Ancestral Lineages and Limit Theorems for Branching Markov Chains in Varying Environment

Abstract

We consider branching processes in discrete time for structured population in varying environment. Each individual has a trait which belongs to some general state space and both its reproduction law and the trait inherited by its offsprings may depend on its trait and the environment. We study the long-time behavior of the population and the ancestral lineage of typical individuals under general assumptions. We focus on the mean growth rate and the trait distribution among the population. The approach relies on many-to-one formulae and the analysis of the genealogy, and a key role is played by well-chosen (possibly non-homogeneous) Markov chains. The applications use large deviations principles and the Harris ergodicity for these auxiliary Markov chains.

Appendix: Fixed Environment and Spectral Gap of the Mean Operator

In this section, we recover standard results in the fixed environment case from the statements of the previous sections. Thus, we set $\mathbb {P}:=\mathbb {P}_{\mathbf{e}}$ and

$$\begin{aligned} m_n(x,.): =m_n(x,\mathbf{e}, .), \qquad m_n(\mu ,.)=\int _{\mathcal {X}}\mu (\mathrm{d}x)m_n(x,.) \end{aligned}$$

for any $x\in \mathcal {X}$ and $\mu \in \mathcal {M}(\mathcal {X})$. We consider a subspace $\mathfrak M$ of $\mathcal M(\mathcal {X})$ stable by addition which contains $\mathcal {M}_1(\mathcal {X})$. We endow $\mathfrak M$ with a norm $\parallel . \parallel _{\mathfrak M}$ and assume that there exists $c>0$ such that $\parallel \mu \parallel _{\mathfrak M}\le c\mu (\mathcal {X})$ for any $\mu \in \mathfrak M$ and that $\mu \rightarrow m_1(\mu ,.)$ is a bounded endomorphism on $(\mathfrak M, \parallel . \parallel _\mathfrak M)$. We denote by $\mathfrak M'$ the topological dual of $\mathfrak M$ and require the following spectral properties.

Assumption 4

There exists $(\lambda , \mu _0, f_0)\in (1,\infty ]\times \mathcal M_1(\mathcal {X}) \times \mathfrak M'$ such that $\inf _{x\in \mathcal {X}} f_0(\delta _x)>0$ and

$$\begin{aligned} m_1(\mu _0,.)=\lambda \mu _0(.), \qquad f_0(m_1(., \mathrm{d}x))=\lambda f_0(.). \end{aligned}$$

Moreover, there exists $a<\lambda$ and $c>0$ such that

$$\begin{aligned} \parallel m_n(\mu ,.)- \lambda ^n f_0(\mu )\mu _0(.)\parallel _\mathfrak M \le ca^n \parallel \mu - f_0(\mu ) \mu _0(.)\parallel _\mathfrak M. \end{aligned}$$

When $\mathcal {X}$ is finite, Perron Frobenius theory ensures that the previous Assumptions hold if the matrix given by the mean operator $m_1$ is aperiodic and irreducible. We refer to [47] for details and extension to a denumerable state space $\mathcal {X}$. Moreover, Krein Rutman Theorem allows to tackle the non-denumerable framework when the mean operator is compact and positive. Let us finally note that several techniques in analysis allow to go beyond these assumptions via the decompositions of the operator, see [43], where an overview of the results in the continuous time framework is given.

The previous assumption ensures that for any nonnegative function f such that $\mu _0(f) \in (0,\infty )$ and any $\mu \in \mathfrak M$,

$$\begin{aligned} m_n(\mu ,f)\sim \lambda ^n f_0(\mu ) \mu _0(f) \qquad \text {and} \qquad m_n(\mu , \mathcal {X})\sim f_0(\mu )\lambda ^n\qquad (n\rightarrow \infty ). \end{aligned}$$

Proposition 4

Let $f\in \mathcal B_b(\mathcal {X})$ and $x_0\in \mathcal {X}$. If Assumption 4 holds and $\sup _{x \in \mathcal {X}}\mathbb {E}(N(x)^2 )<\infty ,$ then

• (i) $Z_n(\mathcal {X})/m_n(x_0,\mathcal {X})$ is bounded in $L^2_{\delta _{x_0}}$;

• (ii) $f_0(Z_n)/\lambda ^n$ converges $\mathbb {P}_{\delta _{x_0}}$ a.s. to $W \in [0,\infty )$ and $\mathbb {P}_{\delta _{x_0}}(W>0)>0$;

• (iii) $Z_n(f)/ Z_n(\mathcal {X}) {\longrightarrow } \mu _0(f)$ as $n\rightarrow \infty$, $\mathbb {P}_{ \delta _{x_0}}$ a.s. on the event $\{W>0\}.$

Proof

First, we recall that $f_0(Z_n)/\lambda ^n$ is martingale since $f_0$ is the eigenvector of the adjoint of the mean operator. Indeed, denoting by

$$\begin{aligned} Z_{1,u}=\sum _{i=1}^{N(u)} \delta _{X(ui)} \end{aligned}$$

for $u \in \mathbb {G}_n$ the punctual measures associated with the offsprings of each individual in generation n and combining the linearity of $f_0$ and $\mathbb {E}$, we get :

$$\begin{aligned} \mathbb {E}(f_0(Z_{n+1}) \ \vert \ \mathcal F_n)= & {} \mathbb {E}\left( \sum _{u\in \mathbb {G}_{n}} f_0(Z_{1,u}) \ \vert \ \mathcal F_n\right) = \sum _{u\in \mathbb {G}_{n}} f_0 (\mathbb {E}(Z_{1,u} \ \vert \ \mathcal F_n)) \\= & {} \sum _{u\in \mathbb {G}_{n}} \lambda f_0(\delta _{X(u)}) = \lambda f_0(Z_n). \end{aligned}$$

Second, we use Assumption 4 and the fact that $0<\inf _{x\in \mathcal {X}}f_0(\delta _x)\le \sup _{x\in \mathcal {X}}f_0(\delta _x)<\infty$ to check that there exists $C>0$ such that for any $x,y\in \mathcal {X}$

$$\begin{aligned} m_n(x,\mathcal {X})\le C m_n(y,\mathcal {X}) \end{aligned}$$

Adding that $m_n(x_0,\mathcal {X})$ grows geometrically with rate $\lambda >1$, we can follow the first part of the proof of Lemma 6 to check that (12) holds and prove (i). Thus, $Z_n(\mathcal {X})/m_n(x,\mathcal {X})$ is bounded in $L^2$ and so does $f_0(Z_n)/\lambda ^n$ since $f_0$ is bounded and $\parallel Z_n \parallel _{\mathfrak M} \le c Z_n(\mathcal {X})$. We deduce that the martingale limit of $f_0(Z_n)/\lambda ^n$ is non-degenerated and $(i)-(ii)$ are proved. Let us now focus on

$$\begin{aligned} Q_n(x,f)=m_n(x,f)/m_n(x,\mathcal {X}). \end{aligned}$$

Using the second part of Assumption 4 and $f_0$ bounded, there exist constants $c',c''$ such that for every $y\in \mathcal {X}$

$$\begin{aligned} \vert Q_n(y,f)-\mu _0(f) \vert \le c'(a/\lambda )^n \parallel \delta _y-f_0(\delta _y)\mu _0\parallel _{\mathfrak M}\le c''(a/\lambda )^n \end{aligned}$$

and

$$\begin{aligned} \vert Q_n(y,f)-Q_n(z,f) \vert \le 2c''(a/\lambda )^n. \end{aligned}$$

It ensures that condition (13) holds since $a<\lambda$ and we can apply Theorem 4 to get

$$\begin{aligned} \frac{Z_n(f)-Z_n(\mathcal {X})\mu _0(f) }{m_n(x,\mathcal {X})} {\mathop {\longrightarrow }\limits ^{n\rightarrow \infty }} 0 \qquad \mathbb {P}_{ \delta _x} \quad \text {a.s}. \end{aligned}$$

Adding that $\liminf _{n\rightarrow \infty } Z_n(\mathcal {X})/m_n(x,\mathcal {X})>0$ a.s. on the event $\{W>0\}$ since $f_0$ is bounded ends up the proof. $\square$

Let us give some additional comments. Checking that the martingale limit W is non-degenerated is delicate in general. Classical $N\log N$ moment assumptions for single-type population in Kesten Stigum theorem (see [38, 39]) can be extended to the case $\# \mathcal {X}<\infty$ [38], and less explicit but more general criteria can be found in [4] for $\#\mathcal {X}=\infty$. Here, we have used the $L^2$ computations of the previous section to get a strong law of large numbers for a general state space $\mathcal {X}$.