Extinction in lower Hessenberg branching processes with ... - arXiv

Extinction in lower Hessenberg branching processes with countably many types arXiv:1706.02919v1 [math.PR] 9 Jun 2017

P. Braunsteins and S. Hautphenne June 12, 2017

Abstract: We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes. These are multitype Galton-Watson processes with typeset X = {0, 1, 2, . . . }, in which individuals of type i may give birth to offspring of type j ≤ i + 1 only. For this class of processes, we study the set S of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector q and whose maximum is the partial extinction probability vector q. ˜ In the case where q˜ = 1, we derive a global extinction criterion which holds under second moment conditions, and when q˜ < 1 we develop necessary and sufficient conditions for q = q. ˜ Keywords: infinite-type branching process; extinction probability; varying environment; fixed point.

1

Introduction

Multitype Galton-Watson branching processes (MGWBPs) are models for the evolution of a population of individuals who live one unit of time and give birth to a random number of offspring that may be of different types. Each type has its own progeny distribution, and individuals behave independently of each other. These processes have long been studied; classical references include the books by Harris [18], Mode [25], Athreya and Ney [2], and Jagers [21]. MGWBPs play an important role in models of population biology including molecular biology, ecology, epidemiology, and evolutionary theory, as well as in particle physics, chemistry, and computer science. Recent books with special emphasis on applications are Axelrod and Kim1

mel [3], and Haccou, Jagers and Vatutin [17]. Applications specific to the infinite-type case include models for the dynamics of escape mutants [30] and models of parasitic infection [4]; see also [3, Chapter 7] and references therein for other biological applications. Characterisation of the extinction probability is one of the main topics of research on branching processes. Let the entries of the vector Zn = (Zn,` )`∈X record the number of type-` individuals alive in generation n of a MGWBP with discrete type space X . The conditional probability that the population eventually becomes empty, given that itP starts with a single individual of type i, is defined as qi = P[limn→∞ `∈X Zn,` = 0 | Z0 = ei ], and we let q = (qi )i∈X . When X is countably infinite, q is referred to as the global extinction probability vector, that is, the probability that the whole population eventually dies out. This is in contrast to another form of extinction event called partial extinction, which is the event that every type eventually disappears. The conditional partial extinction probability is q˜i = P[limn→∞ Zn,` = 0, ∀` ∈ X | Z0 = ei ], and we let q˜ = (˜ qi ). While it is clear that global extinction implies partial extinction, it is however possible for every type to eventually become extinct while the total number of individuals grows without bound; examples where q < q˜ can be found for instance in [19, Section 5.1]. It is well known that both q and q˜ satisfy the fixed point equation s = G(s), where G(s) = (Gi (s)) records the progeny generating function associated with each type. That is, in the infinite-type case, both q and q˜ are elements of the set of fixed points in the infinite-dimensional unit cube, S = {s ∈ [0, 1]X : s = G(s)}.

(1.1)

It is also well established that the vector q is the minimal element of S [26]. In the finite-type irreducible case, the set S contains at most two elements, q = q˜ and 1. A number of authors have derived properties of the set S in a general irreducible infinite-type setting: Moyal [26] provided conditions for S to contain at most a single solution s with supi si < 1, corresponding to q; Spataru [31] stated that S contains at most two elements, q and 1. However, recently Bertacchi and Zucca [7, 8] produced an irreducible example of MGWBP where S contains uncountably many elements with supi si = 1, which acts as a counter-example to Spataru’s claim. Besides the characterisation of the set S of fixed points, another fundamental problem is to provide necessary and sufficient conditions for global and partial extinction [7, 10, 18, 19, 26, 31, 32]. In the finite-type case, it is well known that q = 1 if and only if sp(M ) ≤ 1, where sp(M ) is the spectral radius of the mean progeny matrix M , whose (i, j)th entry is the expected 2

number of type j offspring born to a parent of type i. In the infinite-type case, the analogue of the spectral radius is the convergence norm ν(M ) of M , which provides a partial extinction criterion: q˜ = 1 if and only if ν(M ) ≤ 1, see for example [15]. However, in the case where partial extinction is almost sure, we are still lacking general necessary and sufficient conditions for q = 1. It turns out that there can be no global extinction criterion based solely upon M , as highlighted through an example in [32], but as pointed out by the author, other moment conditions have not been clearly identified. In addition, in the case where q˜ < 1, following the terminology in [7] the process can exhibit strong local survival if q = q˜ < 1, or non-strong local survival if q < q˜ < 1, and it is again challenging to derive a general criterion separating the two cases. The main contribution of this paper is to use a unified probabilistic approach to progress the characterisation of the set of fixed points S and the derivation of global extinction criteria when q˜ = 1 for a class of branching processes with countably infinitely many types called lower Hessenberg branching processes (LHBPs). In these processes, the only constraint (apart from usual regularity conditions) is that type-i individuals can produce offspring of type no larger than i + 1, for all i ≥ 0; as a consequence their (infinite) mean progeny matrix has a lower Hessenberg form. Special cases are skip-free branching random walks and nearest-neighbour branching random walks on Z+ , see for instance [16, 7, 8]. The probabilistic approach we use to study properties of LHBPs relies on a single pathwise argument: we reduce the study of the LHBP to that of a much simpler single-type Galton-Watson process in a varying environment (GWPVE), embedded in the LHBP. GWPVEs are single type Galton-Watson processes whose offspring distributions vary deterministically with the generation. The embedded GWPVE is related to the seed process introduced in [10] which is used for a different purpose in the analysis of general MGWBP with countably many types. In our context, the embedded GWPVE is explosive, in the sense that individuals may have an infinite number of offspring in one generation. In particular, we show the equivalence between the global extinction of the LHBP and the extinction of the embedded GWPVE, and between the partial extinction of the LHBP and the event that all generations of the embedded GWPVE are finite. Based on this relationship, we obtain several results for LHBPs: (i) We prove the existence of a continuum of fixed points solutions s with ˜ whose componentwise minimum and maximum are the q ≤ s ≤ q, global and partial extinction probability vectors, respectively. In the 3

irreducible case, the set S consists of this continuum and the fixed point 1. In the reducible case, under the assumption that Mi,i+1 > 0 for all i ∈ X , we show that there may be an additional countable number of fixed points s such that q˜ ≤ s ≤ 1. (ii) We establish a connection between the growth rates of the embedded GWPVE and the convergence rate of si to 1 as i → ∞ for any s ∈ S \ {q}; this yields a physical interpretation for the fixed points lying ˜ in between q and q. (iii) In the non-trivial case where q˜ = 1, we provide a necessary and sufficient condition for global extinction which holds under some second moment conditions. This is the first extinction criterion for irreducible processes that also applies to cases exhibiting non exponential growth. We illustrate the broad applicability of the criterion through some examples. (iv) Finally, under additional assumptions, we build on the global extinction criterion to derive necessary and sufficient conditions for strong local survival. While there is a vast literature on GWPVEs, the explosive case, which has already been studied for standard Galton-Watson processes [22, 28], is yet to be considered in the context of varying environment. In order to prove our main theorems for LHBPs, we both apply known results on GWPVEs and develop new ones. On the way to studying properties of the embedded GWPVE, we also derive a new partial extinction criterion for LHBPs which is computationally more efficient than other known methods for infinite-type branching processes. The paper is organised as follows. Section 2 provides the background on LHBPs and on two families of related finite-type processes which are constructed pathwise on the same probability space. The later processes play a fundamental role in the construction of the embedded GWPVE with explosion, which is detailed in Section 3 along with its relations with LHBPs and the analysis of its generation-dependent offspring distribution. Section 4 develops (i) and (ii), while Sections 5 and 6 deal with (iii) and (iv), respectively. Possible extensions of our results to more general infinite-type branching processes are discussed in a conclusion section. The proofs related to the illustrative examples are gathered in a final Appendix.

4

2 2.1

Preliminaries Lower Hessenberg infinite-type branching processes

Lower Hessenberg branching processes (LHBPs) form a class of multitype Galton-Watson process with countable type set X = N0 := {0, 1, 2, 3, . . . }, evolving according to the following rules. The population starts with a single individual, whose type is denoted by ϕ0 ; each individual lives for a single generation, and at death, individuals of type i ∈ X give birth to a random number of children of different types not larger than i + 1, described by the vector r = (r0 , r1 , . . . , ri+1 ), where r` denotes the number of offspring of type `. The vector r ∈ (N0 )i+2 is chosen independently of all other individuals according to a probability distribution pi (·) specific to the parental type i. The LHBP is defined on the probability space (Ω, F, P), as formally described in [10]. The progeny generating vector G(·) : [0, 1]X → [0, 1]X of a LHBP has entries Gi (s) = Gi (s0 , s1 , . . . , si+1 ) =

X

r

pi (r)s =

r∈(N0 )i+2

X r∈(N0 )i+2

pi (r)

i+1 Y

srkk , i ∈ X ,

k=0

(2.1) and the mean progeny matrix M is such that Mij = (∂Gi (s)/∂sj )|s=1 , for i, j ∈ X , where Mij is the expected number of type-j children born to a parent of type i. The matrix M is an infinite-dimensional lower Hessenberg matrix, whose row sums are assumed to be finite. Importantly, this means that the event that an individual has an infinite number of offspring has probability zero. To avoid trivialities we further assume that Mi,i+1 > 0 for all i ∈ X , which ensures that every type i can have any type j > i among its descendants, without necessarily assuming irreducibility. We shall refer to the mean progeny representation graph as the oriented graph in which vertices correspond to types and the weight of an edge from i to j corresponds to Mij > 0. For each n ≥ 0, the population size vector Zn = (Zn,0 , Zn,1 , . . .) ∈ (N0 )X has entries Zn,iP counting the number of type-i individuals alive in generation n, and |Zn | = i Zn,i denotes the total population size at generation n. The LHBP {Zn }n≥0 is therefore an infinite-dimensional Markov chain with the single absorbing state 0. The event that the process eventually reaches state 0 is called global extinction, and we denote by q = (q0 , q1 , . . .) the conditional 5

{Zn }

˜n } {Z

(1)

{Zn }

1

1

1

2

1

3 1

1 4

2

2

3

2

1

2

2

(1)

1

2

1

2

1 1

.. .

2 .. .

(1)

˜n } and {Z (1) Figure 2.1: A visualisation of {Zn }, {Z n } for a specific ω ∈ Ω. global extinction probability vector, given the initial type, with   qi = P lim |Zn | = 0 | ϕ0 = i , i ∈ X . n→∞

Global extinction is to be distinguished from partial extinction, which corresponds to the event that every type eventually becomes extinct. The conditional partial extinction probability vector, given the initial type, is denoted by q˜ = (˜ q0 , q˜1 , . . .), with   q˜i = P ∀`∈ X : lim Zn,` = 0 | ϕ0 = i , i ∈ X . n→∞

It is clear that global extinction implies partial extinction, hence q ≤ q˜ (where here and in the sequel the ordering of vectors is componentwise), however the converse does not necessarily hold. The vectors q and q˜ are both solutions to the fixed point equation s = G(s), that is, q and q˜ belong to the set of all fixed points S = {s ∈ [0, 1]X : s = G(s)}.

(2.2)

Due to the lower Hessenberg structure, S is one-dimensional: indeed, for any s ∈ S, the entries si can be expressed as functions of s1 for all i ≥ 2. It will therefore be convenient to consider the one-dimensional projection sets of S, Si = {x ∈ [0, 1] : ∃ s ∈ S, such that si = x},

2.2

i ∈ X.

Related finite-type processes

To compute the infinite extinction probability vectors q and q˜ corresponding to a standard multitype Galton-Watson processes {Zn } with type set X , the 6

authors of [19] defined two families of related finite-type branching processes, constructed pathwise on (Ω, F, P) by modifying the offspring distributions of the original process. We briefly recall this construction here as it will play a significant role in the rest of the paper. ˜n(k) }n≥0,k≥−1 is defined from the paths The first sequence of processes {Z of {Zn } by removing the descendants of all individuals of type i > k. In ˜n(k) } are sterile, that is, other words, all individuals with type i > k in {Z they produce no offspring. The second sequence of processes {Z n(k) }n≥0,k≥−1 is defined from the paths of {Zn } by removing the descendants of all individuals with type i > k and replacing them with an infinite line of descent made up only of type-i individuals. In other words, all individuals with type i > k in {Z (k) n } are immortal , that is, they produce a single type-i offspring with probability 1. ˜n(k) } and {Z (k) For a rigorous construction of {Z n } on the probability space (Ω, F, P) we refer the reader to [10]. An illustration of a specific trajectory of these processes for k = 1 is given in Figure 2.1. Note that, in the LHBP case, for each fixed value of k, the sterile and immortal individuals are necessarily of type k + 1. By construction, for all ω ∈ Ω, (k) (k) Zn,` (ω) = Z˜n,` (ω)

for all n ≥ 0 and 0 ≤ ` ≤ k,

and (k) lim Z (ω) n→∞ n,k+1

=

∞ X

(k) Z˜n,k+1 (ω).

(2.3)

(2.4)

n=0 (k)

The progeny generating vector of {Zn }, denoted by G(k) (s), has entries ( Gi (s), 0≤i≤k (k) (2.5) Gi (s) = sk+1 , i = k + 1. Let q (k) and q˜(k) denote the extinction probability vectors corresponding to (k) ˜n(k) }, respectively. By Equation (2.3) we have, {Zn } and {Z q˜(k) = lim G(k,n) (0, . . . , 0, 1) n→∞

and q (k) = lim G(k,n) (0, . . . , 0, 0), n→∞

where G(k,n) (·) is the n-fold composition of G(k) (·). In [19] it was shown that the sequence {q (k) }k≥−1 is (componentwise) monotonically increasing to the global extinction probability q, and the sequence {q˜(k) }k≥−1 is monotonically decreasing to the partial extinction probability q. ˜

7

3

An embedded GWPVE with explosions

The results in Section 4 on the characterisation of the set (2.2) of fixed points for LHBPs, and the establishment in Section 5 of new global extinction criteria, all rely on a simple argument: we reduce the analysis of the LHBP {Z n } to that of an embedded Galton-Watson process {Yk } in a varying environment subject to explosion. The embedded GWPVE {Yk } is constructed from the paths of the LHBP {Z n } by selecting all individuals whose type is strictly larger than that of all their ancestors, and connecting these individuals to their nearest (in terms of generation) ancestor. More formally, {Yk } is constructed on the same ˜n(k−1) }n,k≥0 and probability space as the families of finite-type processes {Z {Z (k−1) }n,k≥0 . For convenience, we shall assume that if {Z n } starts with n a single individual of type ϕ0 , then {Yk } starts at generation ϕ0 . The kth generation of {Yk } is then made up of all the sterile individuals produced ˜n(k−1) }; if this number is infinite, then we say that the over the lifetime of {Z embedded process experiences explosion at generation k, following the terminology of [22]. Due to the lower Hessenberg assumption, the kth generation of {Yk } contains type-k individuals only, which means that it does indeed evolve marginally as a GWPVE. Definition 1 The embedded GWPVE {Yk } defined on (Ω, F, P) is such that for any ω ∈ Ω and k ≥ ϕ0 , Yk (ω) =

∞ X

(k−1) (k−1) Z˜n,k (ω) = lim Zn,k (ω). n→∞

n=0

(3.1)

The embedded GWPVE {Yk } is equivalent to the seed process defined in [10] but with one difference: whenever the GWPVE explodes, the seed process experiences a total catastrophe; in that paper, properties of the seed process were exploited for a different purpose than here. The construction of {Yk } is also analogous to that of the Galton-Watson process in a stationary ergodic random environment embedded in the nearest-neighbour random walk in [16, Proof of Theorem 2.9]. An illustration of {Yk } (right) compared to the corresponding realisation of {Zn } (left) is given in Figure 3.1. A crucial property of the embedded GWPVE is the fact that the events of global and partial extinction in {Zn } correspond respectively to the events of extinction and non-explosion in {Yk }. This relationship, which is formalised in the next theorem, provides the foundation to many of the results in the 8

{Zn }

{Yk }

0

0

0 0

1 1

1

0

1

2

2

1

2

1

2

2

3

1 2

2

3

3

3

3

1

4

4

0

Figure 3.1: A visualisation of {Zn } and {Yk } for a specific ω ∈ Ω. The highlighted type-k individuals represent the sterile individuals in the corre˜n(k−1) }. sponding realisation of {Z remainder of the paper. In the sequel we use the shorthand notation Pi (·) for P(·|Yi = 1) and Ei (·) for E(·|Yi = 1). Theorem 1 For all 0 ≤ i ≤ k, (k)

qi

= Pi (Yk+1 = 0)

and

(k)

q˜i

= Pi (Yk+1 < ∞) ,

and consequently qi = Pi ( lim Yk = 0) k→∞

q˜i = Pi (∀ k ≥ i, Yk < ∞) .

and

(k)

Proof. The fact that qi = Pi (Yk+1 = 0) follows directly from the con(k) (k) struction of {Yk }. Let E1 = {ω ∈ Ω : limn→∞ Zn,k+1 (ω) < ∞} and (k) ˜n(k) (ω) = 0}. To show q˜i(k) = Pi (Yk+1 < ∞), E2 = {ω ∈ Ω : limn→∞ Z it suffices to prove that (k)

(k)

E2 ⊆ E1 (k)

(k)

(k)

and P(E1 \ E2 ) = 0. (k)

(k)

(3.2)

We first show that E2 ⊆ E1 . Consider ω ∈ E2 , that is, such that ˜n(k) (ω) = 0. Then there exists a generation N such that Z ˜n(k) (ω) = 0 limn→∞ Z for all n ≥ N have a finite number of offspring, P.∞Since(k)individuals PNcan only (k) ˜ ˜ this means n=0 |Zn (ω)| = n=0 |Zn (ω)| < ∞. Thus, by Equation (2.4) P (k) (k) ˜ (k) we have limn→∞ Zn,k+1 (ω) = N n=0 Zn,k+1 (ω) < ∞, therefore ω ∈ E1 . To show   (k) (k) (k) ˜n(k) | > 0) = 0, P E1 \ E2 = P( lim Zn,k+1 < ∞, lim |Z n→∞

n→∞

9

we observe that since Mi,i+1 > 0 for all i, there exists some ε > 0 such that (k) for any initial population vector z˜0 , we have ! k+1 X (k) ˜ (k) | > 0 Z ˜0(k) = z˜0(k) ≤ 1 − ε. P Z˜n,k+1 = 0, |Z (3.3) k+1 n=0 (k) Using (2.4), the fact that Z˜n,k+1 takes integer values, (3.3), and the Markov property, we then have (k) ˜n(k) | > 0) P( lim Zn,k+1 < ∞, lim |Z n→∞

n→∞

≤ P ∃N ≥ 0 :

!

∞ X

(k) ˜n(k) | > 0 ∀n Z˜m,k+1 = 0, |Z

m=N

 = P ∃N ≥ 0 :

∞ \



X

≤

(k) ˜ (k)  Z˜m,k+1 = 0, |Z N +(`+1)(k+1) | > 0

 `=0

∞ X



N +(`+1)(k+1)

m=N +`(k+1)

Y (1 − ε) = 0,

N =0 `≥0

which completes the proof.



To make use of Theorem 1, we need to characterise the possibly defective progeny generating function corresponding to each generation k of the embedded GWPVE, ! ∞ X X (k) gk (s) := Ek (sYk+1 1{Yk+1 < ∞}) = P Z˜ = ` ϕ0 = k s` , n,k+1

`≥0

n=1

(3.4) where s ∈ [0, 1] and gk (1) = Pk (Yk+1 < ∞) = ≤ 1. The generating function of Yk+1 , conditional on Yi = 1 for i ≤ k, is then given by (k) q˜k

gi→k (s) := gi ◦ gi+1 ◦ · · · ◦ gk (s), (k)

s ∈ [0, 1],

(k)

which means that qi = gi→k (0) and q˜i = gi→k (1), and as a consequence, qi = limk→∞ gi→k (0), and q˜i = limk→∞ gi→k (1). The next two lemmas provide respectively an explicit and an implicit relation between the sequence of progeny generating functions {gk (·)} and the progeny generating function G(·) of the original LHBP. The first lemma makes use of the following technical assumption: (k)

Assumption 1 The restriction of {Zn } to the types 0, 1, . . . , k is nonsingular for all k ≥ 0. 10

Lemma 1 Suppose Assumption 1 holds. Then for all k ≥ ϕ0 , the progeny generating function of {Yk } at generation k is given by (k,n)

gk (s) = lim Gk n→∞

s ∈ [0, 1],

(s0 , s1 , . . . , sk , s),

where (s0 , s1 , . . . , sk ) ∈ [0, 1)k+1 and G(k) (·) is defined in (2.5). Proof. By (3.1) and (3.4), we have  n o (k) (k) gk (s) = Ek slimn→∞ Zn,k+1 1 lim Zn,k+1 < ∞ . n→∞

By Assumption 1 and (3.2), we have # " k n o (k) Y Zn,i (k) E 1 lim Zn,k+1 < ∞ − lim si = 0, n→∞ n→∞ i=0

whenever (s0 , . . . , sk ) ∈ [0, 1)k+1 , which implies gk (s) = Ek =

lim s

(k) Zn,k+1

n→∞

k Y

(k)

Z si n,i

!

i=0 (k,n) lim G (s0 , s1 , . . . , sk , s), n→∞ k

(3.5)

where (3.5) follows from the dominated convergence theorem.



Lemma 2 For any k ≥ 0, the progeny generating function gk (·) satisfies gk (s) = Gk (g0→k (s), g1→k (s), . . . , gk (s), s) .

(3.6) (k)

˜n }, Proof: By conditioning on the offspring of a type-k individual in {Z gk (s) h P∞ ˜(k) i P Zn,k+1 (k) ∞ n=1 ˜ = E s 1 n=1 Zn,k+1 mEn(k) (Z˜n(k) ) = 1, and hence

∗

q˜ ≤ q˜(k ) < 1. This completes the proof.



Remark: In the reducible case the same result applies with the notable exception of instances where there exists k ≥ 0 such that `k = 1 and there is no path from k + 1 to k in {Zn }. Every time this occurs, we can then form a new LHBP by ignoring types {0, . . . , k}, which almost surely die out, and considering only types {k +1, k +2, . . . }. This modified process then becomes globally (partially) extinct almost surely if and only if {Zn } becomes globally (partially) extinct almost surely. Lemma 4 demonstrates that in computing the sequences {µk } and {ak } we are also implementing a new partial extinction criterion for LHBP, that is, {µk } and {ak } take finite nonnegative values if and only if q˜ = 1. This criterion has high practical relevance: it is indeed more efficient to compute the sequences {µk } and {ak } using the recursion provided in Lemma 3 than to evaluate the convergence norm of M as the limit of the sequence of spectral radii of the north-west truncations of the mean progeny matrix M (see [29, Theorem 6.8]). In the Q sequel we shall use the shorthand notation mi→k := Ei [Yk+1 ] = 0 gi→k (1) = kj=i µj .

4

Fixed points and extinction probabilities

In this section we characterise the set S of fixed points for LHBPs defined in (2.2). This complements the work of a number of authors who have derived properties of S in a general setting, as discussed in Section 1. The main results rely on the relation between the set of fixed points S and the set S [e] = {s ∈ [0, 1]X : sk = gk (sk+1 ) ∀ k ≥ 0}, 15

which can be seen as the set of fixed points of the embedded GWPVE, and which, like S, is one-dimensional. The next two lemmas establish the relationship between S and S [e] , dealing separately with the irreducible and reducible cases. As most of the results in this section somehow rely on Lemma 1, we shall assume that Assumption 1 generally holds here. For any vector s ∈ S, we shall write s¯(k) := (s0 , s1 , . . . , sk ) for the restriction of s to its first k + 1 entries. Lemma 5 If {Zn } is irreducible then S = S [e] ∪ {1}. Proof: Suppose s ∈ S and s 6= 1. By the irreducibility of {Zn }, we have s < 1. For all k, n ≥ 0, s¯(k) = G(k) (¯ s(k) )= G(k,n) (¯ s(k) ). By Lemma 1 we (k,n) (k) therefore obtain gk (sk+1 ) = limn→∞ Gk (¯ s ) = sk for all k ≥ 0, which [e] implies that s ∈ S . Now suppose s ∈ S [e] , then by Lemma 2, sk = gk (sk+1 ) = Gk (g0→k (sk+1 ), g1→k (sk+1 ), . . . , gk (sk+1 ), sk+1 ) = Gk (s), therefore s ∈ S.



Assume now that {Zn } is reducible, but Mi,i+1 > 0 for all i ≥ 0. In that case, there may be some additional fixed points in S which must be of the form (¯ s(k) , 1) := (¯ s(k) , 1, 1, . . .) for some k ∈ X , where s¯(k) ∈ [0, 1)k+1 . We define the set R containing all types k such that there is no path from k + 1 to k of positive weight in the mean progeny representation graph of {Zn }, and we order its elements as R = {r1 , r2 , . . .} with ri < ri+1 . We write M(k,`) := (Mij )k 1},

(4.1)

and if R contains a largest element rmax , then R∗ = {ri ∈ R : sp(M(ri ,ri+1 ) ) > 1, 1 ≤ i ≤ M − 1, ν(M(rmax ,∞) ) > 1}. ˜ Lemma 6 If {Zn } is reducible then the set of fixed points is S = S [e] ∪S∪{1} where S˜ := {q˜(ri ) : ri ∈ R∗ } contains finitely or countably infinitely many distinct elements.

16

qi = q˜i = 1

qi = q˜i = 1 : 0 qi = q˜i

qi = q˜i < 1 : 0

qi

qi < q˜i = 1 : 0 qi < q˜i < 1 : 0

1 q˜i = 1

qi

q˜i

1

Figure 4.1: A visual representation of possible sets Si in the case where R∗ = ∅. Proof: Any fixed point of the form (¯ s(k) , 1) satisfies (¯ s(k) , 1) = G(¯ s(k) , 1), which implies that each additional fixed point is of the form q˜(ri ) for some ri ∈ R. In the case where R is unbounded, we observe that q˜(ri ) 6= q˜(ri+1 ) if and only if (ri+1 ) i+1 ) (˜ qri +1 , . . . , q˜r(ri+1 ) < 1. (4.2) The left hand side of (4.2) is the extinction probability vector of the ir˜n(ri+1 ) } restricted to types ri + 1, . . . , ri+1 , reducible branching process {Z which has mean progeny matrix M(ri ,ri+1 ) and (4.2) holds if and only if sp(M(ri ,ri+1 ) ) > 1. If R has a maximal element rmax , then q˜(rmax ) < 1 if and only if ν(M(rmax ,∞) ) > 1.  We are now in a position to completely characterise the one-dimensional projection sets Si and identify which elements of S correspond to the global and partial extinction probability vectors. Theorem 2 If S = {1} then q = q˜ = 1; otherwise q = min S

and

q˜ = sup S\(S˜ ∪ {1}).

In particular, (rj )

Si = [qi , q˜i ] ∪ {˜ qi

}rj ∈R∗ ∪ 1

for all i ≥ 0.

Theorem 2 essentially states that, in the irreducible case, S either contains 1, 2 or uncountably many elements. More specifically, it states that q is the componentwise minimal element of S which is the beginning of a continuum of elements whose supremum is reached by q. ˜ Since the set S is one-dimensional, it can be represented by its one-dimensional projection sets. An illustration is given in Figure 4.1. 17

3

3 4/3

0

3 4/3

1

3

3

4/3

2

4/3

3

4/3

4

...

Figure 4.2: The mean progeny representation graph of the reducible branching process described in Example 1. Proof of Theorem 2: We show that q = inf S [e]

and q˜ = sup S [e] ,

[e]

(4.3)

[e]

and for any i ≥ 0, Si = [qi , q˜i ], where Si = {x ∈ [0, 1] : ∃ s ∈ S [e] , such that si = x}. These results follow from the fact that gi (·) and gi−1 (·) are monotone increasing functions, and therefore so are gi→j (·) and −1 −1 gi→j−1 (·) := gj−1 ◦ · · · ◦ gi−1 (·) for j > i. Let s ∈ S [e] , then for all 0 ≤ i < k, (k−1)

qi

(k−1)

=gi→k−1 (0) ≤ si = gi→k−1 (sk ) ≤ gi→k−1 (1) = q˜i

.

Taking the limit as k → ∞ we obtain qi ≤ si ≤ q˜i for all i ≥ 0, which shows (4.3). Now suppose qi ≤ si ≤ q˜i . For any j < i, define sj := gj→i−1 (si ); then qj = gj→i−1 (qi ) ≤ sj ≤ gj→i−1 (˜ qi ) = q˜j . −1 Similarly, for any j > i, define sj := gi→j−1 (si ); then −1 −1 qj = gi→j−1 (qi ) ≤ sj ≤ gi→j−1 (˜ qi ) = q˜j .

This shows that for any i ≥ 0 and for any si ∈ [qi , q˜i ], it is possible to construct a vector s belonging to S [e] .  Theorem 2 implies that in the reducible case, the number of fixed points greater than q˜ and less than 1 is therefore equivalent to the cardinality of the set R∗ , which is countable. Since each element of R∗ corresponds to a sub-process causing {Yk } to have a positive probability of explosion, we may question whether it is possible to have q < q˜ and |R∗ | = ∞

(4.4)

simultaneously. In Example 1 we demonstrate that this is indeed possible. Example 1: Suppose {Zn } has the progeny generating function given by    1 3i+2 1 2 2 1 Gi (s) = s + 1 − i+1 s + , i ≥ 0, 3i+1 i 3 3 i+1 3 18

q0

S0 : 0

q˜0

1

Figure 4.3: A visual representation of the sets S0 in Example 1. whose corresponding mean progeny representation graph is illustrated in Figure 4.2. The proof of the next proposition is given in the Appendix. Proposition 1 For Example 1 (4.4) holds. For this example, there are therefore countably infinitely many fixed points larger than q˜ and uncountably many fixed points smaller than q. ˜ Note that by [19] the fixed points larger than q˜ form a decreasing sequence ˜ In Figure 4.3 we give an illustration of the projection set converging to q. S0 computed numerically using the methods described in [19]. Now that we have thoroughly explored the properties of the onedimensional projection sets Si we move on to deriving results for the infinitedimensional set S. We begin by providing a sufficient condition for S to contain at most a single element, corresponding to q, whose entries do not converge to 1, or more precisely, for lim si = 1,

i→∞

for all s ∈ S\{q}.

(4.5)

In a more general setting, sufficient conditions for (4.5) can be found in Moyal [26, Lemmas 3.3 and 3.4], the most notable being inf qi > 0. The same author also conjectures a more general condition, that is, X (1) (1) where pi = pi (v). sup pi < 1, i

v:|v|=1

In the case of LHBPs we now provide an even stronger result. Theorem 3 If

∞ X

(1)

(1 − pi ) = ∞

(4.6)

i=0

then (4.5) holds. The proof of Theorem 3 requires the following lemma. We state it separately because, for LHBPs, it generalises the conditions of [10, Theorem 1] and goes part way to addressing Remark 1 of the same paper. 19

Lemma 7 If (4.6) holds then P(Yk → 0) + P(Yk → ∞) = 1, which we refer to as the dichotomy property. Proof: We apply [20, Theorem 1] (which follows from [11, Theorem 3]) which states that {Yk } satisfies the dichotomy property if and only if ∞ X (1 − gk0 (0)) = ∞. k=0

While this result was proven in the setting of non-defective generating function {gk (·)}, it can be easily seen that it still holds in the defective case. We assume (4.6) and suppose instead that ∞ X

(1 −

gk0 (0))

0. (4.7)

k=0

Recall that each individual in {Yk } corresponds to an individual in {Zn }. Note that if the corresponding individual in {Zn } has no offspring then neither does the individual in {Yk }, whereas if the corresponding individual in {Zn } has two or more offspring then the individual in {Yk } must have at least two offspring with probability greater than or equal to c2 . Thus, for all k ≥ 0, (1) 1 − gk0 (0) ≥ c2 (1 − pk ), which contradicts (4.6).



Proof of Theorem 3: By Lemma 5, for any s ∈ S we have     (k) Yk Yk s0 = E0 sk 1{Yk < ∞} = q0 + E0 sk ; 0 < Yk < ∞ , for all k ≥ 0. By Lemma 7, {Yk } satisfies the dichotomy property. Thus, if lim inf k sk < 1, then by dominated convergence h  i (k) s0 = lim inf q0 + E0 sYk k ; Yk > 0 = q0 , k

which implies s = q.



Now that we have general sufficient conditions for 1 − si → 0, we investigate properties of this convergence. Our first result shows that when q < q˜ the decay rates of 1−qi and 1− q˜i are different than that of any other element in S. Lemma 8 Under (4.6), for any s ∈ S such that q < s < q, ˜ we have 1 − qi =∞ k→∞ 1 − si lim

and 20

1 − q˜i = 0. k→∞ 1 − si lim

Proof: Note that if {an }n≥0 and {bn }n≥0 are such that an ∈ (0, 1) and bn → ∞ then lim sup abnn = exp{− lim inf bn (1 − an )}

(4.8)

lim inf abnn = exp{− lim sup bn (1 − an )}.

(4.9)

n

n

n

n

For any s ∈ S such that q ≤ s ≤ q, ˜ we have     (k) (k) s0 = q0 + E0 sYk k ; Yk > 0 = q˜0 + E0 sYk k − 1 ; Yk < ∞ . Therefore, since sYk k is uniformly bounded, when q < q˜ (that is, P0 (Yk → ∞, Yk < ∞ ∀ k) > 0), s0 = q0 s0 = q˜0

⇔ ⇔

Yk (1 − sk ) → ∞ w.p. 1 on {Yk → ∞, Yk < ∞ ∀ k} Yk (1 − sk ) → 0 w.p. 1 on {Yk → ∞, Yk < ∞ ∀ k},

leading to the result.



We now give a physical interpretation to the fixed points s such that q < s < q. ˜ The next theorem demonstrates that the rate at which 1 − si decays is closely linked to the asymptotic growth of {Yk }. In this context, we define a growth rate to be a sequence of real numbers, {Ck }k≥0 such that Yk = W ({Ck }) exists a.s., k→∞ Ck lim

where W ({Ck }) is a non-negative, potentially defective, random variable with P(0 < W ({Ck }) < ∞) > 0. We let
gW ({Ck }) (z) = E0 z W ({Ck }) 1 {W ({Ck }) < ∞} . Growth rates of non-defective GWPVEs (q˜ = 1) have been studied by a number of authors. Although it is natural to assume that {m0→k } is a growth rate, it may not always be the case. Sufficient conditions for {m0→k } to be a growth rate are given in [20] and conditions for it to be the only distinct growth rate are discussed in [13, 12]. Examples of GWPVEs with multiple growth rates can be found in [14, 23]. We now show how this feature transfers over to the decay rates of (1 − si ). Theorem 4 Under (4.6), for s ∈ S such that q < s < q˜ and gW ({Ck }) (0) < s0 < gW ({Ck }) (1) for some growth rate {Ck }, we have s0 = gW ({Ck }) (e−c ),

where

lim (1 − sk )Ck = c ∈ (0, ∞).

k→∞

21

Proof: By Lemma 5, if s ∈ S then       Yk /Ck Yk /Ck Ck Ck ; 0 < W ({Ck }) < ∞ ; W ({Ck }) = 0 + E0 sk s0 = E0 sk    Yk /Ck Ck ; W ({Ck }) = ∞ . + E0 sk Since gW ({Ck }) (0) = P0 (W ({Ck }) = 0) < s0 < gW ({Ck }) (1) = P0 (W ({Ck }) < ∞) , we therefore have 0 < lim inf (sk )Ck ≤ lim sup(sk )Ck < 1. k

(4.10)

k

Thus, 
Y /C  s0 = lim sup E0 (sk )Ck k k k→∞  
Ck Yk /Ck = E0 lim sup (sk )

(4.11)

k→∞

 = E0

lim sup(sk )Ck

W ({Ck }) ! ,

(4.12)

k→∞

where (4.11) follows from the dominated convergence theorem, and (4.12) requires (4.10). Repeating the steps with lim inf, and using (4.8) and (4.9), we observe that limk→∞ (sk )Ck = e−c , where c = limk→∞ (1 − sk )Ck , hence
s0 = E0 e−cW ({Ck }) = gW ({Ck }) (e−c ).  Through Lemma 8 and Theorem 4, we create a relationship between the set of fixed-points of {Zn } and the growth rates of {Yk }. Indeed the embedded GWPVE informs us that the set of fixed points S of the LHBP ˜ with their own decay rate, different contains boundary elements, q and q, from that of the intermediate elements which may share one or several decay rates corresponding to the growth rates of {Yk }. On the other hand, the fixed points s such that q < s < q˜ provide the growth rates and corresponding generating functions gW ({Ck }) (z) of the embedded GWPVE. Note that, while the probabilistic interpretations of the vectors q and q˜ are in terms of the extinction of the original LHBP, the physical meaning that we gave to the intermediate fixed-points is in terms of the embedded GWPVE rather than the original process. It then remains to relate the growth of {Yk } back to that of {Zn }, which we do not do here. 22

5

Global survival and extinction

While there are already a number of well established partial extinction criteria, determining a global extinction criterion when q˜ = 1 remains an open question. We shall highlight the difficulty of this problem in Example 2 by providing a class of branching processes for which there exists no global extinction criterion based purely on the mean progeny matrix. In this section we make some progress by constructing a necessary and sufficient condition for global extinction which is valid under some second moment conditions. As demonstrated by Lemma 4, if q˜ = 1 then the first and second moments {µk } and {ak } of the progeny distributions in the GWPVE, computed recursively using Lemma 3, are well defined, and we are in a position to apply established results from the literature on GWPVEs. We choose to apply the following results corresponding to [1, Theorems 1 and 2], which to our knowledge are the most general extinction criteria for GWPVEs. Theorem 5 (Agresti (1975)) For any 1 ≤ i < k, " 1−

1 mi→(k−1)

k−1 1 X gj00 (0) + 2 j=i µj mi→j

#−1

" (k)

≤ qi

≤ 1−

1 mi→(k−1)

+

Theorem 6 (Agresti (1975)) Suppose supk≥0 (ak /µk ) inf k≥n0 (gk00 (0)/µk ) > 0 for some finite n0 , then q=1

⇔

∞ X

1

j=0

m0→j

k−1 X j=i


0 for instance by using the fact that gk00 (0) ≥ 2pk (2ek+1 ). The extinction criterion provided in Theorem 6 translates into a global extinction criterion for LHBP; it should be compared to the extinction criterion for finite-type processes (see for instance [18, Theorem 7.1]) q=1

⇔

sp(M ) ≤ 1,

where sp(M ) is the spectral radius of the finite mean progeny matrix M . For irreducible infinite-type processes, the quantities corresponding to sp(M ) p n are the convergence norm, ν(M ) := lim inf n Ei (Znj ), and the exponential 23

p growth rate of the mean total population size, ξ(M ) := lim inf n n Ei |Zn |. In contrast to the finite-type case where ν(M ) = ξ(M ), we may now have ν(M ) < ξ(M ). While it has been established that q˜ = 1

⇔

µk ∈ [0, ∞), ∀ k ≥ 0

⇔

ν(M ) ≤ 1

(see Lemma 4 and [15]), attempts to use ξ(M ) to construct a global extinction criterion have proven to be more challenging [5, 32]. Given the important role of ξ(M ), we now link it to the global extinction criterion in Theorem 6 by proving the following result. Theorem 7 Assume ν(M ) ≤ 1. If ξ(M ) > 1, then √ lim sup n m0→n ≥ ξ(M ),

(5.1)

n

and and if ξ(M ) < 1, then lim inf

√ n

n

m0→n ≤ lim sup (E0 |Zn |)1/n .

(5.2)

n

√ A consequence of Theorem 7 is that if both limn n m0→n and limn (E0 |Zn |)1/n exist (which will be the case in our illustrative examples), then by the root test, ξ(M ) > 1 ⇒

∞ X

1

j=0

m0→j

< ∞ and ξ(M ) < 1 ⇒

∞ X

1

j=0

m0→j

= ∞.

Thus, if ξ(M ) 6= 1 then the necessary and sufficient condition for almost sure global extinction in Theorem 6 may be replaced by ξ(M ) < 1. One important contribution of Theorem 6, which was motivated in [6], is to provide an extinction criterion applicable even when ξ(M ) = 1. In this sense, it is a more sensitive global extinction criterion capable of classifying processes that exhibit non-exponential, or algebraic, growth, as we shall demonstrate in Example 3. P Proof of Theorem 7: We have m0→(n−1) = nk=0 E0 (Zn,k ) mk→(n−1) , where mn→(n−1) := 1, which gives E0 |Zn | inf mk→(n−1) ≤ m0→(n−1) ≤ E0 |Zn | sup mk→(n−1) . 0≤k≤n

(5.3)

0≤k≤n

Now suppose ξ(M ) > 1. In order to prove (5.1) we need to show that @ n0 < ∞ s.t.

inf mk→(n−1) < 1 ∀ n > n0 .

0≤k≤n

24

(5.4)

Indeed, in that case, lim supn inf 0≤k≤n mk→(n−1) = 1, and thus by (5.3), lim sup

√ n

m0→(n−1)

n

 1/n ≥ lim sup E0 |Zn | inf mk→(n−1) 0≤k≤n

n

1/n

≥ lim inf (E0 |Zn |) n

1/n

 lim sup n

inf mk→(n−1)

0≤k≤n

= ξ(M ).  To show (5.4) assume there exists n0 := sup n : inf 0≤k≤n mk→(n−1) = 1 < ∞, and observe that for any n ≥ 0 the following recursion holds    inf mk→(n−1) = min inf mk→(n−2) µn−1 , µn−1 , 1 . 0≤k≤n

0≤k≤n−1

This implies that for all n > n0 ,  inf mk→(n−1) =

0≤k≤n

 inf

0≤k≤n−1

mk→(n−2) µn−1 ,

and inf 0≤k≤n mk→(n−1) = mn0 →(n−1) , which gives −1

 m0→n

inf mk→n

0≤k≤n

= m0→(n0 −1) ,

for all n > n0 .

By Equation (5.3) we then have E0 |Zn | ≤ m0→(n0 −1) , for all n > n0 , which contradicts the fact that ξ(M ) > 1 and shows (5.4). When ξ(M ) < 1 a similar argument can be used to obtain (5.2).  The next example highlights the fact that the mean progeny matrix M is not sufficient to determine whether q < 1 or q = 1, see Proposition 2. This fact was previously observed in [32, Example 4.4] which contains an inductive proof reliant on explicit expressions of the progeny generating vector. Through this example we provide a streamlined proof which applies to a significantly broader class of branching processes. Example 2: Consider a LHBP {Zn }, with mean progeny matrix   b c 0 0 0 ... a b c 0 0      0 M = 0 a b c , 0 0 a b  c   .. ... ... ... . 25

(5.5)

and progeny generating vector G(s). We assume that a, c > 0 and that Ak,ij ≤ B for all k, i, j ≥ 0, for some positive constant B. We now consider a [u] modification of {Zn }, which we denote by {Zn } for some parameter u ≥ 1, whose progeny generating vector, G[u] (s), is given by   1 1 [u] dui e Gi (si−1 , si , si+1 ) = i Gi (si−1 , si , si+1 )+ 1 − i Gi (si−1 , si , 1), i ≥ 0. du e du e (5.6) This modification decreases the probability that a type-i individual has any type-(i + 1) offspring by a factor of 1/dui e but when the type-i individual does have type-(i + 1) offspring, their number is increased by a factor of dui e, which causes the mean progeny matrix to remain unchanged. Before pro[u] viding results on the extinction of {Zn } we require the following lemma on branching processes with the tridiagonal mean progeny matrix (5.5), whose proof can be found in the Appendix. Lemma 9 Suppose {Zn } has a mean progeny matrix given by (5.5), then q˜ = 1 if and only if b 0 If µ < 1 then q = 1, whereas if µ ≥ 1 then u>µ ⇒ q=1

and

u < µ ⇒ q < 1,

where µ is given in (5.8). An important sub-case of Example 2 is u = 1, the set of unmodified branching processes. In this case, when combined with Lemma 7, Proposition 2 yields 26

3 2 (1

2(1 − γ)

0

1

1

2γ

4 3 (1

− γ)

2

− γ)

...

3

3 2γ

4 3γ

5 4γ

Figure 5.1: The mean progeny representation graph corresponding to Example 3. Corollary 1 If u = 1 and (4.6) holds then q = 1 if and only if µ ≤ 1. Note that the conditions of Theorem 6 do not hold in Example 2, however in the proof of Proposition 2 we demonstrate how Theorem 5 may still be applied in such cases. In the next example, we emphasize the sensitivity of Theorem 6 by applying it to a LHBP exhibiting algebraic growth. Example 3: Let {Zn } be such that M01 = 1, and for i ≥ 0, i+1 i+1 and Mi,i+1 = (1 − γ) , 0 ≤ γ ≤ 1, i i with all remaining entries being 0. The mean progeny representation graph corresponding to this process is illustrated in Figure 5.1. We further assume that inf k pk (2ek+1 ) > 0 and there exists B < ∞ such that Ak,ij ≤ B for all i, j, k ≥ 0. It is not difficult to show that this branching process grows algebraically, that is, ξ(M ) = 1, if and only if ν(M ) ≤ 1 (or equivalently q˜ = 1), which is the case for a range of values of γ, as we shall see. Mi,i−1 = γ

Proposition 3 For the set of branching processes described in Example 3, q=1

if and only if

γ = 0.

In Figure 5.2 we let ( 1 4 s + 1, Gk (s) = 4k+11 4 (γsk−1 + (1 − γ)sk+1 )4 + 4k (8000)

(8000)

3k−1 , 4k

k=0 k ≥ 1,

and we plot q0 ≈ q0 and q˜0 ≈ q˜0 for γ ∈ [0, 1]. Although we have (8000) proven that q0 = 1 when γ = 0 we observe that q0 ≈ 0.95 for this value of γ. This is due to the fact that, by Theorem 5, !−1 k X 1 (k) q0 − q0 ∼ ∼ log−1 (k), ` `=0 27

(8000)

Figure 5.2: The extinction probabilities q0 γ ∈ [0, 1].

(8000)

(solid) and q˜0

(dashed) for

so the convergence of q (k) to q = 1 is slow when γ = 0. For GWPVEs with q < 1, scant attention has been paid to this convergence rate in the literature, so for this example not much can be said when γ > 0. Using Lemmas 3 and 4 we numerically determine that q˜ = 1 if and only if γ ≤ γ ∗ where γ ∗ = max{γ : 0 < µk < ∞ ∀k ≥ 0} ≈ 0.1625. (5.9) Note that in this particular example, due to properties of the sequence {µk }, a sufficient condition for q˜ = 1 is the existence of some k such that µk < µk−1 (see the proof of Proposition 3), therefore γ ∗ can be evaluated particularly (8000) ≤ q0 ≤ q˜0 ≤ q˜(8000) , by visual inspection, the curves efficiently. Given q0 of partial and global extinction seem to merge from some value of γ, however the cut-off is not clear and further analysis is required to pinpoint the precise value. We are also interested in understanding whether this value depends only on the mean progeny matrix or whether other offspring distributions lead to different values. We address these questions in the next section.

6

Strong local survival

Each irreducible infinite-type branching processes falls into one of the four categories q = q˜ = 1, q < q˜ = 1, q < q˜ < 1 or q = q˜ < 1. The results in 28

the previous section deal with the classification of LHBPs with q˜ = 1. The present section builds on these results to provide a method for classifying LHBPs with q˜ < 1, that is, determining whether the process exhibits strong local survival (q = q˜ < 1), or non-strong local survival (q < q˜ < 1). Other attempts at distinguishing between these two cases can be found, for instance, in [7, 8] and [24]. We proceed by defining, for any fixed value of k ≥ 0, a partition of the mean progeny matrix into four components,  (k)  ˜ ¯ 12 M M M= ¯ ˜ , M21 (k) M ˜ (k) is of dimension k ×k and the other three submatrices are infinite. where M We then construct a LHBP branching processes on (Ω, F, P), denoted as ˜ and global and partial extinction ˜n }, with mean progeny matrix (k) M {(k) Z (k) (k) ˜n } are probability vectors q and q, ˜ respectively. Sample paths of {(k) Z constructed from those of {Zn }, by immediately killing all offspring of type i < k, and relabelling the types so that type i ≥ k + 1 becomes i − k − 1. We ˜n } to derive a criterion for strong local survival. now use {(k) Z Theorem 8 Suppose q˜ > 0 and there exists k ∈ N such that     ˜ (k) > 1 and ν (k) M ˜ ≤ 1, sp M

(6.1)

¯ 21 contains a finite number of strictly positive entries. Then there is and M ˜n } becomes globally extinct, strong local survival in {Zn } if and only if {(k) Z that is, q = q˜ < 1 if and only if (k) q = (k) q˜ = 1. (6.2) Proof: We use [8, Theorem 4.2] which we restate using our notation: let (G) (G∗ ) {Zn } and {Zn } be two branching processes on the countable type set X with respective probability generating functions G(·) and G∗ (·), and global extinction probability vectors q and q ∗ . Let A ⊆ X be a nonempty subset of types and denote by q(A) and q ∗ (A) the respective vectors of probability (G) (G∗ ) of local extinction in A. If {Zn } and {Zn } differ on A only, that is, if Gi (s) = G∗i (s) for all i ∈ X \ A and Gi (s) 6= G∗i (s) for all i ∈ A, then q = q(A)

⇔

q ∗ = q ∗ (A). (G)

(6.3) (G∗ )

We apply this result with A = {1, 2, . . . , k}, {Zn } = {Zn }, and {Zn } being such that G∗i (s) = 1 for all i ∈ A, that is, all types in A are sterile. We need to show that (6.3) is equivalent to (6.2). 29

We first observe that q(A) = q˜ since, by (6.1), in {Zn }, types in X \A are only able to survive ∗through the presence of types in A. Next, since types in (G ) A are sterile in {Zn }, q ∗ = q ∗ (A) if and only if qi∗ = qi∗ (A) for all i ≥ k +1. ∗ ∗ It is clear that (qk+1 , qk+2 , . . .) = (k) q by construction. It remains to show that ∗ ∗ ˜n : ϕ0 = `} and the (qk+1 (A), qk+2 (A), . . .) = 1. We couple the process {(k) Z (G∗ ) (G∗ ) process {Zn : ϕ0 = k + `}, ` ≥ 1. Let k + `¯ be the largest type in {Zn } ˜ ) ≤ 1, with probability able to generate offspring in A. Then since ν((k) M (k) ˜ one there exists a generation N such that Zn,1 + . . . + (k) Z˜n,`¯ = 0 for all (G∗ ) (G∗ ) n ≥ N . This implies that with probability one, Zn,k+1 + . . . + Zn,k+`¯ = 0 for ∗ ∗ (A), . . .) = 1.  (A), qk+2 all n ≥ N , which shows (qk+1 Remark: The major condition in Theorem 8 is the existence of some k such ˜ (k) ) > 1 and ν((k) M ˜ ) ≤ 1. If ν((k) M ˜ ) > 1 for all k then we may that sp(M expect q = q˜ < 1, however Example 1 acts as a counter to this claim. To ˜ ) > 1 it may still be possible to apply one of prove q = q˜ < 1 when ν((k) M the sufficient conditions provided in [15]. We are now in a position to answer the questions posed at the end of the previous section. The next proposition is proven in the Appendix. Proposition 4 For the set of branching processes described in Example 3, γ=0 γ ∈ (0, γ ∗ ] γ ∈ (γ ∗ , 1/2) γ ∈ (1/2, 1]

⇒ ⇒ ⇒ ⇒

q q q q

= q˜ = 1 < q˜ = 1 < q˜ < 1 = q˜ < 1,

where γ ∗ is given in (5.9). Proposition 4 demonstrates that the curves for partial and global extinction represented in Figure 5.2 merge at γ = 1/2 and that this value is independent of the particular offspring distributions. At the critical value γ = 1/2 there exists no k satisfying (6.1) causing this case to remain untreated.

7

Conclusion

Besides thoroughly exploring the set of fixed-points for LHBPs, we have introduced a method of classifying LHBPs into one of the categories q = q˜ = 1, q < q˜ = 1, q < q˜ < 1 or q = q˜ < 1. Through Examples 2 and 3 we 30

showed that our results can be used to rigorously determine which category the process falls in; however, in practical situations where rigorous proofs may not be possible, our results can still be applied computationally as a first step in classifying the process. The inherent assumption in LHBPs is the constraint that individuals of type i cannot give birth to offspring whose type is larger than i+m for m = 1. The approach of embedding a GWPVE in the original LHBP can be extended to the case where m takes any finite integer value. The resulting embedded GWPVE then becomes multitype with m types. Results of Section 3 then naturally generalise, but those of Section 4 rely on the characterisation of the m-dimensional projection sets of S, which is more difficult in this case. The global extinction criterion discussed in Section 5 would now build upon extinction criteria for multitype GWPVE, which are less developed in the literature. These questions are the topic of ongoing research.

Appendix: Proofs related to the examples Proof of Proposition 1: From Figure 4.2 we observe that R∗ = X , hence there are countably infinitely many fixed points greater that q. ˜ Now if q˜0 > q0 then there are also uncountably many fixed points less than q. ˜ A sufficient condition for q˜0 > q0 is that there is a positive probability of global survival without any type-i individual ever having a type-i offspring. More specifically, if we let Ai = {ω ∈ Ω : Zi,i−1 = 0},

and B = {ω ∈ Ω : lim |Zn | > 0}, n→∞

then we have ∞ q˜0 −q0 ≥ P0 (∩∞ i=1 Ai ∩ B) = P0 (∩i=1 Ai | B) P0 (B) ≥

∞  Y

1−

i=0

1 3i+1

2i !

q∗,

where q ∗ = 1/2 is the extinction probability of the single-type Galton-Watson process with progeny generating function f (s) = (2/3)s2 + (1/3). Thus, if Q∞ 1 2i ˜0 > q0 . Now, i=0 (1 − 3i+1 ) > 0 then q ∞  Y i=0

1−

2i

1 3i+1

where c = 2/3, and

Q∞

i=0

=

 ∞ Y

1−

i=0 i

c(2/3) = c

P∞

1

≥

3i+1

i i=0 (2/3)

31

3i !(2/3)i

∞ Y

i

c(2/3) ,

i=0

= c3 > 0.



Proof of Lemma 9: Since (5.7) holds, b < 1 and Lemma 3 gives µ0 =

c , 1−b

and

µk =

c 1 − b − aµk−1

for all k ≥ 0.

(7.1)

Since µ1 > µ0 and µk − µk−1 =

c c − , 1 − b − aµk−1 1 − b − aµk−2

we can inductively show that the sequence {µk }k≥0 is strictly positive and increasing. Therefore, under the assumption that a > 0, q˜ = 1 implies that {µk } converges to a finite limit µ, where µ satisfies the equation ax2 − (1 − b)x + c = 0, which has real solutions p 1 − b ± (1 − b)2 − 4ac x± = , 2a since (5.7) holds. When (5.7) holds we have µ0 ≤ x− which, combined with (7.1) and the fact that c x− = , 1 − b − ax− implies µk ≤ x− for all k ≥ 0, hence µk % µ = x− .



Proof of Propostion 2: Let ∆ = (1 − b)2 − 4ac > 0. First, suppose u > µ. We have E0 (Yk ) µk (k) ≤ k−1 , 1 − q0 = E0 (Yk |Yk > 0) u where E0 (Yk ) ≤ µk follows from Lemma 9 and E0 (Yk |Yk > 0) ≥ uk−1 follows from the fact that the minimum number of type-k offspring born to a type(k − 1) parent is duk−1 e. This then implies (k)

µk = 0, k→∞ uk−1

1 − q0 = 1 − lim q0 ≤ lim k→∞

and therefore q = 1 by irreducibility. [u]

Now suppose 1 ≤ u < µ. Observe that Ak,ij = Ak,ij for all i and j with [u]

the exception of Ak,(k+1)(k+1) = duk eAk,(k+1)(k+1) + c(duk e − 1). By Lemma 3

32

we then have aµ2k ak−1 + duk eAk,(k+1)(k+1) + O(1) 1 − b − aµk−1 2 aµ ak−1 + Bduk e + O(1) ≤ 1 − b − aµ

ak =

= ak−1 µ = ak−1 µ

a 1−b−∆ 2a

1/2

1 − b − a 1−b−∆ 2a

1/2

+ B ∗ uk + O(1)

1 − b − ∆1/2 + B ∗ uk + O(1), 1 − b + ∆1/2

for all k ≥ 0 and some B ∗ < ∞, which implies   k ! 1 − b − ∆1/2 ak = O . max u, µ 1 − b + ∆1/2 n o 1/2 By assumption, ∆ > 0 and u < µ which gives max u, µ 1−b−∆ < µ. 1−b+∆1/2 Using the fact that µk % µ, through application of the root test we then obtain ∞ X ak < ∞, µ k m0→k k=0 which, by the right-hand-side of Theorem 5, gives q0 < 1.



Proof of Corollary 1: It remains to show q = 1 when µ = 1. Lemma 7Qimplies P0 (Yk → 0) + P0 (Yk → ∞) = 1 and Lemma 9 implies E0 (Yk ) = k−1  i=0 µi ≤ 1 for all k, leading to P0 (Yk → ∞) = 0 and the result. Proof of Proposition 3: If γ = 0 then µ0 = M01 = 1, and for k ≥ 1, . This gives m0→k = k + 1, and therefore µk = k+1 k ∞ X k=0

1 m0→k

=

∞ X 1 k=1

k

= ∞,

which by Theorem 6, whose conditions are easily shown to be satisfied, implies q = 1. Now suppose γ > 0. By Lemma 3, for k ≥ 1, µk = fk (µk−1 ) :=

k+1 (1 − γ) k . 1 − k+1 γµ k−1 k

(7.2)

If there exists k such that µk > 1/γ, then by Lemma 4 we have q ≤ q˜ < 1. Assume from now on that q˜ = 1, which implies that 0 ≤ µk ≤ 1/γ < ∞ 33

for all k, and γ < 1/2. Since µ0 = M01 = 1, using Equation (7.2) we can inductively show that µk ≥ 1 for all k ≥ 0. We then have, for any k ≥ 1, k+1 (1 − γ) k 1 − k+1 γ k

1 . (7.3) k(1 − γ) P The Raabe-Duhamel test for convergence ensures that ∞ k=0 (1/m0→k ) < ∞, since for k ≥ 1,   (1/m0→(k−1) ) 1 − 1 = k(µk − 1) ≥ > 1. k (1/m0→k ) 1−γ µk ≥

≥ 1+

To complete the proof we must then verify that the conditions of Theorem 6 are satisfied. First, we have inf gk00 (0)/µk ≥ inf 2pk (2ek+1 )γ > 0. Next, by Lemma 3, for all k ≥ 1 and for some constant K < ∞ independent of k, Ak,(k−1)(k−1) µ2k−1 µ2k + 2Ak,(k−1)(k+1) µk−1 µk + Ak,(k+1)(k+1) ak−1 Mk,k−1 µ2k + 1 − Mk,k−1 µk−1 1 − Mk,k−1 µk−1 k+1 2 ak−1 γ k µk + K. ≤ γµk−1 1 − k+1 k

ak =

since, as q˜ = 1, the denominator is uniformly bounded away from 0 and we assumed there exists B < ∞ such that Akij ≤ B for all i, j, k ≥ 0. If µk → 1, then for large k,   γ ak ≤ + o(1) ak−1 + K. 1−γ Since γ < 1/2, we have γ/(1 − γ) < 1, which means that {ak } is a uniformly bounded sequence. Combining this with the fact that µk ≥ 1 for all k implies sup ak /µk < ∞ and Theorem 6 completes the proof. It remains to prove that µk → 1. Observe that (7.2) implies that if µk < µk−1 for some k, then µk+1 < µk , and thus µ = limk→∞ µk exists since 1≤µk ≤ 1/γ for all k. Taking k → ∞ in (7.2) we obtain that µ satisfies µ=

1−γ := f (µ), 1 − γµ

which means µ is either 1 or (1 − γ)/γ > 1. The function f (x) is convex, thus f (x) > x for all x > (1 − γ)/γ; in addition, by (7.2), µk+1 > f (µk ) for all k ≥ 0. These imply that if µk > (1 − γ)/γ for some k, then µk+` becomes negative for some ` > 1, which is a contradiction. So the sequence {µk } lives in the open interval (1, (1 − γ)/γ). Let   s 2 1  k k (k) v± = ± − 4γ(1 − γ) 2γ k + 1 k+1 34

be the solutions of the equation x = fk (x). By the convexity of fk (x) for (K+1) (K+1) all k, if there exists K ≥ 1 such that v− < µK < v+ then {µk }k≥K is a decreasing sequence which converges to 1. Suppose µ = (1 − γ)/γ. (K+1) Then µK ≥ v+ for some K. We can then construct a LHBP, {Zn∗ }, stochastically smaller than {Zn } by selecting a sufficiently large type K and independently killing each type-(K + 1) child born to a type-K parent with ∗(K+1) ∗(K+1) a probability carefully chosen to ensure v− < µ∗K < v+ . For this modified process we have µ∗ = 1, and repeating previous arguments, we obtain q < q ∗ < 1.  Proof of Proposition 4: Given Proposition 3 and Lemmas 3 and 4, it remains to show that q < q˜ for γ ∈ (γ ∗ , 1/2) and q = q˜ for γ ∈ (1/2, 1]. ˜ (k) ) > 1 ∀k ≥ K1 . Note that, in either case, since q˜ < 1, ∃K1 such that sp(M In addition, ∀x > 1, ∃K(x) s.t. Mk,k+1 < x(1 − γ) and Mk,k−1 < xγ ∀k ≥ K(x). Since γ 6= 1/2, we may choose x¯ > 1 small enough so that 1−4¯ x2 (1−γ)γ > 0. ˜ ) < 1 for all k ≥ K ¯ := K(¯ x), and By Lemma 9, this implies that ν((k) M p x2 (1 − γ)γ 1 − 1 − 4¯ ¯ (K) µk ≤ 2¯ xγ ¯

for all k ≥ 0, where {(K) µk }k≥0 is computed using

¯ (K)

˜. M

Assume first that γ ∈ (1/2, 1]. This implies that p 1 − 1 − 4(1 − γ)γ < 1, 2γ ¯ so we may choose x∗ ≤ x¯ small enough, corresponding to K ∗ := K(x∗ ) ≥ K, (K ∗ ) so that µk < 1 − ε for all k ≥ 0 and some ε > 0. Hence there exists K = max{K1 , K ∗ } < ∞ satisfying the conditions of Theorem 8 with (K) q = (K) q˜ = 1. ¯ there exists k1 ≥ 0 such that Now suppose γ ∈ (c, 1/2). If for any K > K (K) µk1 ≥ 1, then by the recursion (7.2) we have (K)

µk ≥ 1 +

1 , k(1 − γ)

∀ k > k1 ,

and the result is derived by repeating the steps that follow Equation (7.3) in ¯ such that the proof of Theorem 3. Suppose instead that there exists K > K (K) µk < 1 for all k ≥ 0. Then by Equation (7.2) we have (K)

µk−1 < 1 − 35

1 , γ(k + 1)

which implies

∞ Y

(K)

µi = 0.

(7.4)

i=0

To demonstrate that this leads to a contradiction we compare M to a matrix ∗ = M ∗ with strictly smaller entries than M . The matrix M ∗ is such that M0,1 1 − γ and ∗ ∗ Mk,k−1 = γ and Mk,k+1 = 1 − γ, ∀ k ≥ 1, Q∞ ∗ with all other entries 0. The value of i=0 µi , with {µ∗i }i≥0 computed Q∞ ∗us∗ ing M , then has a probabilistic interpretation. In particular, i=0 µi is the probability that a simple random walk on the integers, with transition probabilities p+ = 1 − γ > p− = γ, whose initial value is 0, never hits −1. When 1 − γ > γ it is well known that this value is non-zero. By the (K) fact M > M ∗ we then have (K) µi > µ∗i for all i ≥ 0 which implies Q Q∞ that ∞ (K)  µi > i=0 µ∗i > 0, contradicting (7.4). i=0

Acknowledgements The authors would like to acknowledge the support of the Australian Research Council (ARC) through the Centre of Excellence for the Mathematical and Statistical Frontiers (ACEMS). Sophie Hautphenne would further like to thank the ARC for support through Discovery Early Career Researcher Award DE150101044.

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