# On favourite sites of a random walk in moderately sparse random environment

Alicja Kołodziejska

###### Abstract.

We study the favourite sites of a random walk evolving in a sparse random
environment on the set of integers. The walker moves symmetrically apart from
some randomly chosen sites where we impose a random drift. We prove annealed
limit theorems for the time the walk spends in its favourite sites in two
cases. The first one, in which it is the distribution of the drift that
determines the limiting behaviour of the walk, is a generalization of known
results for a random walk in i.i.d. random environment. In the second case a
new behaviour appears, caused by the sparsity of the environment.

Mathematical Institute, University of Wrocław, Pl. Grunwaldzki 2, 50-384
Wrocław, Poland. E-mail<EMAIL_ADDRESS>
Keywords: random walk in random environment, branching process in random
environment, sparse random environment, local times.
MSC2020 subject classifications: primary: 60K37; secondary: 60F05.

##  1\. Introduction

One of the most classic and well studied stochastic processes is a simple
symmetric random walk on the set of integers, which models the movement of a
single particle in one-dimensional, homogeneous medium. The simplicity of the
model allows to analyse it with the help of such classic results as the strong
law of large numbers or the central limit theorem; however, its homogeneity is
not always desired. In many applications one would like to consider some
obstacles or impurities of the medium, possibly placed randomly, that would
have impact on the movement of the particle. One of the ways of defining such
random environment was proposed by Solomon in the seventies [16]. In his
model, called a random walk in a random environment (RWRE), one first samples
the environment by putting random drift independently at every integer, and
then the particle moves in such inhomogeneous, random medium. It soon
transpired that this additional noise leads to behaviour not observed in the
deterministic setting. Various authors described how the distribution of the
environment determines such properties of the walk as its transience and
asymptotic speed [16, 1], limit theorems [11, 13], or large deviations [8, 4].
In particular, under suitable distribution of the drift, the walk may be
transient, but with sub-linear speed, and no longer satisfy the central limit
theorem. This new behaviour is caused, heuristically speaking, by the traps
occurring in the environment, i.e. sites with unfavourable drift; the particle
is forced to make many attempts to cross such a site and this fact has
significant impact on the limiting behaviour of the walk.

The model studied in this article was introduced by Matzavinos, Roitershtein,
and Seol in [14] and is called a random walk in a sparse random environment
(RWSRE). The aim is to consider an environment in which the impurities appear
not at every site, as it is the case in the RWRE, but are put sparsely on the
set of integers. To this end, the environment is sampled by marking some sites
by a two-sided renewal process and putting random drifts only in the marked
points. In the unmarked sites the movement of the particle is symmetric.
Therefore the RWSRE may be seen as an interpolation between the simple
symmetric random walk and the RWRE, and one may expect that, depending on the
distribution of the environment, it should manifest properties resembling one
or the other. Indeed, this dichotomy was already observed in [6, 5, 7] in the
context of limit theorems for the position of the walk and the sequence of
first passage times. Under suitable assumptions on the distribution of the
environment, it is the drift that has major impact on the movement of the
particle and the limit theorems resemble results known for the RWRE. However,
under different assumptions, ones that favour long distances between marked
points, in most sites the walk behaves like a simple symmetric random walk and
this change is visible in the macroscopic scale of the limit theorems.

(a) The case of dominating drift: the particle spends most of its time trying
to cross sites with unfavourable drift.

(b) The case of dominating sparsity: in most of the sites the particle
performs a simple symmetric random walk.

Figure 1.1. Exemplary trajectories of a transient RWSRE. Horizontal lines
indicate marked sites; the darker the line, the stronger the drift to
$-\infty$.

The aim of this article is to study the sequence of maximal local times, i.e.
the amount of time spent by the particle in its favourite sites, in the case
of the transient walk in a sparse random environment. We prove annealed limit
theorems for this sequence under two sets of assumptions. In the first case it
is the drift that drives the limiting behaviour of the walk, and our results
may be seen as a generalization of those obtained by Dolgopyat and Goldsheid
in [10, Theorem 4] for the RWRE. However, the techniques used in [10] were
different from those presented here. In this article we follow the method
proposed by Kesten et al. in [13] when examining the hitting times, that is we
rephrase the question posed for the walk into the setting of the associated
branching process. This method proves useful both in the case of dominating
drift and the complementary case, in which the sparsity of the environment
plays the dominant role in determining the limiting behaviour of the walk.

The article is organized as follows: in the remaining part of this section we
define the examined model formally. Statement of our main results is given in
Section 2. Section 3 introduces the branching process associated with the walk
and presents some of its properties. The proofs of the main theorems are given
in Sections 4 and 5.

###  1.1. Random walk in sparse random environment

Let $\Omega=(0,1)^{\mathbb{Z}}$ and let ${\mathcal{F}}$ be the corresponding
cylindrical $\sigma$-algebra. A random element
$\omega=(\omega_{n})_{n\in{\mathbb{Z}}}$ of $(\Omega,{\mathcal{F}})$
distributed according to a probability measure ${\rm P}$ is called a random
environment. Let $\mathcal{X}={\mathbb{Z}}^{\mathbb{N}}$ be the set of
possible paths of a random walk on ${\mathbb{Z}}$, with corresponding
cylindrical $\sigma$-algebra $\mathcal{G}$. Then any $\omega\in\Omega$ and
$i\in{\mathbb{Z}}$ gives rise to a measure ${\rm P}_{\omega}^{i}$ on
$\mathcal{X}$ such that ${\rm P}_{\omega}^{i}[X_{0}=i]=1$ and

(1.1) ${\rm
P}_{\omega}^{i}\left[X_{n+1}=j|X_{n}=k\right]=\begin{cases}\omega_{k},\quad&\textnormal{if
}j=k+1,\\\ 1-\omega_{k},&\textnormal{if }j=k-1,\\\
0&\textnormal{otherwise,}\end{cases}$

where $X=(X_{n})_{n\in\mathbb{N}}\in\mathcal{X}$. That is, under ${\rm
P}_{\omega}^{i}$, $X$ is a nearest-neighbour random walk starting from $i$
with transition probabilities given by the sequence $\omega$. In particular,
it is a time-homogeneous Markov chain.

Since the environment itself is random, it is natural to consider a measure
$\mathbb{P}^{i}$ on
$(\Omega\times\mathcal{X},{\mathcal{F}}\otimes\mathcal{G})$ such that

(1.2) $\mathbb{P}^{i}\left[F\times G\right]=\int_{F}{\rm
P}_{\omega}^{i}[G]\,{\rm P}(d\omega)$

for any $F\in{\mathcal{F}},G\in\mathcal{G}$. We shall write ${\rm
P}_{\omega}={\rm P}_{\omega}^{0}$ and $\mathbb{P}=\mathbb{P}^{0}$. Observe
that under $\mathbb{P}$ the walk $X$ may exhibit a long-time dependencies and
thus no longer be a Markov chain.

The process $X$ defined above is called a random walk in a random environment
and was introduced by Solomon [16]. A well studied case is $\omega$ being an
i.i.d. sequence, which gives rise to a random walk in i.i.d. random
environment.

We will consider a specific choice of environment that was introduced first by
Matzavinos, Roitershtein, and Seol in [14]. Consider an i.i.d. sequence
$((\xi_{k},\lambda_{k}))_{k\in{\mathbb{Z}}}\in(\mathbb{N}_{+}\times(0,1))^{\mathbb{Z}}$
and define, for any $n,k\in{\mathbb{Z}}$,

(1.3) $S_{n}=\begin{cases}\sum_{j=1}^{n}\xi_{j},\;&n>0,\\\ 0,&n=0,\\\
-\sum_{j={n+1}}^{0}\xi_{j},&n<0;\end{cases}\qquad\omega_{k}=\begin{cases}\lambda_{n+1}\>&\textnormal{
if }k=S_{n}\textnormal{ for some }n\in{\mathbb{Z}},\\\ 1/2&\textnormal{
otherwise.}\end{cases}$

The random walk evolving in an environment $\omega$ defined by (1.3) is called
a random walk in a sparse random environment. We shall refer to the random
sites $S_{n}$ as marked points and write $(\xi,\lambda)$ for a generic element
of the sequence $((\xi_{k},\lambda_{k}))_{k\in{\mathbb{Z}}}$. The environment
is called moderately sparse if ${\rm E}\xi<\infty$ and strongly sparse
otherwise.

Observe that if $\xi=1$ almost surely, then we obtain once again a random walk
in i.i.d. environment. Otherwise the environment is split into blocks of
lengths given by the sequence $(\xi_{k})_{k\in{\mathbb{Z}}}$; within every
block the particle performs a symmetric walk, while the random drift occurs at
the endpoints of blocks. Therefore the RWSRE model may be seen as an
interpolation between a simple symmetric random walk and a walk in i.i.d.
environment, or as a generalization of the latter.

We should remark that the model we consider here is slightly different from
that defined originally in [14]. That is, due to (1.3), we allow for
dependence between the length of the block between marked sites and the drift
at its left end, while originally the dependence was allowed for the drift at
the right end. This change of convention arises naturally from time reversal
coming with the associated branching process which we introduce in Section 3,
and appeared also in [5, 6], where annealed limit theorems for the position of
the walk were proved.

For $k\in{\mathbb{Z}}$, let

$\rho_{k}=\frac{1-\lambda_{k}}{\lambda_{k}}.$

The variables $(\rho_{k})_{k\in{\mathbb{Z}}}$, which quantify the drift in the
environment, appear naturally when examining the properties of the walk. In
particular, as was shown in [14], if

(1.4) ${\rm E}\log\xi<\infty,\quad{\rm E}\log\rho<0,$

then RWSRE is transient to $+\infty$, $\mathbb{P}$-almost surely. From now on
we will assume that conditions (1.4) are satisfied.

##  2\. Annealed limit theorems for maximal local time

Consider a sequence of hitting times

(2.1) $T_{n}=\inf\\{k\geq 0:X_{k}=n\\}$

and let, for $k\leq n$,

(2.2) $L_{k}(n)=|\\{m\leq T_{n}\,:\,X_{m}=k\\}|$

be the local time, i.e. number of times the walk visits $k$ before reaching
$n$. Our object of interest is the limiting behaviour of maximal local time,
that is the variable $\max_{k\leq n}L_{k}(n)$, as $n\to\infty$. We shall
present two cases in which an annealed limit theorem holds for this sequence
of variables, with Fréchet distribution in the limit. We assume that (1.4)
holds, i.e. the walk is transient. Additionally, we consider two sets of
assumptions:

Assumptions $(A)$: For some $\alpha\in(0,2)$,

  * •

${\rm E}\rho^{\alpha}=1$;

  * •

${\rm E}\rho^{\alpha}\log^{+}\rho<\infty$;

  * •

the distribution of $\log\rho$ is non-arithmetic;

  * •

${\rm E}\xi^{(\alpha+\delta)\vee 1}<\infty$ for some $\delta>0$;

  * •

${\rm E}\xi^{\alpha}\rho^{\alpha}<\infty$.

Recall that a distribution is non-arithmetic if it is not concentrated on any
lattice $c{\mathbb{Z}}$, $c>0$. Note that without loss of generality we may
assume that $\alpha+\delta\leq 2$. In this case the limiting behaviour of
maxima is determined mostly by the parameter $\alpha$, that is by properties
of $\rho$; it is a generalization of the result known for the walk in i.i.d.
environment. We shall prove the following:

######  Theorem 2.1.

Under assumptions $(A)$, there is a constant $c_{\alpha}>0$ such that for all
$x>0$,

$\lim_{n\to\infty}\mathbb{P}\left[\frac{\max_{k\leq
n}L_{k}(n)}{n^{1/\alpha}}>x\right]=1-e^{-c_{\alpha}x^{-\alpha}}.$

It turns out that the crucial assumption in this case is that ${\rm
E}\xi^{\alpha+\delta}<\infty$. Different behaviour appears when $\xi$ does not
have high enough moments. Consider the following:

Assumptions $(B)$:  For some $\beta\in[1,2)$,

  * •

${\rm P}[\xi>x]\sim x^{-\beta}\ell(x)$ for some slowly varying $\ell$;

  * •

${\rm E}\rho^{\beta+\delta}<1$ for some $\delta>0$;

  * •

$\xi$ and $\rho$ are independent;

  * •

if $\beta=1$, assume ${\rm E}\xi<\infty$.

In this case we may also assume that $\beta+\delta\leq 2$. Observe that we do
not assume that there exists $\alpha$ such that ${\rm E}\rho^{\alpha}=1$.
However, if it does exist, then $\alpha>\beta$ and ${\rm
E}\xi^{\alpha}=\infty$. Since $\xi$ has regularly varying tails, a good
scaling for maxima of $(\xi_{n})_{n\in\mathbb{N}}$ is a sequence
$(a_{n})_{n\in\mathbb{N}}$ such that

(2.3) $\lim_{n\to\infty}n{\rm P}[\xi>a_{n}]=1.$

It turns out it is also a good scaling for maxima of $L$.

######  Theorem 2.2.

Under assumptions $(B)$, there is a constant $c_{\beta}>0$ such that for all
$x>0$,

$\lim_{n\to\infty}\mathbb{P}\left[\frac{\max_{k\leq
n}L_{k}(n)}{a_{n}}>x\right]=1-e^{-c_{\beta}x^{-\beta}}.$

The exact forms of constants $c_{\alpha},c_{\beta}$ will be given during the
proofs.

##  3\. Auxiliary results

Instead of examining the local times explicitly, we pass to a branching
process associated with RWSRE. In this section we describe the construction of
this process and prove auxiliary lemmas which we will use in both examined
cases.

###  3.1. Associated branching process

An important property of a transient nearest neighbour random walk on
${\mathbb{Z}}$ is its duality with a branching process. Consider a walk
$(X_{n})_{n\in\mathbb{N}}$ such that $X_{0}=0$ and $X_{n}\to\infty$ almost
surely, evolving in an environment $\omega=(\omega_{k})_{k\in{\mathbb{Z}}}$.
Recall that, for $n\in\mathbb{N}$,

$T_{n}=\inf\\{k\in\mathbb{N}\,:\,X_{k}=n\\}$

is the first passage time and, for $k\leq n$,

$L_{k}(n)=|\\{m\leq T_{n}:X_{m}=k\\}|$

is the local time, i.e. the number of times the walk visits site $k$ before
reaching $n$. First of all, note that the transience of the walk implies that,
almost surely, the walk spends only finite time on the negative half-axis.
That is, for any sequence $b_{n}\to\infty$,

$\frac{\max_{k<0}L_{k}(n)}{b_{n}}\to 0\quad\mathbb{P}\textnormal{-a.s.}$

Therefore, when examining the limit theorems, we may restrict our analysis to
the variables $L_{k}(n)$ for $k\geq 0$.

Figure 3.1. Exemplary path of a simple walk and corresponding realization of a
branching process. Immigrants (marked in red) correspond to arrivals to new
sites. The subtrees correspond to the excursions of the walk; the first
excursion from $7$ and its corresponding subtree were marked in blue.

The visits to $k\geq 0$ counted by $L_{k}(n)$ may be split into visits from
the left and from the right, that is,

$\begin{split}L_{k}(n)&=|\\{m\leq T_{n}\,:\,X_{m}=k\\}|\\\ &=|\\{m\leq
T_{n}\,:\,X_{m-1}=k-1,\,X_{m}=k\\}|+|\\{m\leq
T_{n}\,:\,X_{m-1}=k+1,\,X_{m}=k\\}|.\end{split}$

Moreover, since the walk is simple, it makes a step from $k-1$ to $k$ when it
visits site $k$ for the first time. After that, it may make some excursions to
the left from $k$; such an excursion always begins with a step from $k$ to
$k-1$ and ends with a step from $k-1$ to $k$. Therefore, to count all the
visits the walk makes to given sites, it is enough to count its steps to the
left. That is, for fixed $n\in\mathbb{N}$ and $0\leq k\leq n$,

$\begin{split}L_{k}(n)&=1+|\\{m\leq
T_{n}\,:\,X_{m-1}=k,\,X_{m}=k-1\\}|+|\\{m\leq
T_{n}\,:\,X_{m-1}=k+1,\,X_{m}=k\\}|\\\
&=1+\widetilde{Z}_{k-1}+\widetilde{Z}_{k},\end{split}$

where $\widetilde{Z}_{k}=|\\{m\leq T_{n}\,:\,X_{m-1}=k+1,\,X_{m}=k\\}|$ is the
number of visits to point $k$ from the right. The main observation is that the
process given by $Z_{k}=\widetilde{Z}_{n-k}$ has a branching structure. Every
step from $n-k$ to $n-k-1$ occurs either before the walk discovered the site
$n-k+1$, or between consecutive steps from $n-k+1$ to $n-k$. That is,

$Z_{k+1}\overset{{\rm d}}{=}\sum_{j=1}^{Z_{k}+1}G_{n,k}^{(j)},$

where $G_{n,k}^{(j)}$, for $j\leq Z_{k}$, counts the number of steps from
$n-k$ to $n-k-1$ between $j$’th and $j+1$’th step from $n-k+1$ to $n-k$, and
$G_{n,k}^{(Z_{k}+1)}$ counts the number of steps from $n-k$ to $n-k-1$ before
the first visit to $n-k+1$. Observe that, due to the strong Markov property of
the walk, the variables $G_{n,k}^{(j)}$ are i.i.d., independent of $Z_{k}$,
and have geometric distribution with parameter $\omega_{n-k}$, i.e.

${\rm
P}_{\omega}\left[G_{n,k}^{(j)}=m\right]=\omega_{n-k}(1-\omega_{n-k})^{m}\quad\textnormal{for
}m=0,1,2,\dots.$

Therefore, $Z=(Z_{k})_{k\in\mathbb{N}}$ is a branching process in random
environment with unit immigration; note that we do not count the immigrant, so
that $Z_{0}=0$. Moreover, for any fixed $n\in\mathbb{N}$,

(3.1) $\left(L_{k}(n)\right)_{0\leq k\leq n}\overset{{\rm
d}}{=}\left(1+Z_{n-k+1}+Z_{n-k}\right)_{0\leq k\leq n}.$

In particular, if $X$ is a random walk in a sparse random environment, its
associated branching process is a branching process in a sparse random
environment (BPSRE). If in the above construction we consider the walk stopped
upon reaching a marked point $S_{n}$, the branching process starts from one
immigrant and evolves in the environment divided into blocks of lengths given
by $(\xi_{n-k})_{k\in\mathbb{N}}$; within the blocks the reproduction is given
by the law $Geo(1/2)$, while the particles in the $k$’th marked generation are
born with the law $Geo(\lambda_{n-k})$. When examining the process $Z$, it is
convenient – and valid, since the environment is given by an i.i.d. sequence –
to reverse the enumeration, so that the block lengths are given by
$(\xi_{k})_{k\in\mathbb{N}}$ and reproduction law in $k$’th marked point is
$Geo(\lambda_{k})$. The process $Z$ may be then defined formally as follows:
for any fixed environment $\omega$, under ${\rm P}_{\omega}$,

$\displaystyle Z_{0}$ $\displaystyle=0,$ $\displaystyle Z_{k}$
$\displaystyle=\sum_{j=1}^{Z_{k-1}+1}G_{k}^{(j)},$

where the variables $(G_{k}^{(j)})_{j\in\mathbb{N}}$ are independent of
$Z_{k-1}$ and each other, and

$G_{k}^{(j)}\overset{{\rm
d}}{=}Geo(\omega_{k})\quad\textnormal{for}\quad\omega_{k}=\begin{cases}\lambda_{n}\quad&\textnormal{if
$k=S_{n}$ for some $n\in\mathbb{N}$;}\\\
1/2&\textnormal{otherwise.}\end{cases}$

Whenever examining a BPSRE, we will distinct the population at marked
generations with bold letters, that is, for example,
${\mathbb{Z}}_{n}=Z_{S_{n}}$.

${\mathbb{Z}}_{1}$$\xi_{1}$${\mathbb{Z}}_{2}$$\xi_{2}$${\mathbb{Z}}_{3}$$\xi_{3}$${\mathbb{Z}}_{4}$$Z_{S_{4}-1}$$Geo(\lambda_{4})$$Geo(1/2)$$\xi_{4}$${\mathbb{Z}}_{5}$$\xi_{5}$${\mathbb{Z}}_{6}$$\xi_{6}$
Figure 3.2. Schematic picture of the process $Z$. Horizontal blue lines
represent marked generations. Within each block between marked generations,
the triangular area represents progeny of immigrants that arrived in this
block. The coloured region represents process $Y^{4}$.

For $k\in\mathbb{N}$, we will denote by $Y^{k}$ the process counting the
progeny of immigrants from $k$’th block, i.e. those arriving at times
$S_{k-1},S_{k-1}+1,\dots S_{k}-1$. Let, for $j\geq 0$, $Y^{k}_{j}$ denote the
number of descendants of these immigrants present in generation $S_{k-1}+j$.
Observe that the process $Y^{k}$ starts with one immigrant at time $j=0$; it
evolves with unit immigration and $Geo(1/2)$ reproduction law up until time
$j=\xi_{k}-1$. The last immigrant arrives at this time, and the particles at
time $j=\xi_{k}$ are born with the law $Geo(\lambda_{k})$. From there on the
process $Y^{k}$ evolves without immigration (see Figure 3.2).

We will use the convention that $Y^{k}_{j}=0$ for $j<0$, so that

$Z_{n}=\sum_{k\in\mathbb{N}}Y^{k}_{n-S_{k-1}}.$

Observe that the processes $Y^{k}$ are independent under ${\rm P}_{\omega}$
and identically distributed under $\mathbb{P}$.

The branching process in a sparse random environment was studied in [6] for
the purpose of proving annealed limit theorems for the first passage times. An
important observation is that the transience of the walk implies quick
extinctions of the branching process. Let

$\tau_{0}=0,\quad\tau_{n}=\inf\\{k>\tau_{n-1}\,:\,{\mathbb{Z}}_{k}=0\\}$

be the extinction times (note that we only consider the extinctions at marked
generations). Observe that when the extinction occurs, the process starts anew
from one immigrant. Thus the sequence $(\tau_{n}-\tau_{n-1})_{n\geq 1}$ is
i.i.d. under $\mathbb{P}$, and the extinction times split the process $Z$ into
independent epochs. The following is Lemma 4.1 from [6]; it implies that the
extinctions occur rather often in the case of transient RWSRE.

######  Lemma 3.1.

Assume that ${\rm E}\log\rho<0$ and ${\rm E}\log\xi<\infty$. Then
$\mathbb{E}\tau_{1}<\infty$. If additionally ${\rm
E}\rho^{\varepsilon}<\infty$ and ${\rm E}\xi^{\varepsilon}<\infty$ for some
$\varepsilon>0$, then there exists $c>0$ such that
$\mathbb{E}e^{c\tau_{1}}<\infty$.

Observe that due to (3.1) we have, for any $n\in\mathbb{N}$,

(3.2) $\max_{0\leq k\leq S_{n}}L_{k}(S_{n})\overset{{\rm d}}{=}1+\max_{0\leq
k\leq S_{n}}(Z_{k}+Z_{k+1}).$

Therefore, to obtain limit theorems for the sequence of maximal local times
along the marked points, one may examine the maximal generations of the
corresponding branching process. We conclude this section by remarking that in
the setting of moderately sparse environment this is sufficient also to obtain
annealed limit theorems for the sequence $(\max_{k\leq
n}L_{k}(n))_{n\in\mathbb{N}}$. Note that $(a_{n})_{n\in\mathbb{N}}$ given by
(2.3) is regularly varying with index $1/\beta$.

######  Lemma 3.2.

Assume that ${\rm E}\xi<\infty$. If there exist constants $c>0$, $\gamma>0$
and a sequence $(b(n))_{n\in\mathbb{N}}$ which is regularly varying with index
$1/\gamma$ such that for every $x>0$,

$\lim_{n\to\infty}\mathbb{P}\left[\frac{\max_{k\leq
S_{n}}L_{k}(S_{n})}{b(n)}>x\right]=1-e^{-cx^{-\gamma}},$

then for every $x>0$,

$\lim_{n\to\infty}\mathbb{P}\left[\frac{\max_{k\leq
n}L_{k}(n)}{b(n)}>x\right]=1-e^{-(c/{\rm E}\xi)x^{-\gamma}}.$

###### Proof.

Denote, for $n\in\mathbb{N}$,

$\nu_{n}=\inf\\{k>0\,:\,S_{k}>n\\}.$

Then the assumption ${\rm E}\xi<\infty$ and the law of large numbers guarantee
that ${\rm P}$-almost surely

$\frac{\nu_{n}}{n}\xrightarrow{n\to\infty}\frac{1}{{\rm E}\xi}.$

Denote, for $m\in\mathbb{N}$, $M(m)=\max_{k\leq S_{m}}L_{k}(S_{m})$. Since
$S_{\nu_{n}-1}\leq n<S_{\nu_{n}}$, we have, for any $\varepsilon>0$,

$\begin{split}\mathbb{P}\left[b(n)^{-1}\max_{0\leq
k<n}L_{k}(n)>x\right]&\geq\mathbb{P}\left[b(n)^{-1}M(\nu_{n}-1)>x\right]\\\
&\geq\mathbb{P}\left[b(n)^{-1}M(n(1/{\rm
E}\xi-\varepsilon)-1)>x\right]-\mathbb{P}\left[|1/{\rm
E}\xi-\nu_{n}/n|>\varepsilon\right]\\\
&\xrightarrow{n\to\infty}1-\exp(-c(1/{\rm
E}\xi-\varepsilon)x^{-\gamma}),\end{split}$

where we used the fact that

$\frac{b(n(1/{\rm E}\xi-\varepsilon)-1)}{b(n)}\to(1/{\rm
E}\xi-\varepsilon)^{1/\gamma}$

since $b(n)$ is regularly varying. Similarly,

$\begin{split}\mathbb{P}\left[b(n)^{-1}\max_{0\leq
k<n}L_{k}(n)>x\right]&\leq\mathbb{P}\left[b(n)^{-1}M(\nu_{n})>x\right]\\\
&\leq\mathbb{P}\left[b(n)^{-1}M(n(1/{\rm
E}\xi+\varepsilon))>x\right]+\mathbb{P}\left[|1/{\rm
E}\xi-\nu_{n}/n|>\varepsilon\right]\\\
&\xrightarrow{n\to\infty}1-\exp(-c(1/{\rm
E}\xi+\varepsilon)x^{-\gamma}),\end{split}$

which ends the proof since $\varepsilon>0$ is arbitrary. ∎

###  3.2. Estimates of the processes related to the environment

Define

(3.3)
${\bar{R}}_{n}=1+\rho_{n}+\rho_{n}\rho_{n+1}+\dots=\sum_{k=n-1}^{\infty}\Pi_{n,k},$

where $\Pi_{n,k}=\prod_{j=n}^{k}\rho_{j}$ if $n\leq k$ and $\Pi_{n,k}=1$
otherwise. Then the following relation holds:

(3.4) ${\bar{R}}_{n}=1+\rho_{n}{\bar{R}}_{n+1}.$

Moreover, the sequence $({\bar{R}}_{n})_{n\in{\mathbb{Z}}}$ is stationary
under ${\rm P}$. Observe that if ${\rm E}\rho^{\gamma}<1$ for some $\gamma>0$,
then ${\rm E}{\bar{R}}_{1}^{\gamma}<\infty$ (see the proof of Lemma 2.3.1 in
[3]), whereas under $(A)$, the distribution of $\rho$ satisfies the
assumptions of Kesten-Goldie theorem (see [3, Theorem 2.4.4]), thus

${\rm P}[{\bar{R}}_{1}>x]\sim c_{\alpha}x^{-\alpha}$

for some constant $c_{\alpha}$. Therefore

(3.5) ${\rm P}[{\bar{R}}_{1}>x]\leq C_{\gamma}x^{-\gamma}\quad\textnormal{for
some $C_{\gamma}<\infty$ and all $x>0$,}$

whenever either ${\rm E}\rho^{\gamma}<1$, or ${\rm E}\rho^{\gamma}=1$ and
Kesten-Goldie theorem holds for ${\bar{R}}_{1}$. As can be seen in the proofs
of Lemma 6 in [13] and Lemma 5.6 in [6], in the case of dominating drift it is
${\bar{R}}_{1}$ from whom the total population of the process $Z$ (which
corresponds to first passage times of the walk) inherits its annealed tail
behaviour.

Let, for $m\in\mathbb{N}$, the potential $\Psi$ be defined as

(3.6) $\Psi_{m,k}=\Pi_{m,n}\quad\textnormal{for }k\in[S_{n},S_{n+1}).$

As we will see, maxima of the potential determine the limiting behaviour of
maximal generation of $Z$ in the same way as ${\bar{R}}_{1}$ determines the
asymptotics of the total population. Let

(3.7) $M_{\Psi,m}=\max_{k\geq S_{m}-1}(\Psi_{m,k}+\Psi_{m,k+1}).$

Then the sequence $(M_{\Psi,m})_{m\in\mathbb{N}}$ is stationary under ${\rm
P}$; denote by $M_{\Psi}$ its generic element. Observe that

$M_{\Psi,1}\leq 2\max_{k\geq S_{1}-1}\Psi_{1,k}=2\max_{n\geq 0}\Pi_{1,n}\leq
2{\bar{R}}_{1},$

thus

(3.8) ${\rm E}M_{\Psi}^{\gamma}<\infty\quad\textnormal{ whenever }{\rm
E}\rho^{\gamma}<1.$

###  3.3. Auxiliary lemmas

The following lemma, concerning a classic Galton-Watson process, will be used
repeatedly to estimate the growth of BPSRE in the unmarked generations.

######  Lemma 3.3.

Let $(X_{n})_{n\geq 0}$ be a Galton-Watson process with $X_{0}=x_{0}$,
reproduction law $Geo(1/2)$, and no immigrants, and let $(\bar{X}_{n})_{n\geq
0}$ be an analogous process with unit immigration. Then the following hold for
any $N\in\mathbb{N}$:

(3.9) $\mathbb{E}\left[\max_{k\leq N}(X_{k}-x_{0})^{2}\right]\leq 8Nx_{0},$
(3.10) $\mathbb{E}\left[\max_{k\leq N}\bar{X}_{k}^{2}\right]\leq
16(N^{2}+Nx_{0}+x_{0}^{2}).$

###### Proof.

Since the process $(X_{k})_{k\in\mathbb{N}}$ is a martingale with mean
$x_{0}$, Doob’s maximal inequality implies

$\mathbb{E}\left[\max_{k\leq N}(X_{k}-x_{0})^{2}\right]\leq
4\mathbb{E}(X_{N}-x_{0})^{2}=4{\rm Var}X_{N}.$

Now, a standard calculation gives

${\rm Var}X_{N}=2Nx_{0},$

which implies (3.9).

Observe that $\bar{X}_{n}=X^{\prime}_{n}+I_{n}$, where $X^{\prime}$ denotes
the descendants of the initial $x_{0}$ particles, and $I$ denotes the progeny
of immigrants. The processes $I$ and $X^{\prime}$ are independent, and
$X^{\prime}$ has the same distribution as $X$. Moreover, the process
$(\bar{X}_{n})_{n\in\mathbb{N}}$ is a non-negative submartingale, thus by
Doob’s maximal inequality,

$\mathbb{E}\left[\max_{k\leq N}\bar{X}_{k}^{2}\right]\leq
4\mathbb{E}\left[\bar{X}_{N}^{2}\right]=4\left({\rm Var}X^{\prime}_{N}+{\rm
Var}I_{N}+(\mathbb{E}X^{\prime}_{N}+\mathbb{E}I_{N})^{2}\right).$

We have already examined the mean and variance of $X^{\prime}_{N}$. To
calculate moments of $I_{N}$, we may express $I$ as a sum of independent
copies of $X$. Alternatively, we may use the duality of $I$ and a simple
symmetric random walk. It implies that $I_{N}$ equals in distribution to the
number of times the walk hits $0$ from the right when crossing the interval
$[0,N+1]$ for the first time. By the classic gambler’s ruin problem, the
probability that the walk passes from $0$ to $N+1$ without returning to $0$
from the right, is $1/(N+1)$. Therefore $I_{N}\sim Geo(1/(N+1))$, from which
it follows that

$\mathbb{E}I_{N}=N+1,\quad{\rm Var}I_{N}=N^{2}+N.$

Hence

$\mathbb{E}\left[\bar{X}_{N}^{2}\right]=2Nx_{0}+N^{2}+N+(x_{0}+N+1)^{2}\leq
4(N^{2}+Nx_{0}+x_{0}^{2}),$

which ends the proof of (3.10).

∎

The next two lemmas will be of use to us both under assumptions $(A)$ and
$(B)$. Therefore we shall consider the following set of assumptions:

Assumptions $(\Gamma)$: for some $\gamma\leq 2$,

  * •

${\rm E}\rho^{\gamma}\leq 1$ and (3.5) holds,

  * •

${\rm E}\xi^{\gamma/2}<\infty$,

  * •

${\rm E}\rho^{\gamma}\xi^{\gamma/2}<\infty$.

Let $U_{n}$ be the progeny of the first immigrant residing in generation $n$,
with the convention $U_{0}=1$, and denote $\mathbb{U}_{n}=U_{S_{n}}$. For
fixed $N\in\mathbb{N}$, let $U^{k}$ for $k=1,\dots,N$ be copies of the process
$U=(U_{n})_{n\in\mathbb{N}}$, evolving in the same environment and independent
under ${\rm P}_{\omega}$. That is,
$(\sum_{k=1}^{N}U_{n}^{k})_{n\in\mathbb{N}}$ is a BPSRE with $N$ initial
particles evolving without immigration. Although the first part of the
following lemma is analogous to results presented in [13, Lemma 3] and [6,
Lemma 5.6], we provide the full proof as it gives some insight into the
properties of the process $U$.

######  Lemma 3.4.

Assume $(\Gamma)$. Then for some constant $C_{1}$,

(3.11) $\mathbb{P}\left[\sum_{k=1}^{N}\sum_{n\geq
0}\mathbb{U}_{n}^{k}>x\right]\leq C_{1}N^{\gamma}x^{-\gamma},$ (3.12)
$\mathbb{P}\left[\sum_{n\geq
0}\left|\sum_{k=1}^{N}\mathbb{U}_{n}^{k}-N\Pi_{1,n}\right|>x\right]\leq
C_{1}N^{\gamma/2}x^{-\gamma}.$

Moreover,

(3.13) $\mathbb{P}\left[\max_{n\geq 1}\sum_{k=1}^{N}U^{k}_{n}>x\right]\leq
C_{1}N^{\gamma}x^{-\gamma},$ (3.14) $\mathbb{P}\left[\sum_{n\geq
1}\sum_{k=1}^{N}\max_{S_{n-1}\leq
j<S_{n}}|U^{k}_{j}-\mathbb{U}^{k}_{n-1}|>x\right]\leq
C_{1}N^{\gamma/2}x^{-\gamma}.$

###### Proof.

For fixed $n\geq 1$, under ${\rm P}_{\omega}$,

$\mathbb{U}_{n}\overset{{\rm d}}{=}\sum_{k=1}^{U_{S_{n}-1}}G^{(n)}_{k},$

where $G^{(n)}_{k}$ are random variables with law $Geo(\lambda_{n})$,
independent of $U_{S_{n}-1}$ and each other. In particular,

${\rm E}_{\omega}G_{k}^{(n)}=\rho_{n},\quad{\rm
Var}_{\omega}G_{k}^{(n)}=\rho_{n}+\rho_{n}^{2}.$

Since in generations $S_{n-1}+1,\dots S_{n}-1$ the process evolves with
offspring distribution $Geo(1/2)$, standard calculation gives

${\rm E}_{\omega}[U_{S_{n}-1}|\mathbb{U}_{n-1}]=\mathbb{U}_{n-1}\quad{\rm
and}\quad{\rm
Var}_{\omega}(U_{S_{n}-1}|\mathbb{U}_{n-1})=2(\xi_{n}-1)\mathbb{U}_{n-1}.$

This in turn implies

(3.15) $\begin{split}{\rm
E}_{\omega}[\mathbb{U}_{n}|\mathbb{U}_{n-1}]&=\rho_{n}\mathbb{U}_{n-1},\\\
{\rm
E}_{\omega}[(\mathbb{U}_{n}-\rho_{n}\mathbb{U}_{n-1})^{2}|\mathbb{U}_{n-1}]&=(\rho_{n}-\rho_{n}^{2}+2\rho_{n}^{2}\xi_{n})\mathbb{U}_{n-1}.\end{split}$

In particular ${\rm E}_{\omega}\mathbb{U}_{n}=\Pi_{1,n}$.

Observe that the processes $U^{k}$ evolve without immigration and the
extinction time of each $U^{k}$ is stochastically dominated by $\tau_{1}$,
which is finite $\mathbb{P}$-a.s. by Lemma 3.1. In particular, with
probability $1$ the series

$\sum_{k=1}^{N}\sum_{n\geq 0}\mathbb{U}_{n}^{k}$

is indeed a finite sum. Recall the sequence ${\bar{R}}$ defined in (3.3) and
observe that, by (3.4),

$\begin{split}\sum_{k=1}^{N}\sum_{n\geq
0}\mathbb{U}_{n}^{k}&=\sum_{k=1}^{N}\sum_{n\geq
0}\mathbb{U}_{n}^{k}({\bar{R}}_{n+1}-\rho_{n+1}{\bar{R}}_{n+2})\\\
&=\sum_{n\geq
1}\left(\sum_{k=1}^{N}(\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1})\right){\bar{R}}_{n+1}+N{\bar{R}}_{1}\end{split}$

and thus

$\sum_{n\geq
0}\left(\sum_{k=1}^{N}\mathbb{U}_{n}^{k}-N\Pi_{1,n}\right)=\sum_{n\geq
1}\left(\sum_{k=1}^{N}(\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1})\right){\bar{R}}_{n+1}.$

Therefore

$\begin{split}\mathbb{P}\left[\sum_{n\geq
0}\left|\sum_{k=1}^{N}\mathbb{U}_{n}^{k}-N\Pi_{1,n}\right|>x\right]&\leq\mathbb{P}\left[\sum_{n\geq
1}\left|\sum_{k=1}^{N}\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1}\right|{\bar{R}}_{n+1}>x\right]\end{split}$

and

$\begin{split}\mathbb{P}\left[\sum_{k=1}^{N}\sum_{n\geq
1}\mathbb{U}_{n}^{k}>x\right]&\leq\mathbb{P}\left[\sum_{n\geq
1}\left|\sum_{k=1}^{N}(\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1})\right|{\bar{R}}_{n+1}>x/2\right]+\mathbb{P}[N{\bar{R}}_{1}>x/2].\end{split}$

Observe that for any $n\geq 1$, ${\bar{R}}_{n+1}$ is independent of
$(\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1})$. Thus for any $x>0$,

$\mathbb{P}\left[\sum_{n\geq
1}\left|\sum_{k=1}^{N}(\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1})\right|{\bar{R}}_{n+1}>x\right]\leq\sum_{n\geq
1}\mathbb{P}\left[\left|\sum_{k=1}^{N}(\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1})\right|{\bar{R}}_{n+1}>x/2n^{2}\right]\\\
\begin{split}&=\sum_{n\geq 1}\int_{[0,\infty)}{\rm
P}[{\bar{R}}_{n+1}>x/2tn^{2}]\mathbb{P}\left[\left|\sum_{k=1}^{N}(\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1})\right|\in
dt\right]\\\ &\leq C_{\gamma}\,\sum_{n\geq
1}\int_{[0,\infty)}(x/2tn^{2})^{-\gamma}\mathbb{P}\left[\left|\sum_{k=1}^{N}(\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1})\right|\in
dt\right]\\\ &=2^{\gamma}C_{\gamma}\,x^{-\gamma}\sum_{n\geq
1}n^{2\gamma}\mathbb{E}\left|\sum_{k=1}^{N}(\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1})\right|^{\gamma},\end{split}$

where the second inequality follows from (3.5).

The relations (3.15) imply that for any fixed $n$, under ${\rm P}_{\omega}$,
$\sum_{k=1}^{N}(\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1})$ is a sum of
independent centered variables; in particular, using formulae (3.15), we
obtain

$\begin{split}{\rm
E}_{\omega}\left(\sum_{k=1}^{N}(\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1})\right)^{2}&=N{\rm
E}_{\omega}(\mathbb{U}_{n}-\rho_{n}\mathbb{U}_{n-1})^{2}\\\
&=N(\rho_{n}+2\rho_{n}^{2}\xi_{n}-\rho_{n}^{2}){\rm
E}_{\omega}\mathbb{U}_{n-1}\\\
&=N(\rho_{n}+2\rho_{n}^{2}\xi_{n}-\rho_{n}^{2})\Pi_{1,n-1}.\end{split}$

Therefore, conditional Jensen’s inequality and subadditivity of the function
$x\mapsto x^{\gamma/2}$ (recall $\gamma\leq 2$) give

$\begin{split}\sum_{n\geq
1}n^{2\gamma}\mathbb{E}\left|\sum_{k=1}^{N}(\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1})\right|^{\gamma}&\leq\sum_{n\geq
1}n^{2\gamma}{\rm E}\left({\rm
E}_{\omega}\left(\sum_{k=1}^{N}(\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1})\right)^{2}\right)^{\gamma/2}\\\
&=N^{\gamma/2}\sum_{n\geq 1}n^{2\gamma}{\rm
E}((\rho_{n}+2\rho_{n}^{2}\xi_{n}-\rho_{n}^{2})\Pi_{1,n-1})^{\gamma/2}\\\
&\leq N^{\gamma/2}\sum_{n\geq 1}n^{2\gamma}({\rm E}\rho^{\gamma/2}+2{\rm
E}\rho^{\gamma}\xi^{\gamma/2})({\rm E}\rho^{\gamma/2})^{n-1}.\end{split}$

The assumptions of the lemma guarantee that the series is convergent and thus
for some constant $C>0$,

$\begin{split}\mathbb{P}\left[\sum_{n\geq
1}\left|\sum_{k=1}^{N}(\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1})\right|{\bar{R}}_{n+1}>x\right]&\leq
2^{\gamma}C_{\gamma}\,x^{-\gamma}\sum_{n\geq
1}n^{2\gamma}\mathbb{E}\left|\sum_{k=1}^{N}(\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1})\right|^{\gamma}\\\
&\leq CN^{\gamma/2}x^{-\gamma},\end{split}$

which proves (3.12). Invoking (3.5) once again, we conclude that

$\begin{split}\mathbb{P}\left[\sum_{k=1}^{N}\sum_{n\geq
1}\mathbb{U}_{n}^{k}>x\right]&\leq\mathbb{P}\left[\sum_{n\geq
1}\left(\sum_{k=1}^{N}(\mathbb{U}_{n}^{k}-\rho_{n}\mathbb{U}^{k}_{n-1})\right){\bar{R}}_{n+1}>x/2\right]+\mathbb{P}[N{\bar{R}}_{1}>x/2]\\\
&\leq
CN^{\gamma/2}(x/2)^{-\gamma}+C_{\gamma}N^{\gamma}(x/2)^{-\gamma},\end{split}$

which proves (3.11).

To show (3.13), decompose

$\begin{split}\mathbb{P}\left[\max_{j\geq
0}\sum_{k=1}^{N}U^{k}_{n}>x\right]&=\mathbb{P}\left[\max_{n\geq
0}\max_{S_{n}\leq j<S_{n+1}}\sum_{k=1}^{N}U^{k}_{j}>x\right]\\\
&\leq\mathbb{P}\left[\sum_{n\geq 0}\sum_{k=1}^{N}\max_{S_{n}\leq
j<S_{n+1}}U^{k}_{j}>x\right]\\\ &\leq\mathbb{P}\left[\sum_{n\geq
0}\sum_{k=1}^{N}\left(\mathbb{U}^{k}_{n}+\max_{S_{n}\leq
j<S_{n+1}}|U^{k}_{j}-\mathbb{U}^{k}_{n}|\right)>x\right]\\\
&\leq\mathbb{P}\left[\sum_{k=1}^{N}\sum_{n\geq
0}\mathbb{U}^{k}_{n}>x/2\right]+\mathbb{P}\left[\sum_{n\geq
1}\sum_{k=1}^{N}\max_{S_{n-1}\leq
j<S_{n}}|U^{k}_{j}-\mathbb{U}^{k}_{n-1}|>x/2\right],\end{split}$

which means that (3.13) follows from (3.11) and (3.14). To show (3.14), note
that, by Lemma 3.3,

${\rm E}_{\omega}\left[\max_{S_{n-1}\leq
j<S_{n}}|U_{j}-\mathbb{U}_{n-1}|^{2}\right]\leq 8\xi_{n}{\rm
E}_{\omega}\mathbb{U}^{k}_{n-1}=8\xi_{n}\Pi_{1,n-1}.$

Therefore

$\mathbb{P}\left[\sum_{n\geq 1}\sum_{k=1}^{N}\max_{S_{n-1}\leq
j<S_{n}}|U^{k}_{j}-\mathbb{U}^{k}_{n-1}|>x/2\right]\leq\sum_{n\geq
1}\mathbb{P}\left[\sum_{k=1}^{N}\max_{S_{n-1}\leq
j<S_{n}}|U^{k}_{j}-\mathbb{U}^{k}_{n-1}|>x/4n^{2}\right]\\\
\begin{split}\leq&\sum_{n\geq 1}(x/4n^{2})^{-\gamma}N^{\gamma/2}{\rm
E}\left({\rm E}_{\omega}\max_{S_{n-1}\leq
j<S_{n}}|U_{j}-\mathbb{U}_{n-1}|^{2}\right)^{\gamma/2}\\\
\leq&N^{\gamma/2}x^{-\gamma}\sum_{n\geq 1}(4n)^{2\gamma}8^{\gamma/2}{\rm
E}\xi^{\gamma/2}({\rm E}\rho^{\gamma/2})^{n-1}\\\
=&C^{\prime}N^{\gamma/2}x^{-\gamma},\end{split}$

for some constant $C^{\prime}>0$, which proves (3.13) and (3.14). ∎

Let $Y=(Y_{n})_{n\in\mathbb{N}}$ be a copy of the process
$(Y^{1}_{n})_{n\in\mathbb{N}}$. That is, $Y$ starts with one immigrant in
generation $0$ and for the next $\xi_{1}-1$ generations evolves as a Galton-
Watson process with unit immigration and reproduction law $Geo(1/2)$. The last
immigrant arrives in generation $\xi_{1}-1$; particles there reproduce with
distribution $Geo(\lambda_{1})$, giving birth to the first marked generation
$\mathbb{Y}_{1}=Y_{S_{1}}$. From there on the process evolves without
immigration, with particles in each marked generation
$\mathbb{Y}_{n}=Y_{S_{n}}$ being born with $Geo(\lambda_{n})$ distribution,
and $Geo(1/2)$ in consecutive blocks of lengths given by $\xi_{n}-1$ for
$n\geq 2$.

$\mathbb{Y}_{1}$$\xi_{1}$$\mathbb{Y}_{2}$$\xi_{2}$$\mathbb{Y}_{3}$$\xi_{3}$$\mathbb{Y}_{4}$$\xi_{4}$$\mathbb{Y}_{5}$$\xi_{5}$
Figure 3.3. Schematic picture of the process $Y$. Horizontal blue lines
represent marked generations. The immigrants arrive only in the first block.

######  Lemma 3.5.

Assume $(\Gamma)$. Then for some constant $C_{2}$,

(3.16) $\mathbb{P}\left[\max_{n\geq 1}Y_{n}>x\right]\leq
C_{2}x^{-\gamma}\left({\rm E}\left({\rm
E}_{\omega}Y_{\xi_{1}-1}^{2}\right)^{\gamma/2}+\mathbb{E}\mathbb{Y}_{1}^{\gamma}\right).$

If additionally ${\rm E}\xi^{\gamma}<\infty$ and ${\rm
E}\xi^{\gamma}\rho^{\gamma}<\infty$, then for some constant $C_{3}$,

(3.17) $\mathbb{P}\left[\max_{n\geq 1}Y_{n}>x\right]\leq C_{3}x^{-\gamma}.$

###### Proof.

We have

(3.18) $\mathbb{P}\left[\max_{n\geq
1}Y_{n}>x\right]\leq\mathbb{P}\left[\max_{n<S_{1}}Y_{n}>x\right]+\mathbb{P}\left[\max_{n\geq
S_{1}}Y_{n}>x\right].$

For the first $\xi_{1}-1$ generations $Y$ evolves as a Galton-Watson process
with unit immigration and reproduction law $Geo(1/2)$, therefore
$(Y_{n}^{2})_{n<S_{1}}$ is a submartingale with respect to ${\rm P}_{\omega}$.
Using first Markov’s, then Jensen’s, and finally Doob’s maximal inequality, we
obtain

$\mathbb{P}\left[\max_{j<S_{1}}Y_{j}>x\right]\leq
x^{-\gamma}\mathbb{E}\left(\max_{j<S_{1}}Y_{j}\right)^{\gamma}\leq
x^{-\gamma}{\rm E}\left({\rm
E}_{\omega}\max_{n<\xi_{1}}Y_{n}^{2}\right)^{\gamma/2}\leq x^{-\gamma}{\rm
E}\left(4{\rm E}_{\omega}Y_{\xi_{1}-1}^{2}\right)^{\gamma/2}.$

If additionally ${\rm E}\xi^{\gamma}<\infty$, then by Lemma 3.3,

${\rm E}_{\omega}\max_{n<\xi_{1}}Y_{n}^{2}\leq 16\xi_{1}^{2},$

thus

$\mathbb{P}\left[\max_{j<S_{1}}Y_{j}>x\right]\leq 16^{\gamma/2}{\rm
E}\xi^{\gamma}x^{-\gamma}.$

To estimate the second term in (3.18), observe that

$\left(Y_{S_{1}+j}\right)_{j\in\mathbb{N}}\overset{{\rm
d}}{=}\left(\sum_{k=1}^{\mathbb{Y}_{1}}U_{j}^{k}\right)_{j\in\mathbb{N}},$

where $U^{k}$’s are (independent under ${\rm P}_{\omega}$) copies of the
process $U$, independent of $\mathbb{Y}_{1}$ under $\mathbb{P}$. By Lemma 3.4,

$\mathbb{P}\left[\max_{n\geq S_{1}}Y_{n}>x\right]\leq
C_{1}\mathbb{E}\mathbb{Y}_{1}^{\gamma}x^{-\gamma},$

which concludes the proof of the first part of the lemma. If ${\rm
E}\xi^{\gamma}\rho^{\gamma}<\infty$, we may estimate
$\mathbb{E}\mathbb{Y}_{1}^{\gamma}$. Under ${\rm P}_{\omega}$,

$\mathbb{Y}_{1}\overset{{\rm d}}{=}\sum_{k=1}^{Y_{\xi_{1}-1}+1}G_{k},$

where $G_{k}\sim Geo(\lambda_{1})$ are independent of $Y_{\xi_{1}-1}$ and each
other. Moreover, as was explained in the proof of Lemma 3.3,
$Y_{\xi_{1}-1}\sim Geo(1/\xi_{1})$ under ${\rm P}_{\omega}$. Therefore

${\rm E}_{\omega}\mathbb{Y}_{1}^{2}={\rm
E}_{\omega}\left[(Y_{\xi_{1}-1}+1)(2\rho_{1}^{2}+\rho_{1})+(Y_{\xi_{1}-1}^{2}+Y_{\xi_{1}-1})\rho_{1}^{2}\right]=2\xi_{1}^{2}\rho_{1}^{2}+\xi_{1}\rho_{1}.$

Jensen’s inequality and subadditivity of function $x\mapsto x^{\gamma/2}$ give

$\mathbb{E}\mathbb{Y}_{1}^{\gamma}\leq{\rm E}\left({\rm
E}_{\omega}\mathbb{Y}_{1}^{2}\right)^{\gamma/2}\leq 2^{\gamma/2}{\rm
E}\xi^{\gamma}\rho^{\gamma}+{\rm E}\xi^{\gamma/2}\rho^{\gamma/2}<\infty,$

which proves (3.17).

∎

##  4\. Proof of Theorem 2.1

In the proof of Theorem 2.1 we will use the fact that the extinctions divide
process $Z$ into independent epochs. That is, we first determine tail
asymptotics of the maximum up to time $S_{\tau_{1}}$.

For any $A>0$ denote $\sigma=\sigma(A)=\inf\\{n:{\mathbb{Z}}_{n}\geq A\\}$.
The next lemma is an analogue of Lemma 4 in [13] and can be proved the very
same way, that is by examining ${\rm
E}_{\omega}[{\mathbb{Z}}_{k}^{\alpha}|{\mathbb{Z}}_{k-1}]$ using methods we’ve
seen in previous proofs.

######  Lemma 4.1.

For any fixed $A>0$,
$0<\mathbb{E}[{\mathbb{Z}}_{\sigma}^{\alpha}\operatorname{\mathbbm{1}}_{\sigma<\tau_{1}}]<\infty$.

The main proof strategy is as follows: we choose sufficiently big $A$ and
argue that neither the particles living before time $S_{\sigma}$, nor the
descendants of the immigrants arriving after this time contribute
significantly to the examined maximum. Therefore its behavior is determined by
${\mathbb{Z}}_{\sigma}$ particles in the generation $S_{\sigma}$ and their
progeny.

Let us first take care of the particles alive before time $S_{\sigma}$.

######  Lemma 4.2.

For any fixed $A$,

$\mathbb{P}\left[\max_{n<S_{\sigma}\wedge
S_{\tau_{1}}}Z_{n}>x\right]=o(x^{-\alpha}).$

###### Proof.

Fix $A$ and let $x>A$. The only generations before time $S_{\sigma}$ in which
the population size may exceed $x$ are the unmarked ones. However, since
${\mathbb{Z}}_{k}<A$ for $k<\sigma$, the maximum of $Z$ in generations
$S_{k-1}+1,\dots S_{k}-1$ is stochastically dominated by $M_{k}^{A}$, the
maximum of Galton-Watson process with $Geo(1/2)$ offspring distribution, unit
immigration and $A$ initial particles, evolving for time $\xi_{k}$. Observe
that

$\begin{split}\mathbb{P}\left[\max_{n<S_{\sigma}\wedge
S_{\tau_{1}}}Z_{n}>x\right]&\leq\mathbb{P}\left[\max_{k<x^{\delta/2}}M_{k}^{A}>x\right]+\mathbb{P}\left[\tau_{1}>x^{\delta/2}\right]\\\
&\leq
x^{\delta/2}\mathbb{P}\left[M_{1}^{A}>x\right]+\mathbb{P}\left[\tau_{1}>x^{\delta/2}\right].\end{split}$

Since $\alpha+\delta\leq 2$, by Markov’s and Jensen’s inequalities,

$\mathbb{P}\left[M_{1}^{A}>x\right]\leq x^{-\alpha-\delta}{\rm E}\left({\rm
E}_{\omega}(M_{1}^{A})^{2}\right)^{(\alpha+\delta)/2}.$

Lemma 3.3 implies that

${\rm E}_{\omega}(M_{1}^{A})^{2}\leq 16(\xi_{1}^{2}+A\xi_{1}+A^{2})$

and thus, since $x\mapsto x^{(\alpha+\delta)/2}$ is subadditive,

$x^{\delta/2}\mathbb{P}\left[M_{1}^{A}>x\right]\leq
x^{-\alpha-\delta/2}16^{(\alpha+\delta)/2}\left({\rm
E}\xi^{\alpha+\delta}+A^{(\alpha+\delta)/2}{\rm
E}\xi^{(\alpha+\delta)/2}+A^{\alpha+\delta}\right)=o(x^{-\alpha}).$

The second term may be bounded using Lemma 3.1, that is

$\mathbb{P}\left[\tau_{1}>x^{\delta/2}\right]\leq
e^{-cx^{\delta/2}}\mathbb{E}e^{c\tau_{1}}=o(x^{-\alpha}),$

which ends the proof.

∎

The next lemma assures that the contribution of progeny of immigrants arriving
after $S_{\sigma}$ is negligible. Recall that $Y^{k}$ counts the progeny of
immigrants arriving in $k$’th block, that is in generations
$S_{k-1},S_{k-1}+1,\dots S_{k}-1$.

######  Lemma 4.3.

Fix $\varepsilon>0$. There exists $A_{1}(\varepsilon)$ such that for
$A>A_{1}(\varepsilon)$,

(4.1) $\mathbb{P}\left[\sum_{k=\sigma+1}^{\tau_{1}}\max_{n\geq
1}Y^{k}_{n}>\varepsilon x\right]\leq\varepsilon x^{-\alpha}.$

###### Proof.

We have

$\begin{split}\mathbb{P}\left[\sum_{k=\sigma+1}^{\tau_{1}}\max_{n\geq
1}Y^{k}_{n}>\varepsilon
x\right]&=\mathbb{P}\left[\sum_{k=1}^{\infty}\operatorname{\mathbbm{1}}_{\sigma\leq
k<\tau_{1}}\max_{n\geq 1}Y^{k+1}_{n}>\varepsilon x\right]\\\
&\leq\sum_{k=1}^{\infty}\mathbb{P}\left[\sigma\leq k<\tau_{1},\max_{n\geq
1}Y^{k+1}_{n}>\varepsilon x/2k^{2}\right].\end{split}$

Observe that the event $\\{\sigma\leq k<\tau_{1}\\}$ is defined in terms of
$Z_{1},\dots Z_{S_{k}}$, while the process $Y^{k+1}$ evolves in the
environment given by $(\xi_{j},\rho_{j})$ for $j\geq k+1$, hence is
independent of $Z_{1},\dots Z_{S_{k}}$. Moreover, the second part of Lemma 3.5
applied with $\gamma=\alpha$ gives tail bounds on the maximum of $Y^{k+1}$.
That is,

$\begin{split}\sum_{k=1}^{\infty}\mathbb{P}\left[\sigma\leq
k<\tau_{1},\max_{n\geq 1}Y^{k+1}_{n}>\varepsilon
x/2k^{2}\right]&=\sum_{k=1}^{\infty}\mathbb{P}\left[\sigma\leq
k<\tau_{1}\right]\mathbb{P}\left[\max_{n\geq 1}Y^{k+1}_{n}>\varepsilon
x/2k^{2}\right]\\\ &\leq C_{3}\sum_{k=1}^{\infty}\mathbb{P}\left[\sigma\leq
k<\tau_{1}\right](\varepsilon x/2k^{2})^{-\alpha}\\\
&=C_{3}2^{\alpha}(\varepsilon
x)^{-\alpha}\sum_{k=1}^{\infty}k^{2\alpha}\mathbb{P}\left[\tau_{1}\operatorname{\mathbbm{1}}_{\sigma<\tau_{1}}>k\right]\\\
&=C_{3}2^{\alpha}(2\alpha+1)^{-1}\varepsilon^{-\alpha}x^{-\alpha}\mathbb{E}\left[\tau_{1}^{2\alpha+1}\operatorname{\mathbbm{1}}_{\sigma<\tau_{1}}\right]\end{split}$

Since $\mathbb{E}\tau_{1}^{2\alpha+1}<\infty$ and
$\sigma(A)\overset{\mathbb{P}}{\longrightarrow}\infty$ as $A\to\infty$, one
may find $A_{1}(\varepsilon)$ such that for $A>A_{1}(\varepsilon)$ (4.1)
holds.

∎

We already gave bounds on the generations sizes of particles alive before time
$S_{\sigma}$ and those coming from immigrants arriving after that time. What
is left is investigating behaviour of the particles residing exactly in
generation $S_{\sigma}$ and their progeny.

For $k\geq S_{\sigma}$ let $V_{\sigma,k}$ be the number of progeny of the
particles from generation $S_{\sigma}$ residing in generation $k$ and let
${\mathbb{V}}_{\sigma,n}=V_{\sigma,S_{n}}$; in particular,
${\mathbb{Z}}_{\sigma}={\mathbb{V}}_{\sigma,\sigma}$. Recall the variables
$\Psi_{m,k}$ defined in (3.6).

######  Lemma 4.4.

For any $\varepsilon>0$ there exists $A_{2}(\varepsilon)$ such that for
$A>A_{2}(\varepsilon)$,

$\mathbb{P}\left[\left|\max_{k\geq
S_{\sigma}}(V_{\sigma,k}+V_{\sigma,k+1})-{\mathbb{Z}}_{\sigma}\max_{k\geq
S_{\sigma}}(\Psi_{\sigma+1,k}+\Psi_{\sigma+1,k+1})\right|>\varepsilon
x,\sigma<\tau_{1}\right]\leq\varepsilon
x^{-\alpha}\mathbb{E}\left[{\mathbb{Z}}_{\sigma}^{\alpha}\operatorname{\mathbbm{1}}_{\sigma<\tau_{1}}\right].$

###### Proof.

We begin by estimating the difference of maxima within one block. Observe that
the potential $\Psi$ is constant within each block, therefore for any
$n\in\mathbb{N}$,

$\begin{split}&\left|\max_{S_{n}\leq
k<S_{n+1}}(V_{\sigma,k}+V_{\sigma,k+1})-{\mathbb{Z}}_{\sigma}\max_{S_{n}\leq
k<S_{n+1}}(\Psi_{\sigma+1,k}+\Psi_{\sigma+1,k+1})\right|\\\
&\leq\left|\max_{S_{n}\leq
k<S_{n+1}-1}(V_{\sigma,k}+V_{\sigma,k+1})-2{\mathbb{Z}}_{\sigma}\Pi_{\sigma+1,n}\right|\\\
&+|V_{\sigma,S_{n+1}-1}+V_{\sigma,S_{n+1}}-{\mathbb{Z}}_{\sigma}\Pi_{\sigma+1,n}-{\mathbb{Z}}_{\sigma}\Pi_{\sigma+1,n+1}|\end{split}$

Let us estimate the first ingredient. Since

$\max_{S_{n}\leq
k<S_{n+1}-1}(V_{\sigma,k}+V_{\sigma,k+1})=2{\mathbb{V}}_{\sigma,n}+\max_{S_{n}\leq
k<S_{n+1}-1}\left(V_{\sigma,k}+V_{\sigma,k+1}-2{\mathbb{V}}_{\sigma,n}\right),$

we have

$\begin{split}\left|\max_{S_{n}\leq
k<S_{n+1}-1}(V_{\sigma,k}+V_{\sigma,k+1})-2{\mathbb{Z}}_{\sigma}\Pi_{\sigma+1,n}\right|&\leq
2\left(\left|{\mathbb{V}}_{\sigma,n}-{\mathbb{Z}}_{\sigma}\Pi_{\sigma+1,n}\right|+\max_{S_{n}\leq
k<S_{n+1}}|V_{\sigma,k}-{\mathbb{V}}_{\sigma,n}|\right).\end{split}$

The second ingredient may be estimated simply by

$\begin{split}&|V_{\sigma,S_{n+1}-1}+V_{\sigma,S_{n+1}}-{\mathbb{Z}}_{\sigma}\Pi_{\sigma+1,n}-{\mathbb{Z}}_{\sigma}\Pi_{\sigma+1,n+1}|\\\
&\leq|{\mathbb{V}}_{\sigma,n+1}-{\mathbb{Z}}_{\sigma}\Pi_{\sigma+1,n+1}|+|{\mathbb{V}}_{\sigma,n}-{\mathbb{Z}}_{\sigma}\Pi_{\sigma+1,n}|+|V_{\sigma,S_{n+1}-1}-{\mathbb{V}}_{\sigma,n}|,\end{split}$

which gives

$\begin{split}&\left|\max_{S_{n}\leq
k<S_{n+1}}(V_{\sigma,k}+V_{\sigma,k+1})-{\mathbb{Z}}_{\sigma}\max_{S_{n}\leq
k<S_{n+1}}(\Psi_{\sigma+1,k}+\Psi_{\sigma+1,k+1})\right|\\\ &\leq
3|{\mathbb{V}}_{\sigma,n}-{\mathbb{Z}}_{\sigma}\Pi_{\sigma+1,n}|+3\max_{S_{n}\leq
k<S_{n+1}}|V_{\sigma,k}-{\mathbb{V}}_{\sigma,n}|+|{\mathbb{V}}_{\sigma,n+1}-{\mathbb{Z}}_{\sigma}\Pi_{\sigma+1,n+1}|.\end{split}$

Next, in view of

$\begin{split}&\left|\max_{k\geq
S_{\sigma}}(V_{\sigma,k}+V_{\sigma,k+1})-{\mathbb{Z}}_{\sigma}\max_{k\geq
S_{\sigma}}(\Psi_{\sigma+1,k}+\Psi_{\sigma+1,k+1})\right|\\\
&=\left|\max_{n\geq\sigma}\max_{S_{n}\leq
k<S_{n+1}}(V_{\sigma,k}+V_{\sigma,k+1})-\max_{n\geq\sigma}{\mathbb{Z}}_{\sigma}\max_{S_{n}\leq
k<S_{n+1}}(\Psi_{\sigma+1,k}+\Psi_{\sigma+1,k+1})\right|\\\
&\leq\sum_{n\geq\sigma}\left|\max_{S_{n}\leq
k<S_{n+1}}(V_{\sigma,k}+V_{\sigma,k+1})-{\mathbb{Z}}_{\sigma}\max_{S_{n}\leq
k<S_{n+1}}(\Psi_{\sigma+1,k}+\Psi_{\sigma+1,k+1})\right|,\end{split}$

the above estimations give

$\begin{split}\mathbb{P}&\left[\left|\max_{k\geq
S_{\sigma}}(V_{\sigma,k}+V_{\sigma,k+1})-{\mathbb{Z}}_{\sigma}\max_{k\geq
S_{\sigma}}(\Psi_{\sigma+1,k}+\Psi_{\sigma+1,k+1})\right|>\varepsilon
x,\sigma<\tau_{1}\right]\\\
&\leq\mathbb{P}\left[4\sum_{n\geq\sigma}\left|{\mathbb{V}}_{\sigma,n}-{\mathbb{Z}}_{\sigma}\Pi_{\sigma+1,n}\right|>\varepsilon
x/2,\sigma<\tau_{1}\right]\\\
&+\mathbb{P}\left[3\sum_{n\geq\sigma}\max_{S_{n}\leq
k<S_{n+1}}|V_{\sigma,k}-{\mathbb{V}}_{\sigma,n}|>\varepsilon
x/2,\sigma<\tau_{1}\right].\end{split}$

Both ingredients can be estimated by Lemma 3.4 applied with $\gamma=\alpha$.
Conditioned on $(\sigma,Z_{1},\dots Z_{S_{\sigma}})$, the process
$(V_{\sigma,n})_{n\geq S_{\sigma}}$ is a sum of ${\mathbb{Z}}_{\sigma}$
independent copies of the process $U$. We have, on the set
$\\{\sigma<\tau_{1}\\}$,

$\begin{split}\mathbb{P}&\left[4\sum_{n\geq\sigma}\left|{\mathbb{V}}_{\sigma,n}-{\mathbb{Z}}_{\sigma}\Pi_{\sigma+1,n}\right|>\varepsilon
x/2\,\Bigg{|}\,\sigma,Z_{1},\dots Z_{S_{\sigma}}\right]\leq C_{1}(\varepsilon
x/8)^{-\alpha}{\mathbb{Z}}_{\sigma}^{\alpha/2},\end{split}$

which gives

$\begin{split}\mathbb{P}\left[4\sum_{n\geq\sigma}\left|{\mathbb{V}}_{\sigma,n}-{\mathbb{Z}}_{\sigma}\Pi_{\sigma+1,n}\right|>\varepsilon
x/2,\sigma<\tau_{1}\right]\leq C_{1}8^{\alpha}(\varepsilon
x)^{-\alpha}\mathbb{E}\left[{\mathbb{Z}}_{\sigma}^{\alpha/2}\operatorname{\mathbbm{1}}_{\sigma<\tau_{1}}\right].\end{split}$

Similarly,

$\mathbb{P}\left[3\sum_{n\geq\sigma}\max_{S_{n}<k<S_{n+1}}|V_{\sigma,k}-{\mathbb{V}}_{\sigma,n}|>\varepsilon
x/2,\sigma<\tau_{1}\right]\leq C_{1}6^{\alpha}(\varepsilon
x)^{-\alpha}\mathbb{E}\left[{\mathbb{Z}}_{\sigma}^{\alpha/2}\operatorname{\mathbbm{1}}_{\sigma<\tau_{1}}\right].$

Therefore, for some constant $C_{2}$,

$\mathbb{P}\left[\left|\max_{k\geq
S_{\sigma}}(V_{\sigma,k}+V_{\sigma,k+1})-{\mathbb{Z}}_{\sigma}\max_{k\geq
S_{\sigma}}(\Psi_{\sigma+1,k}+\Psi_{\sigma+1,k+1})\right|>\varepsilon
x,\sigma<\tau_{1}\right]\\\ \leq C_{2}(\varepsilon
x)^{-\alpha}\mathbb{E}\left[{\mathbb{Z}}_{\sigma}^{\alpha/2}\operatorname{\mathbbm{1}}_{\sigma<\tau_{1}}\right].$

Finally, for any fixed $\varepsilon>0$, since ${\mathbb{Z}}_{\sigma}\geq A$,
we have

$\mathbb{E}\left[{\mathbb{Z}}_{\sigma}^{\alpha/2}\operatorname{\mathbbm{1}}_{\sigma<\tau_{1}}\right]\leq
A^{-\alpha/2}\mathbb{E}\left[{\mathbb{Z}}_{\sigma}^{\alpha}\operatorname{\mathbbm{1}}_{\sigma<\tau_{1}}\right]$

and one may choose $A_{2}(\varepsilon)$ large enough for the claim to hold.

∎

######  Lemma 4.5.

There exists $c_{\Psi}\in(0,\infty)$ such that for any fixed $A>0$,

(4.2) $\mathbb{P}\left[{\mathbb{Z}}_{\sigma}\max_{k\geq
S_{\sigma}}(\Psi_{\sigma+1,k}+\Psi_{\sigma+1,k+1})>x,\sigma<\tau_{1}\right]\sim
c_{\Psi}\mathbb{E}\left[{\mathbb{Z}}_{\sigma}^{\alpha}\operatorname{\mathbbm{1}}_{\sigma<\tau_{1}}\right]x^{-\alpha}.$

###### Proof.

Since the sequence $\Psi_{\sigma+1,k}$ is constant on the blocks between
marked points, we have

$\max_{k\geq
S_{\sigma}}(\Psi_{\sigma+1,k}+\Psi_{\sigma+1,k+1})=\max_{n\geq\sigma}\left(2\operatorname{\mathbbm{1}}_{\xi_{n+1}>1}\vee(1+\rho_{n+1})\right)\Pi_{\sigma+1,n}.$

Observe that

$\log\left(\left(2\operatorname{\mathbbm{1}}_{\xi_{n+1}>1}\vee(1+\rho_{n+1})\right)\Pi_{1,n}\right)=\sum_{k=1}^{n}\log(\rho_{k})+\log(2\operatorname{\mathbbm{1}}_{\xi_{n+1}>1}\vee(1+\rho_{n+1}))$

is a perturbed random walk. By Theorem 1.3.8 in [12], assumptions $(A)$
guarantee that

$\mathbb{P}\left[\max_{n\geq
0}(2\operatorname{\mathbbm{1}}_{\xi_{n+1}>1}\vee(1+\rho_{n+1}))\Pi_{1,n}>x\right]\sim
c_{\Psi}x^{-\alpha}$

for a constant $c_{\Psi}\in(0,\infty)$ given by

$c_{\Psi}={\rm
E}(2^{\alpha}\operatorname{\mathbbm{1}}_{\xi_{1}>1}\vee(1+\rho_{1})^{\alpha}-\max_{n\geq
2}(2^{\alpha}\operatorname{\mathbbm{1}}_{\xi_{n+1}>1}\vee(1+\rho_{n+1})^{\alpha})\Pi_{1,n}^{\alpha})_{+}.$

Note that the variables
${\mathbb{Z}}_{\sigma}\operatorname{\mathbbm{1}}_{\sigma<\tau_{1}}$ and
$\max_{n\geq\sigma}(2\operatorname{\mathbbm{1}}_{\xi_{n+1}>1}\vee(1+\rho_{n+1}))\Pi_{\sigma+1,n}$
are independent under $\mathbb{P}$. Therefore, by Breiman’s lemma,

$\mathbb{P}\left[{\mathbb{Z}}_{\sigma}\max_{k\geq
S_{\sigma}}(\Psi_{\sigma+1,k}+\Psi_{\sigma+1,k+1})>x,\sigma<\tau_{1}\right]\\\
=\mathbb{P}\left[{\mathbb{Z}}_{\sigma}\operatorname{\mathbbm{1}}_{\sigma<\tau_{1}}\cdot\max_{n\geq\sigma}(2\operatorname{\mathbbm{1}}_{\xi_{n+1}>1}\vee(1+\rho_{n+1}))\Pi_{\sigma+1,n}>x\right]\sim\mathbb{E}\left[{\mathbb{Z}}_{\sigma}^{\alpha}\operatorname{\mathbbm{1}}_{\sigma<\tau_{1}}\right]c_{\Psi}x^{-\alpha}.$

∎

The rest of the proof is standard. First, all the lemmas proven so far allow
us to determine the asymptotics of the maximum in time $[0,S_{\tau_{1}})$.
Then we use the fact that the extinctions divide our process into independent
pieces.

######  Proposition 4.6.

For some constant $c_{M}>0$,

$\mathbb{P}\left[\max_{0\leq n<S_{\tau_{1}}}(Z_{n}+Z_{n+1})>x\right]\sim
c_{M}x^{-\alpha}.$

###### Proof.

Fix $\varepsilon>0$ and take
$A>A(\varepsilon):=\max\\{A_{1}(\varepsilon),A_{2}(\varepsilon)\\}$. First,
observe that

$\mathbb{P}\left[\max_{S_{\sigma}\leq
n<S_{\tau_{1}}}(Z_{n}+Z_{n+1})>x,\sigma<\tau_{1}\right]\leq\mathbb{P}\left[\max_{0\leq
n<S_{\tau_{1}}}(Z_{n}+Z_{n+1})>x\right]\\\
\leq\mathbb{P}\left[\max_{S_{\sigma}\leq
n<S_{\tau_{1}}}(Z_{n}+Z_{n+1})>x,\sigma<\tau_{1}\right]+\mathbb{P}\left[\max_{n<S_{\sigma}\wedge
S_{\tau_{1}}}(Z_{n}+Z_{n+1})>x\right].$

Lemma 4.2 ensures that for large enough $x$,

$\mathbb{P}\left[\max_{n<S_{\sigma}\wedge
S_{\tau_{1}}}(Z_{n}+Z_{n+1})>x\right]\leq\mathbb{P}\left[2\max_{n<S_{\sigma}\wedge
S_{\tau_{1}}}Z_{n}>x\right]\leq\varepsilon x^{-\alpha}.$

Recall that by $Y^{k}=(Y^{k}_{j})_{j\in{\mathbb{Z}}}$ we denoted the process
counting the progeny of immigrants arriving in $k$’th block, with the
convention $Y_{j}^{k}=0$ for $j<0$. For $n\geq S_{\sigma}$,

$Z_{n}=V_{\sigma,n}+\sum_{k=\sigma+1}^{\tau_{1}}Y^{k}_{n-S_{k-1}},$

thus

$\mathbb{P}\left[\max_{S_{\sigma}\leq
n<S_{\tau_{1}}}(V_{\sigma,n}+V_{\sigma,n+1})>x,\sigma<\tau_{1}\right]\leq\mathbb{P}\left[\max_{S_{\sigma}\leq
n<S_{\tau_{1}}}(Z_{n}+Z_{n+1})>x,\sigma<\tau_{1}\right]\\\
\leq\mathbb{P}\left[\max_{S_{\sigma}\leq
n<S_{\tau_{1}}}(V_{\sigma,n}+V_{\sigma,n+1})>(1-\varepsilon)x,\sigma<\tau_{1}\right]+\mathbb{P}\left[2\sum_{k=\sigma+1}^{\tau_{1}}\max_{n\geq
1}Y_{n}^{k}>\varepsilon x\right]$

and (4.1) ensures that

$\mathbb{P}\left[2\sum_{k=\sigma+1}^{\tau_{1}}\max_{n\geq
1}Y_{n}^{k}>\varepsilon x\right]\leq\varepsilon x^{-\alpha}.$

Finally,

$\begin{split}\mathbb{P}&\left[{\mathbb{Z}}_{\sigma}\max_{k\geq
S_{\sigma}}(\Psi_{\sigma,k}+\Psi_{\sigma,k+1})>(1+\varepsilon)x,\sigma<\tau_{1}\right]\\\
&-\mathbb{P}\left[\left|\max_{k\geq
S_{\sigma}}(V_{\sigma,k}+V_{\sigma,k+1})-{\mathbb{Z}}_{\sigma}\max_{k\geq
S_{\sigma}}(\Psi_{\sigma,k}+\Psi_{\sigma,k+1})\right|>\varepsilon
x,\sigma<\tau_{1}\right]\\\ &\leq\mathbb{P}\left[\max_{S_{\sigma}\leq
n<S_{\tau_{1}}}(V_{\sigma,n}+V_{\sigma,n+1})>x,\sigma<\tau_{1}\right]\\\
&\leq\mathbb{P}\left[{\mathbb{Z}}_{\sigma}\max_{k\geq
S_{\sigma}}(\Psi_{\sigma,k}+\Psi_{\sigma,k+1})>(1-\varepsilon)x,\sigma<\tau_{1}\right]\\\
&+\mathbb{P}\left[\left|\max_{k\geq
S_{\sigma}}(V_{\sigma,k}+V_{\sigma,k+1})-{\mathbb{Z}}_{\sigma}\max_{k\geq
S_{\sigma}}(\Psi_{\sigma,k}+\Psi_{\sigma,k+1})\right|>\varepsilon
x,\sigma<\tau_{1}\right],\end{split}$

and by Lemma 4.4,

$\mathbb{P}\left[\left|\max_{k\geq
S_{\sigma}}(V_{\sigma,k}+V_{\sigma,k+1})-{\mathbb{Z}}_{\sigma}\max_{k\geq
S_{\sigma}}(\Psi_{\sigma,k}+\Psi_{\sigma,k+1})\right|>\varepsilon
x,\sigma<\tau_{1}\right]\leq\varepsilon
x^{-\alpha}\mathbb{E}\left[{\mathbb{Z}}_{\sigma}^{\alpha}\operatorname{\mathbbm{1}}_{\sigma<\tau}\right].$

Putting things together and invoking Lemma 4.5 we get that for any
$\varepsilon>0$ such that $\varepsilon(1-\varepsilon)^{\alpha}<c_{\Psi}$ and
for any $A>A(\varepsilon)$,

$0<((1+\varepsilon)^{-\alpha}c_{\Psi}-\varepsilon)\mathbb{E}\left[{\mathbb{Z}}_{\sigma}^{\alpha}\operatorname{\mathbbm{1}}_{\sigma<\tau_{1}}\right]\\\
\leq\liminf_{x\to\infty}x^{\alpha}\mathbb{P}\left[\max_{0\leq
n<S_{\tau_{1}}}(Z_{n}+Z_{n+1})>x\right]\leq\limsup_{x\to\infty}x^{\alpha}\mathbb{P}\left[\max_{0\leq
n<S_{\tau_{1}}}(Z_{n}+Z_{n+1})>x\right]\\\
\leq((1-2\varepsilon)^{-\alpha}c_{\Psi}+\varepsilon)\mathbb{E}\left[{\mathbb{Z}}_{\sigma}^{\alpha}\operatorname{\mathbbm{1}}_{\sigma<\tau_{1}}\right]+2\varepsilon<\infty.$

Observe that this relation implies that both the limits

$\lim_{x\to\infty}x^{\alpha}\mathbb{P}\left[\max_{0\leq
n<S_{\tau_{1}}}(Z_{n}+Z_{n+1})>x\right]\quad\textnormal{and}\quad\lim_{A\to\infty}\mathbb{E}\left[{\mathbb{Z}}_{\sigma(A)}^{\alpha}\operatorname{\mathbbm{1}}_{\sigma(A)<\tau_{1}}\right]$

exist, are positive and satisfy

$\lim_{x\to\infty}x^{\alpha}\mathbb{P}\left[\max_{0\leq
n<S_{\tau_{1}}}(Z_{n}+Z_{n+1})>x\right]=c_{\Psi}\lim_{A\to\infty}\mathbb{E}\left[{\mathbb{Z}}_{\sigma(A)}^{\alpha}\operatorname{\mathbbm{1}}_{\sigma(A)<\tau_{1}}\right]=:c_{M}.$

∎

Due to Lemma 3.2 and the relation (3.2), the next result implies Theorem 2.1.

######  Theorem 4.7.

Under assumptions $(A)$,

$\mathbb{P}\left[n^{-1/\alpha}\max_{0\leq
k<S_{n}}(Z_{k}+Z_{k+1})>x\right]\xrightarrow{n\to\infty}1-\exp\left(-\frac{c_{M}}{\mathbb{E}\tau_{1}}x^{-\alpha}\right).$

###### Proof.

Since the extinctions divide the process $Z$ into independent epochs, an
immediate corollary of Proposition 4.6 is that

$\mathbb{P}\left[n^{-1/\alpha}\max_{0\leq
k<S_{\tau_{n}}}(Z_{k}+Z_{k+1})>x\right]\xrightarrow{n\to\infty}1-\exp(-c_{M}x^{-\alpha}).$

Lemma 3.1 implies that $\mathbb{E}\tau_{1}<\infty$. Therefore passing from the
maximum up to time $S_{\tau_{n}}$ to the maximum up to $S_{n}$ may be done
exactly as in the proof of Lemma 3.2. ∎

##  5\. Proof of Theorem 2.2

As we have seen in the proof of Theorem 2.1, the limiting behaviour of maxima
in case $(A)$ comes from the tail asymptotics of variable $M_{\Psi}$ defined
in (3.7). The assumption ${\rm E}\xi^{\alpha+\delta}<\infty$ implies that for
every $k$, $\max_{j<\xi_{k}}Y^{k}_{j}$ is negligible. In terms of the random
walk, this means that the time the walker spends in a block when crossing it
for the first time is negligible. As we will see, under assumptions $(B)$ it
is not; the maximal local time is obtained when the walker crosses a
particularly long block for the first time, by their visits to sites within
this block and potentially excursions to the left.

Consider a simple symmetric random walk on ${\mathbb{Z}}$ and denote by
$\bar{L}_{k}(n)$ the number of times the walk visits site $k$ before reaching
$n$. Consider $(\bar{L}_{s}(n))_{s\in[0,n]}$ being a piecewise linear
interpolation of $(\bar{L}_{k}(n))_{0\leq k\leq n}$. The Ray-Knight theorem
(see [9, Theorem 2.15]) states that

$\left(\frac{1}{n}\bar{L}_{n(1-t)}(n)\right)_{t\in[0,1]}\overset{{\rm
d}}{\longrightarrow}\left(B_{t}\right)_{t\in[0,1]}$

in $C[0,1]$ as $n\to\infty$, where $B$ is a squared Bessel process which may
be defined as

(5.1) $B_{t}=||W(t)||^{2},$

for $W(t)=(W_{1}(t),W_{2}(t))$ being a standard two-dimensional Brownian
motion with $W(0)=0$. By the continuous mapping theorem,

(5.2) $\left(\frac{1}{n}\max_{k\leq
n}\bar{L}_{k}(n),\frac{1}{n}\bar{L}_{0}(n)\right)\overset{{\rm
d}}{\longrightarrow}(M_{B},B(1)),$

where $M_{B}=\sup\\{B_{t}:t\in[0,1]\\}$.

With this at hand, we may inspect the maximal local time that the RWSRE
obtains when crossing a (long) block between marked points for the first time.
To this end, consider a walk starting at $0$ in the environment that has
marked points only on the non-positive half-line, and stop it when it reaches
point $N$. By Ray-Knight theorem, the limit of maximal local time in the
interval $[1,N]$, where the walk is symmetric, scaled by $N$, is $M_{B}$. As
we have seen in the proof of Theorem 2.1, the number of visits in the negative
half-line should be controlled by the number of visits to $1$ and the maxima
of the potential $\Psi$.

In the associated branching process, the steps of the walk during its first
crossing of a block between marked points are counted by the process $Y$.
Therefore our goal is to understand the growth of maximal generation in the
process $Y$ as the size of the first block – in which the immigrants arrive –
tends to infinity. To this end, for any $N\in\mathbb{N}$ let $Y^{(N)}$ be a
BPSRE evolving in an environment with fixed $\xi_{1}=N$ and such that the
immigrants arrive only in generations up to $N-1$’th.

######  Lemma 5.1.

Under assumptions $(B)$,

(5.3) $\frac{1}{N}\max_{k\geq 0}(Y^{(N)}_{k}+Y^{(N)}_{k+1})\overset{{\rm
d}}{\longrightarrow}M_{\infty}\quad\textnormal{as }N\to\infty,$

where $M_{\infty}\overset{{\rm d}}{=}\max(M_{B},B(1)M_{\Psi}/2)$ and
$M_{\Psi}$ is a copy of the variable defined in (3.7) independent of the
Bessel process $B$.

###### Proof.

To simplify the notation we shall write $Y$ instead of $Y^{(N)}$. Observe that
(5.2) and the duality between branching process and random walk imply

$\left(\frac{1}{N}\max_{k\leq
N-2}(Y_{k}+Y_{k+1}),\frac{1}{N}(Y_{N-1}+Y_{N-2})\right)\overset{{\rm
d}}{\longrightarrow}(M_{B},B(1)).$

However, since the particles in generation $N-1$ are children of those from
$N-2$’th and an immigrant, born with distribution $Geo(1/2)$, we have

$\mathbb{E}\left(Y_{N-1}-Y_{N-2}-1\right)^{2}=\mathbb{E}(Y_{N-1}-\mathbb{E}\left[Y_{N-1}\,|\,Y_{N-2}\right])^{2}=2(\mathbb{E}Y_{N-2}+1)=2(N-1),$

which, together with Chebyshev’s inequality, implies that
$(Y_{N-1}-Y_{N-2})/N\overset{\mathbb{P}}{\longrightarrow}0$ and thus

$\left(\frac{1}{N}\max_{k\leq
N-2}(Y_{k}+Y_{k+1}),\frac{Y_{N-1}}{N}\right)\overset{{\rm
d}}{\longrightarrow}(M_{B},B(1)/2).$

Moreover, the variables $Y_{k}$ for $k\leq N-1$ are independent of the
environment, in particular of $\Psi_{1,n},n\geq 0$.

From here on we proceed as in the proof of Lemma 4.4, to show that the maximum
in generations after $N-1$’th is comparable with $Y_{N-1}M_{\Psi}$. That is,
we use Lemma 3.4 applied with $\gamma=\beta$ to obtain, for some constant
$C>0$,

(5.4) $\mathbb{P}\left[\left|\max_{k\geq
N}(Y_{k}+Y_{k+1})-\mathbb{Y}_{1}\max_{k\geq
N}(\Psi_{2,k}+\Psi_{2,k+1})\right|>x\right]\leq
Cx^{-\beta}\mathbb{E}\mathbb{Y}_{1}^{\beta/2}$

for any $x>0$. The particles in the first marked generation $S_{1}=N$ are born
with distribution $Geo(\lambda_{1})$ from those counted by $Y_{{N-1}}$ and an
immigrant. Therefore we have ${\rm E}_{\omega}\mathbb{Y}_{1}=N\rho_{1}$, and
by Jensen’s inequality,

$\mathbb{E}\mathbb{Y}_{1}^{\beta/2}\leq N^{\beta/2}{\rm E}\rho^{\beta/2}.$

Moreover, we may calculate quenched moments of $\mathbb{Y}_{1}$ conditioned on
$Y_{N-1}$ to get an analogue of (3.15). We obtain

(5.5)
$\begin{split}\mathbb{E}\left|\mathbb{Y}_{1}-\rho_{1}Y_{N-1}\right|^{\beta}&\leq{\rm
E}\left({\rm
E}_{\omega}(\mathbb{Y}_{1}-\rho_{1}Y_{N-1})^{2}\right)^{\beta/2}\\\ &={\rm
E}\left(({\rm
E}_{\omega}Y_{N-1}(\rho_{1}^{2}+\rho_{1})+2\rho_{1}^{2}+\rho_{1}\right)^{\beta/2}\\\
&\leq(N^{\beta/2}+1)(2^{\beta/2}{\rm E}\rho^{\beta}+{\rm
E}\rho^{\beta/2}),\end{split}$

where the last inequality follows from subadditivity of $x\mapsto x^{\beta/2}$
and the fact that ${\rm E}_{\omega}Y_{N-1}=N-1$. Observe that $\max_{k\geq
N}(\Psi_{2,k}+\Psi_{2,k+1})\leq 2+M_{\Psi,2}$ and by (3.8), ${\rm
E}M_{\Psi}^{\beta}<\infty$. Therefore, since
$(Y_{N-1},\mathbb{Y}_{1},\rho_{1})$ is independent of $(\rho_{j})_{j\geq 2}$,
we have

(5.6) $\begin{split}\mathbb{P}&\left[\left|\mathbb{Y}_{1}\max_{k\geq
N}(\Psi_{2,k}+\Psi_{2,k+1})-\rho_{1}Y_{N-1}\max_{k\geq
N}(\Psi_{2,k}+\Psi_{2,k+1})\right|>x\right]\\\ &\leq x^{-\beta}{\rm
E}(2+M_{\Psi})^{\beta}\mathbb{E}|\mathbb{Y}_{1}-\rho_{1}Y_{N-1}|^{\beta}\leq
C^{\prime}x^{-\beta}(N^{\beta/2}+1)\end{split}$

for some constant $C^{\prime}>0$ and any $x>0$.

Observe that (5.4) and (5.6) imply that for any fixed $\varepsilon>0$,

$\begin{split}\mathbb{P}&\left[\left|\max_{k\geq
N}(Y_{k}+Y_{k+1})-Y_{N-1}\max_{k\geq
N}(\Psi_{1,k}+\Psi_{1,k+1})\right|>\varepsilon N\right]\\\
&\leq\mathbb{P}\left[\left|\max_{k\geq
N}(Y_{k}+Y_{k+1})-\mathbb{Y}_{1}\max_{k\geq
N}(\Psi_{2,k}+\Psi_{2,k+1})\right|>\varepsilon N/2\right]\\\
&+\mathbb{P}\left[\left|\mathbb{Y}_{1}\max_{k\geq
N}(\Psi_{2,k}+\Psi_{2,k+1})-\rho_{1}Y_{N-1}\max_{k\geq
N}(\Psi_{2,k}+\Psi_{2,k+1})\right|>\varepsilon N/2\right]\\\ &\leq(\varepsilon
N/2)^{-\beta}\left(CN^{\beta/2}{\rm
E}\rho^{\beta/2}+C^{\prime}(N^{\beta/2}+1)\right)=O(N^{-\beta/2}).\end{split}$

Finally, by (5.5), for any $\varepsilon>0$,

$\mathbb{P}\left[|\mathbb{Y}_{1}-\rho_{1}Y_{N-1}|>\varepsilon
N\right]\leq\varepsilon^{-\beta}(N^{-\beta/2}+N^{-\beta})(2^{\beta/2}{\rm
E}\rho^{\beta}+{\rm E}\rho^{\beta/2})=O(N^{-\beta/2}),$

therefore the weak limit of

$\frac{1}{N}\max_{k\geq 0}(Y_{k}+Y_{k+1})=\frac{1}{N}\max\left(\max_{k\leq
N-2}(Y_{k}+Y_{k+1}),Y_{N-1}+\mathbb{Y}_{1},\max_{k\geq
N}(Y_{k}+Y_{k+1})\right)$

is the same as that of

$\frac{1}{N}\max\left(\max_{k\leq
N-2}(Y_{k}+Y_{k+1}),Y_{N-1}(1+\rho_{1}),Y_{N-1}\max_{k\geq
N}(\Psi_{1,k}+\Psi_{1,k-1})\right)\\\ =\frac{1}{N}\max\left(\max_{k\leq
N-2}(Y_{k}+Y_{k+1}),Y_{N-1}M_{\Psi,1}\right)$

which is $\max(M_{B},B(1)M_{\Psi}/2)$ by the continuous mapping theorem. ∎

######  Remark 5.2.

Under assumptions $(B)$, $\mathbb{E}M_{\infty}^{\beta+\delta}<\infty$. Indeed,
by (5.1),

$M_{B}^{2}=\sup\left\\{\left(W_{1}(t)^{2}+W_{2}(t)^{2}\right)^{2}\,:\,t\in[0,1]\right\\},$

where $W_{1},W_{2}$ are independent one-dimensional Brownian motions. Doob’s
maximal inequality applied to $W_{1},W_{2}$ implies that
$\mathbb{E}M_{B}^{2}<\infty$. Since $\beta+\delta\leq 2$, it follows that
$\mathbb{E}M_{B}^{\beta+\delta}<\infty$. Moreover, by (3.8),
$\mathbb{E}M_{\Psi}^{\beta+\delta}<\infty$, and since $M_{\Psi}$ and $B$ are
independent, we have

$\mathbb{E}M_{\infty}^{\beta+\delta}\leq\mathbb{E}M_{B}^{\beta+\delta}\mathbb{E}(1+M_{\Psi}/2)^{\beta+\delta}<\infty.$

Recall that the process $Y^{k}$ counts the progeny of immigrants arriving in
$k$’th block. Since Lemma 5.1 suggests that the maximum of process $Y^{k}$
should be comparable with $\xi_{k}M_{\infty}$ when $\xi_{k}$ is large, we
begin the proof of Theorem 2.2 by distinguishing large blocks in the
environment. Recall the sequence $(a_{n})_{n\in\mathbb{N}}$ defined in (2.3).
Fix $\varepsilon>0$ and let

$I_{n,\varepsilon}=\\{k\leq n\,:\,\xi_{k}>\varepsilon a_{n}\\},\quad
I_{n,\varepsilon}^{c}=\\{k\leq n\,:\,\xi_{k}\leq\varepsilon a_{n}\\}.$

For fixed $n$ and $k\leq n$, we will say that $k$’th block is large if $k\in
I_{n,\varepsilon}$, and small otherwise.

It follows from the definition of the sequence $(a_{n})_{n\in\mathbb{N}}$ and
regular variation of the tails of $\xi$ that for any $x>0$,

(5.7) $n{\rm P}[\xi>xa_{n}]\to x^{-\beta},\quad n\to\infty.$

Therefore, by Proposition 3.21 in [15],

(5.8) $\sum_{k=1}^{n}\delta_{(\xi_{k}/a_{n},k/n)}\overset{{\rm
d}}{\longrightarrow}P_{\mu},$

where $P_{\mu}$ is a Poisson point process on $(0,\infty]\times[0,\infty)$
with intensity measure $d\mu(x,t)=\beta x^{-\beta-1}dxdt$. In particular, as
$n\to\infty$, the sequence of variables $|I_{n,\varepsilon}|$, which count the
number of large blocks, converges weakly to Poisson distribution with
parameter $\varepsilon^{-\beta}$.

We begin by showing that all the progeny of immigrants arriving in small
blocks is negligible.

######  Proposition 5.3.

There is a constant $C_{5}$ such that for any $\varepsilon>0$ and
$\bar{\varepsilon}>0$,

$\limsup_{n\to\infty}\mathbb{P}\left[\max_{j\geq 1}\sum_{k\in
I_{n,\varepsilon}^{c}}Y^{k}_{j-S_{k-1}}>\bar{\varepsilon}a_{n}\right]\leq
C_{5}\bar{\varepsilon}^{-\beta-\delta}\varepsilon^{\delta}.$

###### Proof.

We will use the fact that the extinction times divide our process into i.i.d.
pieces. Let

$\eta_{n}=\inf\\{k>0:\tau_{k}>n\\}.$

Since $\mathbb{E}\tau_{1}<\infty$ by Lemma 3.1, the strong law of large
numbers implies $\eta_{n}/n\to\eta:=1/\mathbb{E}\tau$ as $n\to\infty$,
$\mathbb{P}$-a.s. We have

$\begin{split}\mathbb{P}\left[\max_{j\geq 1}\sum_{k\in
I_{n,\varepsilon}^{c}}Y^{k}_{j-S_{k-1}}>\bar{\varepsilon}a_{n}\right]&\leq\mathbb{P}\left[\max_{j\geq
1}\sum_{k\leq\tau_{2n\eta}}Y^{k}_{j-S_{k-1}}\operatorname{\mathbbm{1}}_{\xi_{k}\leq\varepsilon
a_{n}}>\bar{\varepsilon}a_{n}\right]\\\
&+\mathbb{P}\left[|\eta-\eta_{n}/n|>\eta\right].\end{split}$

The second term tends to $0$ as $n\to\infty$. Since the extinctions divide our
process into i.i.d. pieces, we have

$\begin{split}\mathbb{P}\left[\max_{j\geq
1}\sum_{k\leq\tau_{2n\eta}}Y^{k}_{j-S_{k-1}}\operatorname{\mathbbm{1}}_{\xi_{k}\leq\varepsilon
a_{n}}>\bar{\varepsilon}a_{n}\right]&\leq\sum_{m=1}^{2n\eta}\mathbb{P}\left[\max_{j\geq
1}\sum_{k=\tau_{m-1}}^{\tau_{m}}Y^{k}_{j-S_{k-1}}\operatorname{\mathbbm{1}}_{\xi_{k}\leq\varepsilon
a_{n}}>\bar{\varepsilon}a_{n}\right]\\\ &=2n\eta\,\mathbb{P}\left[\max_{j\geq
1}\sum_{k=0}^{\tau_{1}}Y^{k}_{j-S_{k-1}}\operatorname{\mathbbm{1}}_{\xi_{k}\leq\varepsilon
a_{n}}>\bar{\varepsilon}a_{n}\right]\\\ &\leq
2n\eta\,\mathbb{P}\left[\sum_{k=0}^{\tau_{1}}\max_{j\geq
1}Y^{k}_{j}\operatorname{\mathbbm{1}}_{\xi_{k}\leq\varepsilon
a_{n}}>\bar{\varepsilon}a_{n}\right]\\\
&=2n\eta\,\mathbb{P}\left[\sum_{k=1}^{\infty}\operatorname{\mathbbm{1}}_{k\leq\tau_{1}}\max_{j\geq
1}Y^{k}_{j}\operatorname{\mathbbm{1}}_{\xi_{k}\leq\varepsilon
a_{n}}>\bar{\varepsilon}a_{n}\right]\\\ &\leq
2n\eta\sum_{k=1}^{\infty}\mathbb{P}\left[\tau_{1}\geq
k\right]\mathbb{P}\left[\max_{j\geq
1}Y^{k}_{j}\operatorname{\mathbbm{1}}_{\xi_{k}\leq\varepsilon
a_{n}}>\bar{\varepsilon}a_{n}/2k^{2}\right],\end{split}$

where in the last line we used the fact that $\\{\tau_{1}\geq k\\}$ and the
process $Y^{k}$ are independent.

Since the environment is given by an i.i.d. sequence, it is enough to estimate
the tails of the maximum of the process
$(Y_{j}\operatorname{\mathbbm{1}}_{\xi_{1}\leq\varepsilon a_{n}})_{j\geq 1}$.
By Lemma 3.5 applied with $\gamma=\beta+\delta$,

$\mathbb{P}\left[\max_{j\geq
1}Y_{j}\operatorname{\mathbbm{1}}_{\xi_{1}\leq\varepsilon a_{n}}>x\right]\leq
C_{2}x^{-\gamma}\left({\rm E}\left({\rm
E}_{\omega}Y_{\xi_{1}-1}^{2}\operatorname{\mathbbm{1}}_{\xi_{1}\leq\varepsilon
a_{n}}\right)^{\gamma/2}+\mathbb{E}\mathbb{Y}_{1}^{\gamma}\operatorname{\mathbbm{1}}_{\xi_{1}\leq\varepsilon
a_{n}}\right).$

As we have calculated in the proof of Lemma 3.5,

${\rm
E}_{\omega}Y_{\xi_{1}-1}^{2}\operatorname{\mathbbm{1}}_{\xi_{1}\leq\varepsilon
a_{n}}=\xi_{1}(\xi_{1}-1)\operatorname{\mathbbm{1}}_{\xi_{1}\leq\varepsilon
a_{n}},\quad{\rm
E}_{\omega}\mathbb{Y}_{1}^{2}=(2\xi_{1}^{2}\rho_{1}^{2}+\xi_{1}\rho_{1})\operatorname{\mathbbm{1}}_{\xi_{1}\leq\varepsilon
a_{n}},$

therefore

${\rm E}\left({\rm
E}_{\omega}Y_{\xi_{1}-1}^{2}\operatorname{\mathbbm{1}}_{\xi_{1}\leq\varepsilon
a_{n}}\right)^{\gamma/2}\leq{\rm
E}\xi^{\gamma}\operatorname{\mathbbm{1}}_{\xi\leq\varepsilon a_{n}}$

and

$\mathbb{E}\mathbb{Y}_{1}^{\gamma}\operatorname{\mathbbm{1}}_{\xi_{1}\leq\varepsilon
a_{n}}\leq{\rm E}\left({\rm
E}_{\omega}\mathbb{Y}_{1}^{2}\operatorname{\mathbbm{1}}_{\xi_{1}\leq\varepsilon
a_{n}}\right)^{\gamma/2}\leq\left(2^{\gamma/2}{\rm E}\rho^{\gamma}+{\rm
E}\rho^{\gamma/2}\right){\rm
E}\xi^{\gamma}\operatorname{\mathbbm{1}}_{\xi\leq\varepsilon a_{n}}.$

Putting things together, for some constant $C>0$ and any $x>0$,

$\mathbb{P}\left[\max_{j\geq
1}Y_{j}\operatorname{\mathbbm{1}}_{\xi_{1}\leq\varepsilon a_{n}}>x\right]\leq
Cx^{-\gamma}{\rm E}\xi^{\gamma}\operatorname{\mathbbm{1}}_{\xi\leq\varepsilon
a_{n}}\leq Cx^{-\gamma}\int_{0}^{\varepsilon a_{n}}t^{\gamma-1}{\rm
P}[\xi>t]dt.$

By Karamata’s theorem ([2], Theorem 1.5.11) and (5.7),

$\int_{0}^{\varepsilon a_{n}}t^{\gamma-1}{\rm
P}[\xi>t]dt\sim\frac{1}{\gamma+\beta}(\varepsilon a_{n})^{\gamma}{\rm
P}[\xi>\varepsilon
a_{n}]\sim\frac{1}{\gamma+\beta}\varepsilon^{\gamma-\beta}a_{n}^{\gamma}n^{-1}.$

Using those estimates, we obtain, for some constants $C,C^{\prime}>0$,

$\begin{split}\mathbb{P}\left[\max_{j\geq
1}\sum_{k\leq\tau_{2n\eta}}Y^{k}_{j-S_{k-1}}\operatorname{\mathbbm{1}}_{\\{\xi_{k}\leq\varepsilon
a_{n}\\}}>\bar{\varepsilon}a_{n}\right]&\leq
Cn\sum_{k=1}^{\infty}\mathbb{P}[\tau_{1}\geq
k]\left(\bar{\varepsilon}a_{n}/2k^{2}\right)^{-\gamma}\varepsilon^{\gamma-\beta}a_{n}^{\gamma}n^{-1}\\\
&\leq
C^{\prime}\bar{\varepsilon}^{-\gamma}\varepsilon^{\gamma-\beta}\mathbb{E}\tau_{1}^{2\gamma+1},\end{split}$

which finishes the proof since $\gamma=\beta+\delta$ and
$\mathbb{E}\tau_{1}^{2\gamma+1}<\infty$ by Lemma 3.1. ∎

The next step is to investigate the maximal generations among the progeny of
immigrants from large blocks. Although it may happen that the descendants of
particles from several large blocks coexist in one generation of the process
$Z$, we will show later that it is unlikely, so that we may begin by
investigating the maxima of $|I_{n,\varepsilon}|$ independent processes, each
representing progeny of immigrants from a large block. To this end, assume
that our probability space contains variables
$\left\\{(Y^{j,(N)}_{k})_{k\in\mathbb{N}}\,:\,j,N\in\mathbb{N}\right\\}$ such
that

  * •

the processes $(Y^{j,(N)}_{k})_{k\in\mathbb{N}}$ are i.i.d. copies of
$(Y^{(N)}_{k})_{k\in\mathbb{N}}$,

  * •

the family
$\left\\{(Y^{j,(N)}_{k})_{k\in\mathbb{N}}\,:\,j,N\in\mathbb{N}\right\\}$ is
independent of the environment
$\\{(\xi_{k},\lambda_{k})\\}_{k\in{\mathbb{Z}}}$.

For any $j,N\in\mathbb{N}$ denote $M_{N}^{j}=\max_{k\geq
0}(Y^{j,(N)}_{k}+Y^{j,(N)}_{k+1})$ and let
$\bar{D}_{j,n}=\\{\mathbb{Y}^{j,(\xi_{j})}_{\sqrt{n}}=0\\}$. Observe that the
event $\bar{D}_{j,n}$ means that the process $Y^{j,(\xi_{j})}$ went extinct at
most at its $\sqrt{n}$’th marked generation.

######  Proposition 5.4.

Fix $\varepsilon>0$ and let $A_{n}\in\sigma(I_{n,\varepsilon})$ be such that
${\rm P}[A_{n}]\to 1$ as $n\to\infty$. For any $x>0$,

$\lim_{n\to\infty}\mathbb{P}\left[\max_{j\in
I_{n,\varepsilon}}M^{j}_{\xi_{j}}\operatorname{\mathbbm{1}}_{\bar{D}_{j,n}}>xa_{n},A_{n}\right]=1-\exp\left(-x^{-\beta}\mathbb{E}M_{\infty}^{\beta}\operatorname{\mathbbm{1}}_{M_{\infty}<x/\varepsilon}-\varepsilon^{-\beta}\mathbb{P}[M_{\infty}\geq
x/\varepsilon]\right).$

###### Proof.

Observe that due to our assumptions the event $\bar{D}_{j,n}$ depends only on
$\xi_{j}$ and the process $Y^{j,(\xi_{j})}$. Therefore we investigate a
maximum of variables which are i.i.d. under $\mathbb{P}$.

Recall that $|I_{n,\varepsilon}|$ converges in distribution to
$Poiss(\varepsilon^{-\beta})$. Moreover, conditioning on
$|I_{n,\varepsilon}|=k$, the examined maximum is a maximum of $k$ independent
variables with distribution given by

$\mathbb{P}\left[M_{\xi}\operatorname{\mathbbm{1}}_{\bar{D}_{n}}\in\cdot\,\Big{|}\,\xi>\varepsilon
a_{n}\right],$

for $\xi$ independent of $\\{Y^{(N)},M_{N}\,:\,N\in\mathbb{N}\\}$ and
$\bar{D}_{n}=\\{\mathbb{Y}^{(\xi)}_{\sqrt{n}}=0\\}$. In particular,

(5.9) $\begin{split}\mathbb{P}\left[\max_{k\in
I_{n,\varepsilon}}M^{k}_{\xi_{k}}\operatorname{\mathbbm{1}}_{\bar{D}_{k,n}}>xa_{n}\right]&=1-{\rm
E}\left[(1-\mathbb{P}\left[M_{\xi}>xa_{n},\,\bar{D}_{n}\,|\,\xi>\varepsilon
a_{n}\right])^{|I_{n,\varepsilon}|}\operatorname{\mathbbm{1}}_{A_{n}}\right]\\\
&=1-{\rm
E}\left[(1-\mathbb{P}\left[M_{\xi}>xa_{n},\,\bar{D}_{n}\,|\,\xi>\varepsilon
a_{n}\right])^{|I_{n,\varepsilon}|}\right]+o(1),\end{split}$

where the second equality follows from

${\rm
E}\left[(1-\mathbb{P}\left[M_{\xi}>xa_{n},\,\bar{D}_{n}\,|\,\xi>\varepsilon
a_{n}\right])^{|I_{n,\varepsilon}|}\operatorname{\mathbbm{1}}_{A_{n}^{c}}\right]\leq{\rm
P}[A_{n}^{c}].$

Note that, since the extinction time of the process $Y^{(\xi)}$ is dominated
by $\tau_{1}$, Lemma 3.1 implies

$\mathbb{P}[\bar{D}_{n}^{c}]\leq\mathbb{P}[\tau_{1}\geq\sqrt{n}]\leq
e^{-c\sqrt{n}}\mathbb{E}e^{c\tau_{1}},$

and by (5.7),

$\mathbb{P}[\bar{D}_{n}^{c}\,|\,\xi>\varepsilon a_{n}]\leq
e^{-c\sqrt{n}}\mathbb{E}e^{c\tau_{1}}{\rm P}[\xi>\varepsilon
a_{n}]^{-1}\sim\mathbb{E}e^{c\tau_{1}}\varepsilon^{\beta}ne^{-c\sqrt{n}}\to 0$

as $n\to\infty$. Therefore, for any fixed $\bar{\varepsilon}>0$, for $n$ large
enough,

(5.10) $\mathbb{P}\left[M_{\xi}>xa_{n},\,\bar{D}_{n}^{c}\,|\,\xi>\varepsilon
a_{n}\right]\leq\bar{\varepsilon}.$

By Lemma 5.1, $M_{N}/N\overset{{\rm d}}{\longrightarrow}M_{\infty}$ as
$N\to\infty$. Observe that the distribution of $M_{\infty}$ is continuous and
thus appropriate cumulative distribution functions converge uniformly; in
particular, for large enough $n$,

(5.11) $\sup_{y>0}\left|\mathbb{P}\left[M_{\xi}>y\,|\,\xi>\varepsilon
a_{n}\right]-\mathbb{P}\left[M_{\infty}>y/\xi\,|\,\xi>\varepsilon
a_{n}\right]\right|<\bar{\varepsilon}$

for $M_{\infty}$ independent of $\xi$. Observe that

$\begin{split}\mathbb{P}&\left[M_{\infty}>xa_{n}/\xi\,|\,\xi>\varepsilon
a_{n}\right]=\frac{\mathbb{P}[\xi M_{\infty}>xa_{n},\xi>\varepsilon
a_{n}]}{{\rm P}[\xi>\varepsilon a_{n}]}\\\ &=\frac{1}{{\rm P}[\xi>\varepsilon
a_{n}]}\left(\int_{[0,x/\varepsilon)}{\rm
P}[\xi>xa_{n}/t]\mathbb{P}[M_{\infty}\in dt]+\int_{[x/\varepsilon,\infty)}{\rm
P}[\xi>\varepsilon a_{n}]\mathbb{P}[M_{\infty}\in dt]\right)\\\
&=\int_{[0,x/\varepsilon)}\frac{{\rm P}[\xi>xa_{n}/t]}{{\rm P}[\xi>\varepsilon
a_{n}]}\mathbb{P}[M_{\infty}\in dt]+\mathbb{P}[M_{\infty}\geq
x/\varepsilon].\end{split}$

By uniform convergence theorem for regularly varying functions (see (B.1.2) in
[3]), for $n$ large enough,

$\sup_{c\geq 1}\left|\frac{{\rm P}[\xi>c\varepsilon a_{n}]}{{\rm
P}[\xi>\varepsilon a_{n}]}-c^{-\beta}\right|<\bar{\varepsilon},$

which means that

$\left|\int_{[0,x/\varepsilon)}\frac{{\rm P}[\xi>xa_{n}/t]}{{\rm
P}[\xi>\varepsilon
a_{n}]}-\left(\frac{x}{t\varepsilon}\right)^{-\beta}\mathbb{P}[M_{\infty}\in
dt]\right|<\bar{\varepsilon}.$

Hence

$\left|\mathbb{P}[M_{\infty}>xa_{n}/\xi\,|\,\xi>\varepsilon
a_{n}]-\left(x^{-\beta}\varepsilon^{\beta}\mathbb{E}M_{\infty}^{\beta}\operatorname{\mathbbm{1}}_{M_{\infty}<x/\varepsilon}+\mathbb{P}[M_{\infty}\geq
x/\varepsilon]\right)\right|<\bar{\varepsilon},$

which together with (5.11) implies

(5.12) $\left|\mathbb{P}\left[M_{\xi}>xa_{n}\,|\,\xi>\varepsilon
a_{n}\right]-\left(x^{-\beta}\varepsilon^{\beta}\mathbb{E}M_{\infty}^{\beta}\operatorname{\mathbbm{1}}_{M_{\infty}<x/\varepsilon}+\mathbb{P}[M_{\infty}\geq
x/\varepsilon]\right)\right|<2\bar{\varepsilon}.$

Putting the estimates (5.10) and (5.12) to (5.9) and using the fact that
$|I_{n,\varepsilon}|\overset{{\rm
d}}{\longrightarrow}Poiss(\varepsilon^{-\beta})$, we obtain

$\begin{split}1-&\exp\left(-\varepsilon^{-\beta}\left(x^{-\beta}\varepsilon^{\beta}\mathbb{E}M_{\infty}^{\beta}\operatorname{\mathbbm{1}}_{M_{\infty}<x/\varepsilon}+\mathbb{P}[M_{\infty}\geq
x/\varepsilon]-3\bar{\varepsilon}\right)\right)\\\
&\leq\liminf_{n\to\infty}\mathbb{P}\left[\max_{k\leq
n}M^{k}_{\xi_{k}}\operatorname{\mathbbm{1}}_{\xi_{k}>\varepsilon
a_{n}}>xa_{n}\right]\leq\limsup_{n\to\infty}\mathbb{P}\left[\max_{k\leq
n}M^{k}_{\xi_{k}}\operatorname{\mathbbm{1}}_{\xi_{k}>\varepsilon
a_{n}}>xa_{n}\right]\\\ &\leq
1-\exp\left(-\varepsilon^{-\beta}\left(x^{-\beta}\varepsilon^{\beta}\mathbb{E}M_{\infty}^{\beta}\operatorname{\mathbbm{1}}_{M_{\infty}<x/\varepsilon}+\mathbb{P}[M_{\infty}\geq
x/\varepsilon]+3\bar{\varepsilon}\right)\right),\end{split}$

which finishes the proof since $\bar{\varepsilon}$ is arbitrary. ∎

We are now ready to prove Theorem 2.2, rephrased into the setting of the
associated branching process.

######  Theorem 5.5.

Under assumptions $(B)$,

$\mathbb{P}\left[a_{n}^{-1}\max_{0\leq
k<S_{n}}(Z_{k}+Z_{k+1})>x\right]\xrightarrow{n\to\infty}1-\exp\left(-\mathbb{E}M_{\infty}^{\beta}x^{-\beta}\right).$

###### Proof.

Fix $\varepsilon>0$. For any $\bar{\varepsilon}>0$,

(5.13) $\mathbb{P}\left[\max_{j\geq 1}\sum_{k\in
I_{n,\varepsilon}}(Y^{k}_{j-S_{k-1}}+Y^{k}_{j-S_{k-1}+1})>xa_{n}\right]\leq\mathbb{P}\left[\max_{j<S_{n}}(Z_{j}+Z_{j+1})>xa_{n}\right]\\\
\leq\mathbb{P}\left[2\max_{j\geq 1}\sum_{k\in
I_{n,\varepsilon}^{c}}Y^{k}_{j-S_{k-1}}>\bar{\varepsilon}a_{n}\right]+\mathbb{P}\left[\max_{j\geq
1}\sum_{k\in
I_{n,\varepsilon}}(Y^{k}_{j-S_{k-1}}+Y^{k}_{j-S_{k-1}+1})>(x-\bar{\varepsilon})a_{n}\right].$

Note that because of (5.8) we expect that for large $n$ the set
$I_{n,\varepsilon}$ should be distributed rather uniformly on $\\{1,\dots
n\\}$, so that the large blocks are far from each other. Indeed, since
$nP[\xi>\varepsilon a_{n}]\to\varepsilon^{-\beta}$, for any sequence $b_{n}$
such that $b_{n}=o(n)$,

${\rm P}\left[\left(\exists k,l\in I_{n,\varepsilon}\right)\,k\neq l,|k-l|\leq
b_{n}\right]\leq n{\rm P}[\xi>\varepsilon a_{n}]\cdot b_{n}{\rm
P}[\xi>\varepsilon a_{n}]\to 0\quad\textnormal{ as }n\to\infty.$

That is, with high probability, large blocks are at distance at least $b_{n}$
from each other. On the other hand, we know that the extinction occurs very
often in our process, which should mean that as the process evolves, no two
bloodlines of immigrants from large blocks coexist at one time. Let

$D_{k,n}=\left\\{\mathbb{Y}^{k}_{\sqrt{n}}=0\right\\}$

be an event that the progeny of immigrants from $k$’th block does not survive
more than $\sqrt{n}$ blocks. Then, by Lemma 3.1,

$\mathbb{P}\left[\bigcup_{k\leq n}D_{k,n}^{c}\right]\leq
n\mathbb{P}[\tau_{1}>\sqrt{n}]\leq ne^{-c\sqrt{n}}\mathbb{E}e^{c\tau_{1}}\to
0$

as $n\to 0$. Therefore the probability of the set

$D_{n}=\bigcap_{k\leq n}D_{k,n}$

converges to $1$ as $n\to\infty$ and so does the probability of

$A_{n}=\\{\left(\forall k,l\in I_{n,\varepsilon}\right)k\neq
l\implies|k-l|>2\sqrt{n}\\}.$

Moreover, on the set $A_{n}\cap D_{n}$, the progeny of immigrants from each
large block dies out before the next large block occurs. That is, $\max_{j\geq
1}\sum_{k\in I_{n,\varepsilon}}(Y^{k}_{j-S_{k-1}}+Y^{k}_{j-S_{k-1}+1})$ is
really a maximum of independent maxima of $Y^{k}$ such that $k\in
I_{n,\varepsilon}$. Therefore,

$\mathbb{P}\left[\max_{j\geq 1}\sum_{k\in
I_{n,\varepsilon}}(Y^{k}_{j-S_{k}}+Y^{k}_{j-S_{k}+1})>xa_{n},\,A_{n}\cap
D_{n}\right]=\mathbb{P}\left[\max_{k\in
I_{n,\varepsilon}}M^{k}_{\xi_{k}}\operatorname{\mathbbm{1}}_{\bar{D}_{k,n}}>xa_{n},\,A_{n}\right].$

By Proposition 5.4, this quantity converges to

$1-\exp\left(-x^{-\beta}\mathbb{E}M_{\infty}^{\beta}\operatorname{\mathbbm{1}}_{M_{\infty}<x/\varepsilon}-\varepsilon^{-\beta}\mathbb{P}[M_{\infty}\geq
x/\varepsilon]\right)$

as $n\to\infty$. Going back to (5.13), we have

(5.14)
$1-\exp\left(-x^{-\beta}\mathbb{E}M_{\infty}^{\beta}\operatorname{\mathbbm{1}}_{M_{\infty}<x/\varepsilon}-\varepsilon^{-\beta}\mathbb{P}[M_{\infty}\geq
x/\varepsilon]\right)\leq\liminf_{n\to\infty}\mathbb{P}\left[\max_{j<S_{n}}(Z_{j}+Z_{j+1})>xa_{n}\right].$

On the other hand, by Proposition 5.3,

$\limsup_{n\to\infty}\mathbb{P}\left[2\max_{j\geq 1}\sum_{k\in
I_{n,\varepsilon}^{c}}Y^{k}_{j-S_{k-1}}>\bar{\varepsilon}a_{n}\right]\leq
C_{5}(\bar{\varepsilon}/2)^{-\beta-\delta}\varepsilon^{\delta},$

which means that

(5.15)
$\begin{split}\limsup_{n\to\infty}\mathbb{P}&\left[\max_{j<S_{n}}(Z_{j}+Z_{j+1})>xa_{n}\right]\leq
C_{5}(\bar{\varepsilon}/2)^{-\beta-\delta}\varepsilon^{\delta}\\\
&+1-\exp\left(-(x-\bar{\varepsilon})^{-\beta}\mathbb{E}M_{\infty}^{\beta}\operatorname{\mathbbm{1}}_{M_{\infty}<(x-\bar{\varepsilon})/\varepsilon}-\varepsilon^{-\beta}\mathbb{P}[M_{\infty}\geq(x-\bar{\varepsilon})/\varepsilon]\right).\end{split}$

Observe that, since $\mathbb{E}M_{\infty}^{\beta+\delta}<\infty$ (see Remark
5.2), we have

$\varepsilon^{-\beta}\mathbb{P}[M_{\infty}\geq
x/\varepsilon]\leq\varepsilon^{\delta}x^{-\beta-\delta}\mathbb{E}M_{\infty}^{\beta+\delta}\to
0\quad\textnormal{ as }\varepsilon\to 0,$

while by the monotone convergence theorem,

$\mathbb{E}M_{\infty}^{\beta}\operatorname{\mathbbm{1}}_{M_{\infty}<x/\varepsilon}\to\mathbb{E}M_{\infty}^{\beta}\quad\textnormal{
as }\varepsilon\to 0.$

Therefore passing with $\varepsilon$ to $0$ in (5.14) gives

$1-\exp\left(-x^{-\beta}\mathbb{E}M_{\infty}^{\beta}\right)\leq\liminf_{n\to\infty}\mathbb{P}\left[\max_{j<S_{n}}(Z_{j}+Z_{j+1})>xa_{n}\right],$

and similarly in (5.15),

$\limsup_{n\to\infty}\mathbb{P}\left[\max_{j<S_{n}}(Z_{j}+Z_{j+1})>xa_{n}\right]\leq
1-\exp\left(-(x-\bar{\varepsilon})^{-\beta}\mathbb{E}M_{\infty}^{\beta}\right),$

which ends the proof since $\bar{\varepsilon}>0$ is arbitrary. ∎

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## Acknowledgements

The research was supported by the National Science Center, Poland (Opus, grant
number 2020/39/B/ST1/00209).