# Distance-dependent chase-escape on trees

Sarai Hernandez-Torres<EMAIL_ADDRESS>,  Matthew Junge
<EMAIL_ADDRESS>,  Naina Ray  and  Nidhi Ray

###### Abstract.

We give a necessary and sufficient condition for species coexistence in a
parasite-host growth process on infinite $d$-ary trees. The novelty of this
work is that the spreading and death rates for hosts depend on the distance to
the nearest parasite.

Hernandez-Torres was supported by ISF grant 1692/17. Junge was partially
supported by NSF Grant #2115936.

##  1\. Introduction

Chase-escape is an interacting particle system inspired by certain parasite-
host dynamics [RKW95, Kee95, Met00]. The same dynamics have been reinterpreted
in a variety of applications: predator-prey systems, rumor scotching,
infection spread, and malware repair in a device network [Bor14, Bor08,
dALRR15, HJCW19, BHJR22]. Red (host) particles occupy and spread to adjacent
sites of a graph according to exponential clocks while facing the threat of
blue (parasite) particles. When a host is infected by a parasite, the host
perishes. The site where this occurs is occupied by the parasite for all time
thereafter.

The authors of [BCE+21] introduced a variant called _chase-escape with death_
in which red particles die at a given rate, independently of the spread of
parasites. They studied this process on infinite $d$-ary trees and
characterized the phase behavior. Much of the analysis in [BCE+21] relied on a
novel connection to weighted Catalan numbers. We deepen this connection by
generalizing to the setting in which the host spreading and death rates depend
on the distance to the nearest parasite.

We begin by defining _generalized chase-escape_ on a $d$-ary tree. Fix $d\geq
1$ and let $\mathbb{T}_{d}$ denote the infinite, rooted $d$-ary tree in which
every vertex has $d$ children. Denote the root by $\mathbf{0}$ and let
$\mathcal{T}_{d}$ be $\mathbb{T}_{d}$ augmented with an additional vertex
$\mathfrak{b}$ attached to $\mathbf{0}$. Vertices of $\mathcal{T}_{d}$ are in
one of four states $\\{w,b,r,\dagger\\}$. State $w$ is a “white” unoccupied
site, state $b$ is a “blue” site occupied by a parasite, state $r$ is a “red”
site occupied by a host, and state $\dagger$ is a site containing a “dead”
host. Given vertices $u,v\in\mathcal{T}_{d}$, define $|u-v|$ to be the number
of edges on the unique shortest length path connecting $u$ and $v$. We write
$\sigma(u)$ to be set of vertices on the shortest path connecting
$\mathfrak{b}$ to $u$.

The dynamics of _distance-dependent chase-escape_ are as follows. Vertex
$\mathfrak{b}$ is initially in state $b$. The root is initially in state $r$.
All other vertices begin in state $w$. Adjacent sites in states $(b,r)$
transition to $(b,b)$ according to a rate 1 Poisson process i.e., the time for
each event is an exponential random variable with mean 1. To specify the red
spreading and death rates we take vectors

(1) $\displaystyle{\vec{\lambda}}$
$\displaystyle=(\lambda_{1},\lambda_{2},\ldots)\text{ and
}{\vec{\rho}}=(\rho_{1},\rho_{2},\ldots)$

of nonnegative real numbers. Given a vertex $u\in\mathcal{T}$, define the
distance to the nearest vertex in state $b$ as

(2) $\displaystyle\ell(u)$ $\displaystyle=\min\\{|u-v|\colon
v\in\sigma(u)\text{ and $v$ is in state $b$}\\}.$

Note that $\ell$ depends on the current configuration of the tree. Adjacent
vertices $(u,v)$ with $u$ in state $r$ and $v$ in state $w$ have $v$
transition to state $r$ according to a Poisson process with intensity
$\lambda_{\ell(u)}$ i.e., after exponentially distributed times with mean
$1/\lambda_{\ell(u)}$. Meanwhile, a vertex $u$ in state $r$ transitions to
state $\dagger$ according to a Poisson process with intensity
$\rho_{\ell(u)}$. Unless the dependence is important to highlight, we will
typically write $\mathbf{P}(A)$ rather than
$\mathbf{P}_{{\vec{\lambda}},{\vec{\rho}}}(A)$ for events $A$ pertaining to
distance-dependent chase-escape.

In words, blue may only spread to red sites and does so at rate 1. Red may
only spread to white sites and does so at a rate that depends on the distance
to the nearest blue site. Similarly, the rate that a red particle dies depends
its distance to the nearest blue particle. The chase-escape with death process
from [BCE+21] is the special case of our model with $\lambda_{i}\equiv\lambda$
and $\rho_{i}\equiv\rho$ for all $i\geq 1$ with $\lambda,\rho>0$.

We are interested in the persistence of both species. Let $\mathcal{B}$ equal
the sites that are at some time colored blue. Since blue may only occupy sites
that were at some time red, we say that _coexistence_ occurs if
$\mathbf{P}(|\mathcal{B}|=\infty)>0$. We say that _expected coexistence_
occurs if the weaker condition $\mathbf{E}[|\mathcal{B}|]=\infty$ occurs. The
work [BCE+21] provided a necessary and sufficient condition for coexistence to
occur and described the behavior at the phase transition in detail. For the
sake of a simpler, less technical generalization of a main idea from [BCE+21],
we focus on expected coexistence (rather than coexistence).

$u(0)$$v(0)$$u(0)$$u(1)$$u(2)$$v(2)$$u(2)$$v(2)$$v(1)$$v(0)$ Figure 1. A Dyck
path of length $10$ with weight $u(0)^{2}v(0)^{2}u(1)v(1)u(2)^{2}v(2)^{2}.$

As observed in [BCE+21], the phase structure of chase-escape with death is
connected to weighted Catalan numbers. We generalize this connection. Let
$D_{j}=\sum_{i=1}^{j}\rho_{i}$ and define the weights

(3) $\displaystyle u(j)=\frac{\lambda_{j+1}}{1+\lambda_{j+1}+D_{j+1}}\text{
and }v(j)=\frac{1}{1+\lambda_{j+2}+D_{j+2}}.$

Given a lattice path $\gamma$ consisting of unit rise and fall steps, each
rise step from $(x,j)$ to $(x+1,j+1)$ has weight $u(j)$, while a fall step
from $(x,j+1)$ to $(x+1,j)$ has weight $v(j)$. The _weight_ $\omega(\gamma)$
of a Dyck path $\gamma$ (nonnegative lattice path starting at $(0,0)$
consisting of $k$ rise and $k$ fall steps) is the product of the rise and fall
step weights along $\gamma$. See Figure 1. The corresponding _weighted Catalan
number_ is

(4) $\displaystyle
C_{k}^{{\vec{\lambda}},{\vec{\rho}}}=\sum_{\gamma\in\Gamma_{k}}\omega(\gamma)$

where $\Gamma_{k}$ is the set of all Dyck paths of length $2k$. Denote the
generating function by

(5) $\displaystyle g(z)$ $\displaystyle=\sum_{k\geq
0}C_{k}^{{\vec{\lambda}},{\vec{\rho}}}z^{k}.$

Let $M$ be the largest value such that $|g(z)|<\infty$ for all complex numbers
$|z|<M$.

The weights at (3) make it so the $C_{k}^{{\vec{\lambda}},{\vec{\rho}}}$
correspond to the probability that a renewal, where blue is once again
adjacent to the rightmost red site, occurs in a version of distance-dependent
chase-escape on the nonnegative integers (see (9)). In the setting from
[BCE+21], it is proven that the radius of convergence of $g$ relative to the
degree of the tree determines the phase. This continues to hold in distance-
dependent chase-escape.

######  Theorem 1.

thm:main Suppose that there exist constants $c>0$ and $m\geq 1$ such that

(6)
$\displaystyle\prod_{i=3}^{\ell-2}\left(1+\frac{\lambda_{i}}{1+D_{i}}\right)\leq
c\ell^{m}\quad\text{ for all $\ell\geq 5$},$

and that for $u$ and $v$ as at (3)

(7) $\displaystyle\lim_{j\to\infty}u(j)v(j)=0.$

Then for $d\geq 2$, expected coexistence on $\mathcal{T}_{d}$ occurs if and
only if $M\leq d$ with $M$ the radius of convergence of the generating
function defined at (5).

The hypotheses (6) and (7) are consequences of generalizing the main result of
[BCE+21]. It is unclear how much (6) can be relaxed. It is fairly robust. For
example, (6) and (7) hold so long as $\lambda_{i}/D_{i}=O(i^{-\epsilon})$ for
some $\epsilon>0$. The hypothesis at (7) is essential to our argument as it
allows us to apply thm:worpitzky. Note that (6) and (7) are easily verified in
the case that ${\vec{\lambda}}$ and ${\vec{\rho}}$ are constant. Since we
focus on the weaker requirement of expected coexistence, we are able to
sidestep many technical difficulties. As a result, we give a streamlined
presentation that clarifies and builds on some of the main ideas from
[BCE+21].

In Section 2, we analyze distance-dependent chase-escape on the non-negative
integers. This lets us connect renewal events in the one-dimensional process
to weighted Catalan numbers. In lem:bigger, we prove that the probability blue
reaches beyond a given distance is comparable to the probability a renewal
occurs at that distance. Section 3 contains the proof of thm:main. lem:bigger
lets us upper bound $\mathbf{E}[|\mathcal{B}|]$ in terms of $g(d)$. This lets
us deduce that $d<M$ implies $\mathbf{E}[|\mathcal{B}|]<\infty$. When
$\mathbf{E}[|\mathcal{B}|]<\infty$, it is easy to see that $d\leq M$. To
handle the boundary case $M=d$, we employ two classical results. One is
Worpitsky’s Circle Theorem, which we apply via a continued fraction
characterization of $g(z)$. We then apply Pringsheim’s Theorem to deduce that
$\mathbf{E}[|\mathcal{B}|]=\infty$ when $M=d$.

##  2\. Distance-dependent chase-escape on the integers

We begin by defining distance-dependent chase-escape on the non-negative
integers, which is equivalent to the case $\mathcal{T}_{1}$ with
$\mathfrak{b}=0$ and $\mathbf{0}=1$. We indicate the state of the vertex $n$
at time $t$ by $s_{t}(n)\in\\{w,b,r,{\dagger}\\}$, which indicates if the
vertex is white, blue, red or dead. Initially, $s_{0}(0)=b$, $s_{0}(1)=r$ and
$s_{0}(n)=w$ for any $n>1$, and the process follows the dynamics of distance
dependent-chase escape with rates ${\vec{\lambda}}$ and ${\vec{\rho}}$. For
each time $t\geq 0$, we write $B_{t}=\sup\\{n\colon s_{t}(n)=b\\}$ and
$R_{t}=\sup\\{n\colon s_{t}(n)=r\\}$. Define the maximum integer reached by
$B_{t}$ as

(8) $\displaystyle Y$ $\displaystyle=\sup\\{B_{t}\colon t\geq 0\\}.$

As in [BCE+21], we are mainly interested in times at which the process renews.
For each vertex $k\geq 0$, call

$\mathfrak{R}_{k}=\\{B_{t}=k,\;R_{t}=k+1\text{ and
}s_{t}(n)\neq{\dagger}\text{ for all }n>k+1\\}$

a _renewal event at the vertex $k$_. At these points, the process exhibits its
initial conditions with a translation by $k$. For $t\geq 0$, define the event
$A_{t}=\\{s_{t}(k)\neq{\dagger}\text{ for all }k\\}$ that there are no killed
red sites at time $t$.

We follow the evolution of distance-dependent chase-escape with a Markov
chain. Let $S_{t}=R_{t}-B_{t}$ be the distance between the rightmost blue and
red particles at time $t\geq 0$. We define a discrete version of
$(S_{t})_{t\geq 0}$ by considering the collection of times where a particle
changes its state. Let $\tau(0)=0$ and

$\tau(i)=\inf\\{t\geq\tau(i-1)\,:\,S_{t}\neq S_{\tau(i-1)}\text{ or
}\mathds{1}(A_{t})=0\\}.$

The _jump chain_ $J=(J_{i})_{i\in\mathbb{Z}_{+}}$ of $(S_{t})_{t\geq 0}$ is
defined by

$J_{i}=\begin{cases}S_{\tau(i)},&\mathds{1}(A_{\tau(i)})=1\\\
0&\text{otherwise}\end{cases}.$

We say that a jump chain is _living at step $n$_ if $J_{i}>0$ for all $0\leq
i\leq n$. The transition probabilities of the jump Markov chain are, for each
$j>0$,

$p_{j,j+1}=\frac{\lambda_{j}}{1+\lambda_{j}+D_{j}},\quad
p_{j,j-1}=\frac{1}{1+\lambda_{j}+D_{j}},\quad
p_{j,0}=\frac{D_{j}}{1+\lambda_{j}+D_{j}},$

where $D_{j}=\sum_{i=1}^{j}\rho_{i}$.

On the event $\mathfrak{R}_{k}$, the path of the jump chain $J$ (up to the
renewal time) can be identified with a Dyck path of length $2k$ translated by
one vertical unit. Moreover, the weights that we considered in (3) correspond
to the transition probabilities of the jump chain: $p_{j,j+1}=u(j-1)$ and
$p_{j,j-1}=v(j-2)$. It follows that

(9)
$\displaystyle\mathbf{P}_{{\vec{\lambda}},{\vec{\rho}}}(\mathfrak{R}_{k})=C_{k}^{{\vec{\lambda}},{\vec{\rho}}}$

the weighted Catalan number defined at (4) that uses the weights at (3).

We use ideas from [BCE+21, Lemma 2.2] to prove that $\mathbf{P}(Y\geq k)$ is
comparable to $\mathbf{P}(\mathfrak{R}_{k})$. The difficulty is that the event
$\\{Y\geq k\\}$ includes _all_ realizations for which blue reaches $k$, while
$\mathfrak{R}_{k}$ only includes realizations which have a renewal at $k$.

######  Lemma 2.

lem:bigger Let $C$ and $m$ be as in (6). There exists $c_{0}$ that does not
depend on $k$ such that

$\mathbf{P}(Y\geq k)\leq c_{0}k^{1+m}\mathbf{P}(\mathfrak{R}_{k})$

for all $k\geq 1$.

###### Proof.

Define the _height profile_ of a (jump chain) path
$J=(J_{0},J_{1},\ldots,J_{m})$ to be the vector
$h(J)=(h_{1}(J),\ldots,h_{m+1}(J))$ which indicates the frequency of each
height reached by $J$. Formally,
$h_{i}(J)\coloneqq\sum_{\ell=0}^{m}\mathds{1}(J_{\ell}=i)$. The probability
that the distance-dependent chase-escape follows the path $J$ in its jump
chain is

$p(J)=\prod_{i=1}^{k}\lambda_{i}^{h_{i}(J)}\prod_{j=1}^{m+1}\left(\frac{1}{1+\lambda_{j}+D_{j}}\right)^{h_{j}(J)}.$

It is necessary and sufficient for the occurrence of $\\{Y\geq k\\}$ to
_first_ have the rightmost red particle reach $k$, and _second_ have the
rightmost blue particle reach $k$ after $\ell$ steps. An advantage of this
perspective is that the occurrence of second stage only depends on $\ell$. In
particular, it does not depend on the behavior of red beyond $k$. We now
formally decompose jump chains corresponding to $\\{Y\geq k\\}$ into these two
stages.

081415 Figure 2.  A path $\gamma\in\Gamma_{5}$ when $k=7$.

For a jump chain $\gamma$ corresponding to a configuration from $\\{Y\geq
k\\}$, let $\ell\in\\{1,\ldots,k-1\\}$ be the value of the jump chain when red
arrives to $k$. After this, we enter the second stage of $\\{Y\geq k\\}$; the
rightmost blue particle must advance from $\ell$ to $k$. So, the probability
of the second stage is

(10)
$\sigma(\ell)=\prod_{n=1}^{\ell}\mathbf{P}\left(\begin{array}[]{c}\text{blue
advances 1 step before}\\\ \text{any site $k-\ell+n,\ldots,k$
dies}\end{array}\right)=\prod_{n=1}^{\ell}\frac{1}{1+D_{n}}.$

Let $\Gamma_{\ell}$ be the set of all living jump chain paths of length
$2k-\ell-1$ that start at $(0,1)$ and end with an upward step to
$(2k-\ell-1,\ell)$. See Figure 2.

For $\gamma\in\Gamma_{\ell}$, the probability that $Y\geq k$ and the first
$2k-\ell-1$ steps of the jump chain follow $\gamma$ is $q(\gamma)\coloneqq
p(\gamma)\sigma(\ell)$. Setting
$q_{\ell}\coloneqq\sum_{\gamma\in\Gamma_{\ell}}q(\gamma)$, we arrive at the
following decomposition of $\\{Y\geq k\\}$:

(11) $\displaystyle\mathbf{P}(Y\geq
k)=\sum_{\ell=2}^{k}\sum_{\gamma\in\Gamma_{\ell}}p(\gamma)\sigma(\ell)=\sum_{\ell=2}^{k}q_{\ell}.$

A subset of $\mathfrak{R}_{k}$ is the collection of processes which follow
jump chains in $\Gamma_{2}$, and subsequently have blue advance by one, then
red advance by one, followed by blue advancing one. This gives the bound

$q_{2}\frac{\lambda_{2}}{(1+\lambda_{1}+D_{1})(1+\lambda_{2}+D_{2})^{2}}\leq\mathbf{P}(\mathfrak{R}_{k}).$

We will now prove that $q_{\ell}$ for $\ell\geq 2$ is comparable to $q_{2}$.

081415$\star$$\star$$\star$$\star$$+$$+$$+$ Figure 3.  Let $k=7$. The black
line with dots is a path $\gamma\in\Gamma_{5}$. The blue line with stars is
the modified path $\tilde{\gamma}\in\Gamma_{2}$. The red line with pluses is
the extension of $\tilde{\gamma}$ to a jump chain in $\mathfrak{R}_{7}$.

Given $\gamma\in\Gamma_{\ell}$, we obtain $\tilde{\gamma}\in\Gamma_{2}$
inserting $\ell-2$ downward steps before the last upward step. See Figure 3.
The paths $\gamma$ and $\tilde{\gamma}$ agree on the first $2k-\ell-2$ steps,
so

(12) $\displaystyle q(\gamma)$
$\displaystyle=\frac{\sigma(\ell)}{\sigma(2)\prod_{i=1}^{\ell-2}(1+\lambda_{i}+D_{i})^{-1}}q(\tilde{\gamma})$
(13)
$\displaystyle=\frac{\prod_{n=1}^{\ell}\frac{1}{1+D_{n}}}{\prod_{n=1}^{2}\frac{1}{1+D_{n}}\prod_{i=1}^{\ell-2}(1+\lambda_{i}+D_{i})^{-1}}q(\tilde{\gamma})$

Simplifying gives

(14) $\displaystyle q(\gamma)$
$\displaystyle=q(\tilde{\gamma})\frac{(1+D_{1}+\lambda_{1})(1+D_{2}+\lambda_{2})}{(1+D_{\ell})(1+D_{\ell-1})}\prod_{i=3}^{\ell-2}\left(1+\frac{\lambda_{i}}{1+D_{i}}\right).$

Taking $C$ and $m$ as at (6) set

$c_{0}=c\frac{(1+D_{1}+\lambda_{1})(1+D_{2}+\lambda_{2})}{(1+D_{\ell})(1+D_{\ell-1})}.$

It follows from our hypothesis (6) that for all $\gamma\in\Gamma_{\ell}$

$q(\gamma)\leq c_{0}\ell^{m}q(\tilde{\gamma}).$

As the map $\gamma\in\Gamma_{\ell}\mapsto\tilde{\gamma}\in\Gamma_{2}$ is
injective, we have

(15) $\displaystyle
q_{\ell}\leq\sum_{\gamma\in\Gamma_{\ell}}c_{0}\ell^{m}q(\tilde{\gamma})\leq
c_{0}\ell^{m}\sum_{\gamma\in\Gamma_{2}}q(\gamma)=c_{0}\ell^{m}q_{2}.$

Applying (15) at (11) and summing completes the lemma.

∎

##  3\. Proof of thm:main

###  3.1. Preliminaries

This treatment is similar to that in [BCE+21, Section 3]. For completeness, we
reproduce and generalize the necessary elements. Given a sequence
$(c_{n})_{n\geq 0}$, define the formal continued fraction

(16) $\displaystyle
K[c_{0},c_{1},\ldots]:=\cfrac{c_{0}}{1-\cfrac{c_{1}}{1-\ddots}}.$

Let

(17) $\displaystyle
a_{j}:=u(j)v(j)=\frac{\lambda_{j+1}}{(1+\lambda_{j+1}+D_{j+1})(1+\lambda_{j+2}+D_{j+2})}.$

It follows from [GJ85, Chapter 5] that

(18) $\displaystyle g(z)=f(z):=K[1,a_{0}z,a_{1}z,\ldots]$

for all $|z|<M$. A classical theorem of Worpitzky lets us prove that $f$ is
_meromorphic_ i.e. holomorphic outside of a set of isolated poles.

######  Theorem 3 (Worpitzky Circle Theorem).

thm:worpitzky Let $c_{j}\colon D\to\\{|w|<1/4\\}$ be a family of analytic
functions over a domain $D\subseteq\mathbb{C}$. Then
$K[1,c_{0}(z),c_{1}(z),\ldots]$ converges uniformly for $z$ in any compact
subset of $D$.

######  Corollary 4.

lem:meromorphic Assuming (7), it holds that $f$ is meromorphic on
$\mathbb{C}$.

###### Proof.

We will prove that $f$ is meromorphic for all $z\in\Delta=\\{|z|<r_{0}\\}$
with $r_{0}>0$ arbitrary. Let $T_{j}(z):=K[a_{j}z,a_{j+1}z,\ldots]$ be the
tail of the continued fraction so that
$f(z)=K[1,a_{0}z,\ldots,a_{j-1}z,T_{j}(z)]$. By the hypothesis (7), we have
$|a_{j}|\downarrow 0$ as $j\to\infty$. It follows that for some $j=j(r_{0})$
large enough, $|a_{k}z|\leq 1/4$ for all $k\geq j$ and $z\in\Delta$.
thm:worpitzky ensures that $|T_{j}(z)|<\infty$ and the partial continued
fractions $K[a_{j}z,\ldots,a_{n}z]$ are analytic (again by thm:worpitzky) and
converge uniformly to $T_{j}$ for $z\in\Delta$. Thus, $T_{j}$ is a uniform
limit of analytic functions and is therefore analytic on $\Delta$. We can then
write $f(z)=K[1,a_{0}z,\ldots,a_{j-1}z,T_{j}(z)].$ Since each $a_{i}z$ is a
linear function in $z$, $f$ is a quotient of two analytic functions. ∎

Our next lemma requires a classical theorem from complex variable theory (see
[FS09, Theorem IV.6] for example).

######  Theorem 5 (Pringsheim’s Theorem).

thm:pringsheim If $\varphi(z)$ is representable at the origin by a power
series $\varphi(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ that has real coefficients
$a_{n}\geq 0$ and radius of convergence $M$, then the point $z=M$ is a
singularity of $\varphi(z)$.

######  Lemma 6.

lem:M_regimes Let $\rho>0$. Then $M\leq d$ if and only if $g(d)=\infty$.

###### Proof.

We first note that the implication “$M<d$ implies $g(d)=\infty$” as well as
the reverse direction “$g(d)=\infty$ implies $M\leq d$” both follow
immediately from the definition of the radius of convergence. It remains to
show that $M=d$ implies $g(d)=\infty$. lem:meromorphic proves that $f$ is a
meromorphic function. Since $g=f$ for $|z|<M$, $f(x)>0$ for $x\in(0,d)$ we
have

$g(d)=\lim_{x\uparrow d}g(x)=\lim_{x\uparrow d}f(x).$

Moreover, thm:pringsheim gives $z=d$ is a singularity. Monotone convergence
and nonnnegativity of the coefficients then ensure that
$g(d^{-})=f(d^{-})=\infty$. ∎

###  3.2. Proof of thm:main

###### Proof.

Index the $d^{k}$ vertices at distance $k$ from the root of $\mathcal{T}_{d}$
by $(v_{k,i})_{i=1}^{d^{k}}$. Self-similarity of the tree ensures that

(19) $\displaystyle\mathbf{E}[|\mathcal{B}|]$
$\displaystyle=1+\mathbf{E}\left[\sum_{k=0}^{\infty}\sum_{i=1}^{d^{k}}\mathbf{1}{\\{\text{$v_{k,i}$
is eventually blue}\\}}\right]$ (20)
$\displaystyle=1+\sum_{k=0}^{\infty}\mathbf{P}(Y=k)d^{k}.$

Now, suppose that $M>d$. Using (20) and the comparison in lem:bigger gives

(21) $\displaystyle E[|\mathcal{B}|]$ $\displaystyle\leq
1+\sum_{k=1}^{\infty}ck^{1+m}\mathbf{P}(\mathfrak{R}_{k})d^{k}.$

Since $M>d$, the sum on the right converges even with the polynomial
prefactor. Thus, $\mathbf{E}[|\mathcal{B}|]<\infty$.

Next, suppose that $\mathbf{E}[|\mathcal{B}|]<\infty$. Let $Y$ be as at (8).
Since $\\{Y=k\\}$ contains the event that a renewal occurs at $k$ followed by
blue advancing one step, we have

$\mathbf{P}(\mathcal{R}_{k-1})\frac{1}{1+\lambda_{1}+\rho_{1}}\leq\mathbf{P}(Y=k).$

Applying this bound to (20) and reindexing the sum gives

$\mathbf{E}[|\mathcal{B}|]\geq\frac{d}{1+\lambda_{1}+\rho_{1}}\sum_{k=0}^{\infty}\mathbf{P}(\mathfrak{R}_{k})d^{k}=\frac{1}{1+\lambda_{1}+\rho_{1}}g(d).$

Hence, $g(d)<\infty$, which gives $M>d$ by lem:M_regimes. ∎

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