{t\in\mathbb{R}_{+}}$ on the same space, then we use the notation $(\boldsymbol{\mathcal{Y}}_{t}^{(n)})_{t\in\mathbb{R}_{+}}\stackrel{{\scriptstyle\mathcal{L}}}{{\longrightarrow}}(\boldsymbol{\mathcal{Y}}_{t})_{t\in\mathbb{R}_{+}}$ as $n\to\infty$.

Let us define a sequence of random step processes $(\boldsymbol{\mathcal{Z}}_{t}^{(n)})_{t\in\mathbb{R}_{+}}$, $n\in\mathbb{N},$ using the controlled $p$–type branching process $({\boldsymbol{Z}}_{k})_{k\in\mathbb{Z}_{+}}$, as

Recall that, under Hypothesis 5, $\tilde{\boldsymbol{u}},\tilde{\boldsymbol{v}}\in\mathbb{R}_{+}^{p}$ are the right and left eigenvectors of $\tilde{\mathsf{m}}$ corresponding to eigenvalue $\tilde{\rho}=1$, respectively.