1 Introduction and main results
The branching process is a random process starting with one individual, the initial ancestor, which produces offspring according to some random rule. The collection of offspring constitutes the first generation. Each individual of the first generation gives birth to a random number of children with the same offspring distribution as for the initial ancestor. The numbers of offspring of different individuals (including the initial ancestor) are independent. This process continues forever or until the population dies out. An interesting problem is the behavior of the long-time evolution of the process. Plainly, it depends on a particular rule that regulates giving birth to offspring. In the simplest case, when the offspring distribution is the same for all generations, the branching process is called the Galton–Watson process. We refer to [2] for numerous results concerning, for instance, long-term survival or extinction of such a process, the growth rate of the population, fluctuations of population sizes. Thanks to a simple tree structure, not only does the Galton–Watson process find numerous applications as a model of biological reproduction processes, but also in many other fields including computer science and physics.
The homogeneity of the Galton–Watson process reduces its applicability. In some cases it may happen that the population evolution conditions change randomly over time. This leads to the notion of branching process in random environment (BPRE) introduced by Smith and Wilkinson [13]. The BPRE is a population growth process, in which the individuals reproduce independently of each other with the offspring distribution picked randomly at each generation. More precisely, let ν be a random measure on the set of nonnegative integers ${\mathbb{N}_{0}}:=\{0,1,2,\dots ,\}$. Then a sequence ${({\nu _{n}})_{n\ge 1}}$ of independent copies of ν can be interpreted as a random environment. The BPRE is then the sequence ${Z^{\prime }}={({Z^{\prime }_{n}})_{n\ge 0}}$ defined by the recursive formula ${Z^{\prime }_{n+1}}={\textstyle\sum _{k=1}^{{Z^{\prime }_{n}}}}{\xi _{k}^{(n)}}$, where, given ${({\nu _{n}})_{n\ge 1}}$, ${({\xi _{k}^{(n)}})_{k\ge 1}}$ are independent identically distributed (iid) and independent of ${Z^{\prime }_{n}}$ with the common distribution ${\nu _{n+1}}$. We refer to the recent monograph by Kersting and Vatutin [11] for an overview of fundamental properties of this process.
We intend to study here branching processes in sparse random environment (BPSRE), in which homogeneity of the environment is modified on a sparse subset of $\mathbb{N}$. This is an intermediate model between Galton–Watson processes and the BPRE. To give a precise definition, let μ be a fixed probability measure on ${\mathbb{N}_{0}}$ and $\mathbb{Q}={(({d_{k}},{\nu _{k}}))_{k\ge 1}}$ a sequence of independent copies of a random vector $(d,\nu )$, where d is a positive integer-valued random variable and ν is a random measure on ${\mathbb{N}_{0}}$ independent of d. First we choose a subset of integers marked by the positions of a standard random walk ${({S_{k}})_{k\ge 0}}$ defined by
and then we impose random measures at the marked sites. The branching process in sparse random environment $\mathbb{Q}$ (BPSRE) is formally defined as follows:
\[ {Z_{0}}=1,\hspace{1em}{Z_{n+1}}={\sum \limits_{j=1}^{{Z_{n}}}}{\xi _{j}^{(n)}},\hspace{1em}n\in {\mathbb{N}_{0}},\]
where, if $n={S_{k}}$ for some $k\in \mathbb{N}$, then, given $\mathbb{Q}$, ${\xi _{j}^{(n)}}$ are independent random variables with the common distribution ${\nu _{k+1}}$, which are also independent of ${Z_{n}}$. Otherwise, if $n\notin \{{S_{0}},{S_{1}},{S_{2}},\dots \}$, then ${\xi _{j}^{(n)}}$ are independent random variables with the common distribution μ, which are also independent of ${Z_{n}}$. The process ${({Z_{n}})_{n\ge 0}}$ behaves like the Galton–Watson process, with the exception of some randomly chosen generations in which the offspring distribution is random.We intend to describe how the additional randomness of the environment affects the behavior of the BPSRE. To this end, we focus on Yaglom-type results. For the Galton–Watson process in the critical case, that is, when the expected number of offspring is 1 (see (A2) below), it is known that the probability of survival up to the generation n is of the order $1/n$ and the population size conditioned to the survival set converges weakly to an exponential distribution (Section 9 in [2]). In contrast, in the critical case for the BPRE, that is, when the expectation of the logarithm of the number of offspring is 0 (see (A1)), the probability of survival up to the generation n is asymptotically $1/\sqrt{n}$, and the process conditioned to the survival event converges weakly to a Rayleigh distribution. We prove below in Theorems 1 and 2 that, although the environment is random on a sparse subset only, the behavior the BPSRE reminds that of a BPRE.
To close the introduction, we mention that closely related random walks in a sparse random environment, which is an intermediate model between the simple random walk and the random walk in a random environment, have been recently investigated in [4–6].
1.1 Notation and assumptions
Given a deterministic or random probability measure θ on ${\mathbb{N}_{0}}$, define the generating function
Denote by
its mean and by
\[ {\sigma _{\theta }}:=\frac{{f^{\prime\prime }_{\theta }}(1)}{{({f^{\prime }_{\theta }}(1))^{2}}}=\frac{1}{{A_{\theta }^{2}}}{\sum \limits_{j=2}^{\infty }}j(j-1)\theta (\{j\})\]
its normalized second factorial moment. We shall also use a standardized truncated second moment defined by
\[ \kappa ({f_{\theta }};a):=\frac{1}{{A_{\theta }^{2}}}{\sum \limits_{j=a}^{\infty }}{j^{2}}\theta (\{j\}),\hspace{1em}a\in {\mathbb{N}_{0}}.\]
To simplify our notation we shall write, for $k\ge 1$, ${A_{k}}$ and ${\sigma _{k}}$ instead of ${A_{{\nu _{k}}}}$ and ${\sigma _{{\nu _{k}}}}$, respectively. Thus, in our setting ${({A_{k}})_{k\ge 1}}$ and ${({\sigma _{k}})_{k\ge 1}}$ are two (dependent) sequences of iid random variables. As usual, ${x^{+}}=\max (x,0)$ and ${x^{-}}=\max (-x,0)$ for $x\in \mathbb{R}$.Throughout the paper we impose the following assumptions:
-
(A1) $\mathbb{E}\log {A_{1}}=0$, ${\mathfrak{v}^{2}}:=\mathrm{Var}\hspace{0.1667em}(\log {A_{1}})\in (0,\infty )$ and $\mathbb{E}{({\log ^{-}}{A_{1}})^{4}}\lt \infty $;
-
(A2) ${A_{\mu }}=1$;
-
(A3) $\mathbb{E}{d^{3/2}}\lt \infty $ and we put $\mathtt{m}:=\mathbb{E}d$;
-
(A4) $\mathbb{E}{({\log ^{+}}\kappa ({f_{\nu }};a))^{4}}\lt \infty $ for some $a\in \mathbb{N}$.
1.2 Main results
Let ${\tau _{\mathrm{Sparse}}}\in (0,\infty ]$ be the extinction time of ${({Z_{n}})_{n\ge 0}}$, that is,
The following observation is almost immediate.
Proposition 1.
Under the assumptions (A1)–(A2), $\mathbb{P}\{{\tau _{\mathrm{Sparse}}}\lt \infty \}=1$.
In this paper we focus on the annealed analysis of BPSRE ${({Z_{n}})_{n\ge 0}}$, that is, on the behavior of ${({Z_{n}})_{n\ge 0}}$ averaged over all realizations of the environment. Our first main result is concerned with the (annealed) tail behavior of $\mathbb{P}\{{\tau _{\mathrm{Sparse}}}\gt n\}=\mathbb{P}\{{Z_{n}}\gt 0\}$ as $n\to \infty $.
Our next result is an (annealed) Yaglom-type functional limit theorem for the process $({Z_{n}})$. Recall that a Brownian meander, see [8], is a stochastic process ${({B_{+}}(t))_{t\in [0,1]}}$ defined as follows. Let ${(B(t))_{t\in [0,1]}}$ be a standard Brownian motion and $\zeta :=\sup \{t\in [0,1]:B(t)=0\}$ be its last visit to 0 on $[0,1]$. Then
Theorem 2.
Assume (A1)–(A4). Then with ${({B_{+}}(t))_{t\in [0,1]}}$ being a Brownian meander
\[ \mathrm{Law}\left({\left(\frac{\log {Z_{\lfloor nt\rfloor }}}{\mathfrak{v}\sqrt{{\mathtt{m}^{-1}}n}}\right)_{t\in [0,1]}}\hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{n}}\gt 0\right)\hspace{2.5pt}\Longrightarrow \hspace{2.5pt}\mathrm{Law}\left({({B_{+}}(t))_{t\in [0,1]}}\right),\hspace{1em}n\to \infty ,\]
weakly on the space of probability measures on $D[0,1]$ endowed with the Skorokhod ${J_{1}}$-topology.
Using formula (1.1) in [8] we obtain the following one-dimensional result.
Corollary 1.
Assume (A1)–(A4). Then, for every fixed $t\in (0,1]$,
The random variable ${B_{+}}(t)$ has an absolutely continuous distribution with a bounded nonvanishing density on $[0,\infty )$. Furthermore,
so ${B_{+}}(1)$ has the Rayleigh distribution.
(2)
\[ \underset{n\to \infty }{\lim }\mathbb{P}\left\{\frac{\log {Z_{\lfloor nt\rfloor }}}{\mathfrak{v}\sqrt{{\mathtt{m}^{-1}}n}}\ge x\hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{n}}\gt 0\right\}=\mathbb{P}\{{B_{+}}(t)\ge x\},\hspace{1em}x\ge 0.\]Remark 1.
The assumption (A4) and the last part of the assumption (A1) can be weakened without changing the formulations of the main results. A version of (A4) appears as Assumption (C) in [11, Chapter 5]. It is a convenient general condition allowing for an (asymptotically) closed form of the survival probability and also validity of a functional limit theorem for a critical branching process in iid random environment. A more general version of Assumption (C) can be found in [1]. However, we prefer to sacrifice generality in favor of transparency and simplicity of the formulations.
2 Proofs
The proof of our main results consists of three steps. First, we analyze an embedded process ${({Z_{{S_{n}}}})_{n\ge 0}}$ by finding its survival asymptotic and proving a counterpart of Theorem 2. Second, we deduce from the results obtained for ${({Z_{{S_{n}}}})_{n\ge 0}}$ the corresponding statements for a randomly stopped process ${({Z_{{S_{\vartheta (n)}}}})_{n\ge 0}}$, where ${(\vartheta (n))_{n\ge 0}}$ is the first passage time process for the random walk ${({S_{k}})_{k\ge 0}}$. At the last step, we show that ${({Z_{{S_{\vartheta (n)}}}})_{n\ge 0}}$ is uniformly close to ${({Z_{n}})_{n\ge 0}}$.
2.1 Analysis of the embedded process
Observe that ${({Z_{{S_{k}}}})_{k\ge 0}}$ is a branching process in iid random environment $\widetilde{\mathbb{Q}}={({\widetilde{\nu }_{k}})_{k\ge 1}}$ which can be explicitly described as follows. Let ${({({\widetilde{Z}_{j}^{(i)}})_{j\ge 0}})_{i\ge 0}}$ be a sequence of independent copies of a critical Galton–Watson process ${({\widetilde{Z}_{j}})_{j\ge 0}}$ in deterministic environment with the offspring distribution μ and ${\widetilde{Z}_{0}}=1$. Suppose that ${({({\widetilde{Z}_{j}^{(i)}})_{j\ge 0}})_{i\ge 0}}$ is independent of the environment $\mathbb{Q}$. Then
Let $\widetilde{\nu }$ be a generic copy of iid random measures ${({\widetilde{\nu }_{k}})_{k\ge 1}}$. Put
where d is assumed independent of ${({\widetilde{Z}_{k}})_{k\ge 0}}$. Equality (3) entails that the generating function of the random measure $\widetilde{\nu }$ is given by
Since ${\widetilde{g}^{\prime }}(1)=\mathbb{E}{\widetilde{Z}_{d-1}}=1$, the latter formula immediately implies that
Further,
see, for instance, Chapter I.2 in [2] for the last equality. Note that (4) guarantees that
(3)
\[ {\widetilde{\nu }_{k}}(\{j\})={\sum \limits_{l=0}^{\infty }}{\nu _{k}}(\{l\})\mathbb{P}\left\{{\sum \limits_{i=1}^{l}}{\widetilde{Z}_{{d_{k}}-1}^{(i)}}=j\right\},\hspace{1em}k,j\in {\mathbb{N}_{0}}.\](4)
\[ {A_{{\widetilde{\nu }_{k}}}}={f^{\prime }_{{\widetilde{\nu }_{k}}}}(1)={f^{\prime }_{{\nu _{k}}}}(1)={A_{{\nu _{k}}}}={A_{k}},\hspace{1em}k\in {\mathbb{N}_{0}}.\]
\[\begin{array}{c}\displaystyle {\sigma _{{\widetilde{\nu }_{k}}}}=\frac{{f^{\prime\prime }_{{\widetilde{\nu }_{k}}}}(1)}{{({f^{\prime }_{{\widetilde{\nu }_{k}}}}(1))^{2}}}=\frac{{f^{\prime\prime }_{{\nu _{k}}}}(1)+{f^{\prime }_{{\nu _{k}}}}(1){\widetilde{g}^{\prime\prime }}(1)}{{({f^{\prime }_{{\nu _{k}}}}(1))^{2}}}\\ {} \displaystyle ={\sigma _{{\nu _{k}}}}+\frac{{\sigma _{\mu }}(\mathbb{E}d-1)}{{A_{{\nu _{k}}}}}={\sigma _{k}}+\frac{{\sigma _{\mu }}(\mathbb{E}d-1)}{{A_{k}}},\hspace{1em}k\in {\mathbb{N}_{0}},\end{array}\]
where we have used that
(5)
\[ {\widetilde{g}^{\prime\prime }}(1)=\mathbb{E}{\widetilde{Z}_{d-1}}({\widetilde{Z}_{d-1}}-1)={\sigma _{\mu }}(\mathbb{E}d-1),\]
\[ \mathbb{E}\log {A_{{\widetilde{\nu }_{1}}}}=\mathbb{E}\log {A_{{\nu _{1}}}}=\mathbb{E}\log {A_{1}}=0,\]
which means that the embedded process ${({Z_{{S_{n}}}})_{n\ge 0}}$ is critical. In particular,
Recall that we denote by $\kappa ({f_{\theta }};a)$ the truncated second moment of a measure θ.
Lemma 1.
Let ${a_{\ast }}\in {\mathbb{N}_{0}}$ and assume that
\[ \mathbb{E}{({\log ^{+}}\kappa ({f_{\nu }};{a_{\ast }}))^{4}}\lt \infty \hspace{1em}\textit{and}\hspace{1em}\mathbb{E}{({\log ^{-}}{A_{\nu }})^{4}}\lt \infty .\]
Then $\mathbb{E}{({\log ^{+}}\kappa ({f_{\widetilde{\nu }}};{a_{\ast }}))^{4}}\lt \infty $.
Proof.
We start by writing
\[\begin{array}{c}\displaystyle \kappa ({f_{\widetilde{\nu }}};a)=\frac{1}{{A_{\widetilde{\nu }}^{2}}}{\sum \limits_{j=a}^{\infty }}{j^{2}}{\sum \limits_{l=0}^{\infty }}\nu (\{l\})\mathbb{P}\left\{{\sum \limits_{i=1}^{l}}{\widetilde{Z}_{d-1}^{(i)}}=j\right\}\\ {} \displaystyle =\frac{1}{{A_{\nu }^{2}}}{\sum \limits_{l=0}^{\infty }}\nu (\{l\})\mathbb{E}\left({\left({\sum \limits_{i=1}^{l}}{\widetilde{Z}_{d-1}^{(i)}}\right)^{2}}{1_{\left\{{\textstyle\textstyle\sum _{i=1}^{l}}{\widetilde{Z}_{d-1}^{(i)}}\ge a\right\}}}\right).\end{array}\]
In view of (5), for all $a\in \mathbb{N}$,
\[ \mathbb{E}\left({\left({\sum \limits_{i=1}^{l}}{\widetilde{Z}_{d-1}^{(i)}}\right)^{2}}{1_{\left\{{\textstyle\textstyle\sum _{i=1}^{l}}{\widetilde{Z}_{d-1}^{(i)}}\ge a\right\}}}\right)\le \mathbb{E}{\left({\sum \limits_{i=1}^{l}}{\widetilde{Z}_{d-1}^{(i)}}\right)^{2}}\le {C_{1}}\mathtt{m}{l^{2}},\]
where $\mathtt{m}=\mathbb{E}d$ and ${C_{1}}\gt 0$ is a constant. Thus,
\[\begin{array}{c}\displaystyle \frac{1}{{A_{\nu }^{2}}}{\sum \limits_{l={a_{\ast }}}^{\infty }}\nu (\{l\})\mathbb{E}\left({\left({\sum \limits_{i=1}^{l}}{\widetilde{Z}_{d-1}^{(i)}}\right)^{2}}{1_{\left\{{\textstyle\textstyle\sum _{i=1}^{l}}{\widetilde{Z}_{d-1}^{(i)}}\ge {a_{\ast }}\right\}}}\right)\\ {} \displaystyle \le \frac{{C_{1}}\mathtt{m}}{{A_{\nu }^{2}}}{\sum \limits_{l={a_{\ast }}}^{\infty }}{l^{2}}\nu (\{l\})={C_{1}}\mathtt{m}\kappa ({f_{\nu }};{a_{\ast }}).\end{array}\]
Since $\mathbb{E}{({\log ^{+}}\kappa ({f_{\nu }};{a_{\ast }}))^{4}}\lt \infty $, it suffices to check that
\[ \mathbb{E}{\left({\log ^{+}}\frac{1}{{A_{\nu }^{2}}}{\sum \limits_{l=0}^{{a_{\ast }}}}\nu (\{l\})\mathbb{E}\left({\left({\sum \limits_{i=1}^{l}}{\widetilde{Z}_{d-1}^{(i)}}\right)^{2}}{1_{\left\{{\textstyle\textstyle\sum _{i=1}^{l}}{\widetilde{Z}_{d-1}^{(i)}}\ge {a_{\ast }}\right\}}}\right)\right)^{4}}\lt \infty .\]
The inner expectation is equal to 0 if $l=0$ and uniformly bounded by a constant ${C_{2}}\gt 0$ for all $l=1,\dots ,{a_{\ast }}$. It remains to note that
\[\begin{array}{c}\displaystyle \mathbb{E}{\left({\log ^{+}}\frac{{C_{2}}}{{A_{\nu }^{2}}}{\sum \limits_{l=1}^{{a_{\ast }}}}\nu (\{l\})\right)^{4}}\le \mathbb{E}{\left({\log ^{+}}\frac{{C_{2}}}{{A_{\nu }^{2}}}{\sum \limits_{l=1}^{\infty }}l\nu (\{l\})\right)^{4}}\le \mathbb{E}{\left({\log ^{+}}\frac{{C_{2}}}{{A_{\nu }}}\right)^{4}}\\ {} \displaystyle \le {C_{3}}\mathbb{E}{\left({\log ^{+}}\frac{1}{{A_{\nu }}}\right)^{4}}+{C_{4}}={C_{3}}\mathbb{E}{\left({\log ^{-}}{A_{\nu }}\right)^{4}}+{C_{4}}\lt \infty \end{array}\]
for some ${C_{3}}\gt 0$ and ${C_{4}}\ge 0$. □Using Theorem 5.1 on p. 107 in [11] we obtain the following result.
Furthermore, Theorem 5.6 on p. 126 in [11] entails the proposition.
Proposition 3.
Assume (A1), (A2), (A4) and $\mathbb{E}d\lt \infty $ and $\mathbb{E}{(\kappa ({f_{\nu }};a))^{4}}\lt \infty $ for some $a\in {\mathbb{N}_{0}}$. Then, with ${({B_{+}}(t))_{t\in [0,1]}}$ being the Brownian meander,
\[ \mathrm{Law}\left({\left(\frac{\log {Z_{{S_{\lfloor nt\rfloor }}}}}{\mathfrak{v}\sqrt{n}}\right)_{t\in [0,1]}}\hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{{S_{n}}}}\gt 0\right)\hspace{2.5pt}\Longrightarrow \hspace{2.5pt}\mathrm{Law}\left({({B_{+}}(t))_{t\in [0,1]}}\right),\hspace{1em}n\to \infty ,\]
weakly on the space of probability measures on $D[0,1]$ endowed with the Skorokhod ${J_{1}}$-topology.
The corollary given next follows from formula (1.1) in [8].
Corollary 2.
Under the assumptions of Proposition 3, for every fixed $t\in (0,1]$,
The random variable ${B_{+}}(t)$ has an absolutely continuous distribution with a bounded nonvanishing density on $[0,\infty )$.
(7)
\[ \underset{n\to \infty }{\lim }\mathbb{P}\left\{\frac{\log {Z_{{S_{\lfloor nt\rfloor }}}}}{\mathfrak{v}\sqrt{n}}\ge x\hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{{S_{n}}}}\gt 0\right\}=\mathbb{P}\{{B_{+}}(t)\ge x\},\hspace{1em}x\ge 0.\]2.2 Proof of Proposition 1 and Theorem 1
Recall that ${\tau _{\mathrm{Embed}}}=\inf \{k\ge 0:{Z_{{S_{k}}}}=0\}$ is the extinction time of the embedded process ${({Z_{{S_{k}}}})_{k\ge 0}}$ and note that
\[ \mathbb{P}\{{\tau _{\mathrm{Sparse}}}\lt \infty \}\ge \mathbb{P}\{{\tau _{\mathrm{Embed}}}\lt \infty \}=1,\]
where the equality is justified by (6). This proves Proposition 1.For $n\in {\mathbb{N}_{0}}$, define the first passage time $\vartheta (n)$ by
Note that
Checking of relation (9) is clearly sufficient for a proof of Theorem 1. Furthermore, (9) would demonstrate that
\[ \mathbb{P}\{{Z_{{S_{\vartheta (n)}}}}\gt 0\}\le \mathbb{P}\{{Z_{n}}\gt 0\}\le \mathbb{P}\{{Z_{{S_{\vartheta (n)-1}}}}\gt 0\},\hspace{1em}n\in {\mathbb{N}_{0}}.\]
In view of the strong law of large numbers for $\vartheta (n)$, which reads
\[ \frac{\vartheta (n)}{n}\hspace{2.5pt}\to \hspace{2.5pt}\frac{1}{\mathtt{m}},\hspace{1em}n\to \infty ,\hspace{1em}\text{a.s.},\]
and Proposition 2, it is natural to expect that
(9)
\[ \mathbb{P}\{{Z_{{S_{\vartheta (n)}}}}\gt 0\}\hspace{2.5pt}\sim \hspace{2.5pt}\frac{{\mathtt{m}^{1/2}}{C_{\mathrm{Embed}}}}{\sqrt{n}}\hspace{2.5pt}\sim \hspace{2.5pt}\mathbb{P}\{{Z_{{S_{\vartheta (n)-1}}}}\gt 0\},\hspace{1em}n\to \infty .\]Observe that
\[\begin{array}{c}\displaystyle \mathbb{P}\{{Z_{{S_{\vartheta (n)}}}}\gt 0\}=\mathbb{P}\{{\tau _{\mathrm{Embed}}}\gt \vartheta (n)\}\\ {} \displaystyle =\mathbb{P}\{{\tau _{\mathrm{Embed}}}-1\ge \vartheta (n)\}=\mathbb{P}\{{S_{{\tau _{\mathrm{Embed}}}-1}}\gt n\},\end{array}\]
and, similarly,
\[\begin{array}{c}\displaystyle \mathbb{P}\{{Z_{{S_{\vartheta (n)-1}}}}\gt 0\}=\mathbb{P}\{{\tau _{\mathrm{Embed}}}\gt \vartheta (n)-1\}\\ {} \displaystyle =\mathbb{P}\{{\tau _{\mathrm{Embed}}}\ge \vartheta (n)\}=\mathbb{P}\{{S_{{\tau _{\mathrm{Embed}}}}}\gt n\}.\end{array}\]
The desired relation (9) follows from Theorem 3.1 in [12] applied with $r=3/2$ provided we can check that
\[ n\mathbb{P}\{d\gt n\}=o(\mathbb{P}\{{\tau _{\mathrm{Embed}}}\gt n\})=o(\mathbb{P}\{{Z_{{S_{n}}}}\gt 0\}),\hspace{1em}n\to \infty .\]
By Proposition 2, this is equivalent to
which is secured by assumption (A3). This completes the proof of Theorem 1. Remark 2.
It is plausible that the assumption $\mathbb{E}{d^{3/2}}\lt \infty $ in Theorem 1 can be replaced by $\mathbb{E}d\lt \infty $. According to Proposition 2 we still have in this case
For the claim of Theorem 1 to be true it is sufficient that
The major difficulty in proving that (11) together with $\mathbb{E}d\lt \infty $ implies (12) is dependence of ${\tau _{\mathrm{Embed}}}$ and ${({S_{k}})_{k\in \mathbb{N}}}$. If these quantities were independent, then (12) would hold, see Proposition 4.3 in [9]. In our setting, ${\tau _{\mathrm{Embed}}}$ and ${({S_{k}})_{k\in \mathbb{N}}}$ are not independent but ${\tau _{\mathrm{Embed}}}$ does not depend on the future of ${({S_{k}})_{k\in \mathbb{N}}}$ in the sense of [7], see Section 7 therein. However, we have not been able to prove any version of the aforementioned Proposition 4.3 in [9] suitable for our purposes.
(11)
\[ \mathbb{P}\{{\tau _{\mathrm{Embed}}}\gt n\}\hspace{2.5pt}\sim \hspace{2.5pt}\frac{{C_{\mathrm{Embed}}}}{\sqrt{n}},\hspace{1em}n\to \infty .\](12)
\[ \mathbb{P}\{{S_{{\tau _{\mathrm{Embed}}}}}\gt n\}\hspace{2.5pt}\sim \hspace{2.5pt}\frac{{C_{\mathrm{Sparse}}}}{\sqrt{n}},\hspace{1em}n\to \infty .\]Remark 3.
The tail behavior of ${\tau _{\mathrm{Sparse}}}$ is rather elusive in the case where $\mathbb{E}d=\infty $ and/or $\mathbb{E}{(\log {A_{1}})^{2}}=\infty $. We shall only sketch difficulties that arise in this scenario. Let ${({\widehat{T}_{n}})_{n\in \mathbb{N}}}$ be a standard zero-delayed random walk with steps $\log {A_{k}}$, $k\in \mathbb{N}$, see (4). It is known, see Section 5.8 in [11], that the existence of the limit
and some further technical assumptions akin to (A4) imply that
The random variable W has the same distribution as ${\mathcal{S}_{\alpha }^{-\alpha }}$, where ${\mathcal{S}_{\alpha }}$ is a random variable with an α-stable distribution concentrated on the positive halfline. Since $\mathbb{E}{\mathcal{S}_{\alpha }^{r}}\lt \infty $ for all $r\in (0,\alpha )$, we infer $\mathbb{E}{W^{\rho -1}}\lt \infty $. Hence, the right-hand side of (14) is regularly varying at ∞ of index $(\rho -1)\alpha $. However, this result looks quite dubious, since for an extremely sparse environment we expect the survivial probability to be of order close to $1/n$, as for the critical Galton–Watson process. The problem here is again the dependence of ${\tau _{\mathrm{Embed}}}$ and $\vartheta (n)$ which makes the ‘natural’ asymptotics ≈ in (14) doubtful. We do not have any reasonable conjecture for the asymptotic behavior of $\mathbb{P}\{{\tau _{\mathrm{Embed}}}\ge \vartheta (n)\}$.
\[ \mathbb{P}\{{\tau _{\mathrm{Embed}}}\gt n\}\hspace{2.5pt}\sim \hspace{2.5pt}\frac{\ell (n)}{{n^{1-\rho }}},\hspace{1em}n\to \infty ,\]
for some ℓ slowly varying at infinity. Assume further that the distribution of d belongs to the domain of attraction of an α-stable law, $\alpha \in (0,1)$. Recall that $\mathbb{P}\{d\gt n\}\vartheta (n)$ converges in distribution as $n\to \infty $ to a random variable W, say, with the Mittag-Leffler distribution with parameter α, see, for instance, Theorem 7 in [10]. Therefore, one is tempted to write
(14)
\[\begin{array}{cc}& \displaystyle \mathbb{P}\{{Z_{n}}\gt 0\}\hspace{2.5pt}\sim \hspace{2.5pt}\mathbb{P}\{{\tau _{\mathrm{Embed}}}\ge \vartheta (n)\}\hspace{2.5pt}\sim \hspace{2.5pt}\mathbb{P}\{{\tau _{\mathrm{Embed}}}\gt \vartheta (n)\}\\ {} & \displaystyle \approx \hspace{2.5pt}\frac{\ell (1/\mathbb{P}\{d\ge n\})}{{(\mathbb{P}\{d\ge n\})^{\rho -1}}}\mathbb{E}{W^{\rho -1}},\hspace{1em}n\to \infty .\end{array}\]2.3 Proof of Theorem 2
We start by noting that $\mathtt{m}\lt \infty $ together with the strong law of large numbers for $(\vartheta (n))$ imply
weakly on the space of probability measures on $D[0,1]$. We have replaced log by ${\log ^{+}}$ in (15) because the event $\{{Z_{{S_{\lfloor {\mathtt{m}^{-1}}n\rfloor }}}}\gt 0\}$ does not entail the event
By Proposition 2, Theorem 1 and (10),
Thus, the limit relation (16) follows once we can prove that
In view of (17), it suffices to show that
\[ \underset{t\in [0,1]}{\sup }\left|\frac{\vartheta (\lfloor nt\rfloor )-1}{n}-\frac{t}{\mathtt{m}}\right|\hspace{2.5pt}\to 0,\hspace{1em}n\to \infty ,\hspace{1em}\text{a.s.}\]
Thus, the weak convergence claimed in Proposition 3 can be strengthened to the joint convergence
\[\begin{array}{c}\displaystyle \mathrm{Law}\left(\left({\left(\frac{\log {Z_{{S_{\lfloor {\mathtt{m}^{-1}}nt\rfloor }}}}}{\mathfrak{v}\sqrt{{\mathtt{m}^{-1}}n}},\frac{\vartheta (\lfloor nt\rfloor )-1}{{\mathtt{m}^{-1}}n}\right)_{t\in [0,1]}}\hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{{S_{\lfloor {\mathtt{m}^{-1}}n\rfloor }}}}\gt 0\right)\right)\\ {} \displaystyle \hspace{2.5pt}\Longrightarrow \hspace{2.5pt}\mathrm{Law}\left({\left({B_{+}}(t),t\right)_{t\in [0,1]}}\right),\hspace{1em}n\to \infty ,\end{array}\]
which holds weakly on the space of probability measures on $D[0,1]\times D[0,1]$ endowed with the product ${J_{1}}$-topology. Using the continuous mapping theorem in combination with continuity of the composition (see, for instance, Theorem 13.2.2 in [14]) we infer
(15)
\[\begin{array}{cc}& \displaystyle \mathrm{Law}\left(\left({\left(\frac{{\log ^{+}}{Z_{{S_{\vartheta (\lfloor nt\rfloor )-1}}}}}{\mathfrak{v}\sqrt{{\mathtt{m}^{-1}}n}}\right)_{t\in [0,1]}}\hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{{S_{\lfloor {\mathtt{m}^{-1}}n\rfloor }}}}\gt 0\right)\right)\\ {} & \displaystyle \Longrightarrow \hspace{2.5pt}\mathrm{Law}\left({({B_{+}}(t))_{t\in [0,1]}}\right),\hspace{1em}n\to \infty ,\end{array}\]
\[ \Big\{{Z_{{S_{\vartheta (\lfloor nt\rfloor )-1}}}}\gt 0\hspace{2.5pt}\text{for all}\hspace{2.5pt}t\in [0,1]\Big\}.\]
Now we check that (15) secures
(16)
\[\begin{array}{cc}& \displaystyle \mathrm{Law}\left(\left({\left(\frac{\log {Z_{{S_{\vartheta (\lfloor nt\rfloor )-1}}}}}{\mathfrak{v}\sqrt{{\mathtt{m}^{-1}}n}}\right)_{t\in [0,1]}}\hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{n}}\gt 0\right)\right)\\ {} & \displaystyle \Longrightarrow \hspace{2.5pt}\mathrm{Law}\left({({B_{+}}(t))_{t\in [0,1]}}\right),\hspace{1em}n\to \infty .\end{array}\](17)
\[ \mathbb{P}\{{Z_{n}}\gt 0\}\hspace{2.5pt}\sim \hspace{2.5pt}\mathbb{P}\{{Z_{{S_{\lfloor {\mathtt{m}^{-1}}n\rfloor }}}}\gt 0\}\hspace{2.5pt}\sim \hspace{2.5pt}\frac{{C_{\mathrm{Sparse}}}}{\sqrt{n}},\hspace{1em}n\to \infty .\](18)
\[ \underset{n\to \infty }{\lim }\sqrt{n}\mathbb{P}\{{Z_{{S_{\lfloor {\mathtt{m}^{-1}}n\rfloor }}}}\gt 0,{Z_{n}}=0\}=\underset{n\to \infty }{\lim }\sqrt{n}\mathbb{P}\{{Z_{{S_{\lfloor {\mathtt{m}^{-1}}n\rfloor }}}}=0,{Z_{n}}\gt 0\}=0.\]
\[ \underset{n\to \infty }{\lim }\mathbb{P}\{{Z_{n}}\gt 0\hspace{0.1667em}|\hspace{0.1667em}{Z_{{S_{\lfloor {\mathtt{m}^{-1}}n\rfloor }}}}\gt 0\}=1.\]
Fix any $\varepsilon \gt 0$. The assumption (A3) implies that
by Theorem 4 in [3]. Thus,
\[\begin{aligned}{}& \hspace{-14.22636pt}\mathbb{P}\{{Z_{n}}\gt 0\hspace{0.1667em}|\hspace{0.1667em}{Z_{{S_{\lfloor {\mathtt{m}^{-1}}n\rfloor }}}}\gt 0\}\\ {} & =\mathbb{P}\{{Z_{n}}\gt 0,{S_{\lfloor {\mathtt{m}^{-1}}(1+\varepsilon )n\rfloor }}\gt n\hspace{0.1667em}|\hspace{0.1667em}{Z_{{S_{\lfloor {\mathtt{m}^{-1}}n\rfloor }}}}\gt 0\}+o(1)\\ {} & \ge \mathbb{P}\{{Z_{{S_{\lfloor {\mathtt{m}^{-1}}(1+\varepsilon )n\rfloor }}}}\gt 0,{S_{\lfloor {\mathtt{m}^{-1}}(1+\varepsilon )n\rfloor }}\gt n\hspace{0.1667em}|\hspace{0.1667em}{Z_{{S_{\lfloor {\mathtt{m}^{-1}}n\rfloor }}}}\gt 0\}+o(1)\\ {} & =\mathbb{P}\{{Z_{{S_{\lfloor {\mathtt{m}^{-1}}(1+\varepsilon )n\rfloor }}}}\gt 0\hspace{0.1667em}|\hspace{0.1667em}{Z_{{S_{\lfloor {\mathtt{m}^{-1}}n\rfloor }}}}\gt 0\}+o(1)\\ {} & =\frac{\mathbb{P}\{{Z_{{S_{\lfloor {\mathtt{m}^{-1}}(1+\varepsilon )n\rfloor }}}}\gt 0\}}{\mathbb{P}\{{Z_{{S_{\lfloor {\mathtt{m}^{-1}}n\rfloor }}}}\gt 0\}}+o(1)\\ {} & \to {(1+\varepsilon )^{-1/2}},\hspace{1em}n\to \infty ,\end{aligned}\]
where we have used Proposition 2 for the last passage. Sending $\varepsilon \to 0$ gives (18).To finish the proof of Theorem 2 it remains to check that, for all $\varepsilon \gt 0$,
To this end, we need an auxiliary lemma.
(19)
\[ \underset{n\to \infty }{\lim }\mathbb{P}\left\{\underset{t\in [0,1]}{\sup }\left|\frac{\log {Z_{\lfloor nt\rfloor }}-\log {Z_{{S_{\vartheta (\lfloor nt\rfloor )-1}}}}}{\mathfrak{v}\sqrt{{\mathtt{m}^{-1}}n}}\right|\gt \varepsilon \hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{n}}\gt 0\right\}=0.\]In order to prove (19) we first show that
Note that
where ${({\widetilde{Z}_{k}^{(j)}}(m))_{k\ge 0}}$ is the Galton–Watson process initiated by the j-th individual in the generation m. On the event $\{{Z_{n}}\gt 0\}$,
By Markov’s inequality in combination with $\mathbb{P}\{{Z_{n}}\gt 0\}\ge (1/{C_{5}}){n^{-1/2}}$ for some ${C_{5}}\gt 0$ and large n,
(20)
\[ \underset{n\to \infty }{\lim }\mathbb{P}\left\{\underset{t\in [0,1]}{\sup }\frac{\log {Z_{\lfloor nt\rfloor }}-\log {Z_{{S_{\vartheta (\lfloor nt\rfloor )-1}}}}}{\mathfrak{v}\sqrt{{\mathtt{m}^{-1}}n}}\gt \varepsilon \hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{n}}\gt 0\right\}=0.\](21)
\[ {Z_{\lfloor nt\rfloor }}={\sum \limits_{j=1}^{{Z_{{S_{\vartheta (\lfloor nt\rfloor )-1}}}}}}{\widetilde{Z}_{\lfloor nt\rfloor -{S_{\vartheta (\lfloor nt\rfloor )-1}}}^{(j)}}({S_{\vartheta (\lfloor nt\rfloor )-1}}),\hspace{1em}t\in [0,1],\hspace{1em}n\in \mathbb{N},\]
\[ \frac{{Z_{\lfloor nt\rfloor }}}{{Z_{{S_{\vartheta (\lfloor nt\rfloor )-1}}}}}=\frac{{\textstyle\textstyle\sum _{j=1}^{{Z_{{S_{\vartheta (\lfloor nt\rfloor )-1}}}}}}{\widetilde{Z}_{\lfloor nt\rfloor -{S_{\vartheta (\lfloor nt\rfloor )-1}}}^{(j)}}({S_{\vartheta (\lfloor nt\rfloor )-1}})}{{Z_{{S_{\vartheta (\lfloor nt\rfloor )-1}}}}},\hspace{1em}t\in [0,1],\hspace{1em}n\in \mathbb{N},\]
and thereupon
\[ \underset{t\in [0,1]}{\sup }\frac{{Z_{\lfloor nt\rfloor }}}{{Z_{{S_{\vartheta (\lfloor nt\rfloor )-1}}}}}\le \underset{1\le k\le \vartheta (n)}{\sup }\frac{{\textstyle\textstyle\sum _{j=1}^{{Z_{{S_{k-1}}}}}}{\max _{0\le i\le {d_{k}}}}{\widetilde{Z}_{i}^{(j)}}({S_{k-1}})}{{Z_{{S_{k-1}}}}},\hspace{1em}n\in \mathbb{N}.\]
Instead of (20), we shall prove a stronger relation
(22)
\[ \underset{n\to \infty }{\lim }\mathbb{P}\left\{\sum \limits_{1\le k\le \vartheta (n)}\frac{{\textstyle\textstyle\sum _{j=1}^{{Z_{{S_{k-1}}}}}}{\max _{0\le i\le {d_{k}}}}{\widetilde{Z}_{i}^{(j)}}({S_{k-1}})}{{Z_{{S_{k-1}}}}}\gt \varepsilon {n^{3}}\hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{n}}\gt 0\right\}=0.\]
\[\begin{aligned}{}& \hspace{-5.69054pt}\mathbb{P}\left\{\sum \limits_{1\le k\le \vartheta (n)}\frac{{\textstyle\textstyle\sum _{j=1}^{{Z_{{S_{k-1}}}}}}{\max _{0\le i\le {d_{k}}}}{\widetilde{Z}_{i}^{(j)}}({S_{k-1}})}{{Z_{{S_{k-1}}}}}\gt \varepsilon {n^{3}}\hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{n}}\gt 0\right\}\\ {} & \le {\varepsilon ^{-1}}{n^{-3}}{\sum \limits_{k=1}^{\infty }}\mathbb{E}\left({1_{\{{S_{k-1}}\le n\}}}\frac{1}{{Z_{{S_{k-1}}}}}{\sum \limits_{j=1}^{{Z_{{S_{k-1}}}}}}\underset{0\le i\le {d_{k}}}{\max }{\widetilde{Z}_{i}^{(j)}}({S_{k-1}})\hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{n}}\gt 0\right)\\ {} & \le {C_{5}}{\varepsilon ^{-1}}{n^{-5/2}}{\sum \limits_{k=1}^{\infty }}\mathbb{E}\left({1_{\{{S_{k-1}}\le n,\hspace{0.1667em}{Z_{n}}\gt 0\}}}\frac{1}{{Z_{{S_{k-1}}}}}{\sum \limits_{j=1}^{{Z_{{S_{k-1}}}}}}\underset{0\le i\le {d_{k}}}{\max }{\widetilde{Z}_{i}^{(j)}}({S_{k-1}})\right)\\ {} & \le {C_{5}}{\varepsilon ^{-1}}{n^{-5/2}}{\sum \limits_{k=1}^{\infty }}\mathbb{E}\left({1_{\{{S_{k-1}}\le n,\hspace{0.1667em}{Z_{{S_{k-1}}}}\gt 0\}}}\frac{1}{{Z_{{S_{k-1}}}}}{\sum \limits_{j=1}^{{Z_{{S_{k-1}}}}}}\underset{0\le i\le {d_{k}}}{\max }{\widetilde{Z}_{i}^{(j)}}({S_{k-1}})\right)\\ {} & ={C_{5}}{\varepsilon ^{-1}}{n^{-5/2}}\left(\mathbb{E}\underset{0\le i\le d}{\max }{\widetilde{Z}_{i}}\right){\sum \limits_{k=1}^{\infty }}\mathbb{P}\{{S_{k-1}}\le n\}=O({n^{-3/2}}),\hspace{1em}n\to \infty .\end{aligned}\]
To justify the penultimate equality, observe that, given $({Z_{{S_{k-1}}}},{S_{k-1}})$, the sequences
\[ {({\widetilde{Z}_{i}^{(1)}}({S_{k-1}}))_{i\ge 0}},\dots ,{({\widetilde{Z}_{i}^{({Z_{{S_{k-1}}}})}}({S_{k-1}}))_{i\ge 0}}\]
are independent copies of the critical Galton–Watson process ${({\widetilde{Z}_{i}})_{i\ge 0}}$. The last equality is a consequence of Lemma 2 and the elementary renewal theorem which states that
We shall now check that
Using again decomposition (21), we write on the event $\{{Z_{n}}\gt 0\}$
Since $\mathbb{P}\{{Z_{{S_{\vartheta (n)}}}}\gt 0\}\sim \mathbb{P}\{{Z_{n}}\gt 0\}$ as $n\to \infty $, by (9), and $\{{Z_{{S_{\vartheta (n)}}}}\gt 0\}$ entails $\{{Z_{n}}\gt 0\}$, relation (24) is equivalent to
Observe that on the event $\{{Z_{{S_{\vartheta (n)}}}}\gt 0\}$
Let $\widetilde{p}:=\mathbb{P}\{{\widetilde{Z}_{d}}\gt 0\}$ be the probability of the event that the critical Galton–Watson process ${({\widetilde{Z}_{k}})_{k\ge 0}}$ survives up to random time d independent of ${({\widetilde{Z}_{k}})_{k\ge 0}}$. Obviously, $\widetilde{p}\in (0,1)$. Given $({S_{k-1}},{Z_{{S_{k-1}}}})$, the sum
(23)
\[ \underset{n\to \infty }{\lim }\mathbb{P}\left\{\underset{t\in [0,1]}{\inf }\frac{\log {Z_{\lfloor nt\rfloor }}-\log {Z_{{S_{\vartheta (\lfloor nt\rfloor )-1}}}}}{\mathfrak{v}\sqrt{{\mathtt{m}^{-1}}n}}\lt -\varepsilon \hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{n}}\gt 0\right\}=0.\]
\[\begin{aligned}{}\underset{t\in [0,1]}{\inf }\frac{{Z_{\lfloor nt\rfloor }}}{{Z_{{S_{\vartheta (\lfloor nt\rfloor )-1}}}}}& =\underset{t\in [0,1]}{\inf }\frac{{\textstyle\textstyle\sum _{j=1}^{{Z_{{S_{\vartheta (\lfloor nt\rfloor )-1}}}}}}{\widetilde{Z}_{\lfloor nt\rfloor -{S_{\vartheta (\lfloor nt\rfloor )-1}}}^{(j)}}({S_{\vartheta (\lfloor nt\rfloor )-1}})}{{Z_{{S_{\vartheta (\lfloor nt\rfloor )-1}}}}}\\ {} & \ge \underset{1\le k\le \vartheta (n)}{\inf }\frac{{\textstyle\textstyle\sum _{j=1}^{{Z_{{S_{k-1}}}}}}{\min _{0\le i\le {d_{k}}}}{\widetilde{Z}_{i}^{(j)}}({S_{k-1}})}{{Z_{{S_{k-1}}}}}\\ {} & \ge \underset{1\le k\le \vartheta (n)}{\inf }\frac{{\textstyle\textstyle\sum _{j=1}^{{Z_{{S_{k-1}}}}}}{1_{\{{\widetilde{Z}_{{d_{k}}}^{(j)}}({S_{k-1}})\gt 0\}}}}{{Z_{{S_{k-1}}}}},\hspace{1em}n\in \mathbb{N}.\end{aligned}\]
As in the proof of (20), we shall prove a relation which is stronger than (23), namely,
(24)
\[ \underset{n\to \infty }{\lim }\mathbb{P}\left\{\underset{1\le k\le \vartheta (n)}{\inf }\frac{{\textstyle\textstyle\sum _{j=1}^{{Z_{{S_{k-1}}}}}}{1_{\{{\widetilde{Z}_{{d_{k}}}^{(j)}}({S_{k-1}})\gt 0\}}}}{{Z_{{S_{k-1}}}}}\lt \varepsilon {n^{-3}}\hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{n}}\gt 0\right\}=0.\](25)
\[ \underset{n\to \infty }{\lim }\mathbb{P}\left\{\underset{1\le k\le \vartheta (n)}{\inf }\frac{{\textstyle\textstyle\sum _{j=1}^{{Z_{{S_{k-1}}}}}}{1_{\{{\widetilde{Z}_{{d_{k}}}^{(j)}}({S_{k-1}})\gt 0\}}}}{{Z_{{S_{k-1}}}}}\lt \varepsilon {n^{-3}}\hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{{S_{\vartheta (n)}}}}\gt 0\right\}=0.\]
\[ {\sum \limits_{j=1}^{{Z_{{S_{k-1}}}}}}{1_{\{{\widetilde{Z}_{{d_{k}}}^{(j)}}({S_{k-1}})\gt 0\}}}\gt 0,\hspace{1em}k\le \vartheta (n),\]
since otherwise the population does not survive up to time ${S_{\vartheta (n)}}$. Using this and the union bound yields
\[\begin{array}{c}\displaystyle \mathbb{P}\left\{\underset{1\le k\le \vartheta (n)}{\inf }\frac{1}{{Z_{{S_{k-1}}}}}{\sum \limits_{j=1}^{{Z_{{S_{k-1}}}}}}{1_{\{{\widetilde{Z}_{{d_{k}}}^{(j)}}({S_{k-1}})\gt 0\}}}\lt \varepsilon {n^{-3}}\hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{{S_{\vartheta (n)}}}}\gt 0\right\}\\ {} \displaystyle \le \sum \limits_{k\ge 1}\mathbb{P}\left\{0\lt \frac{1}{{Z_{{S_{k-1}}}}}{\sum \limits_{j=1}^{{Z_{{S_{k-1}}}}}}{1_{\{{\widetilde{Z}_{{d_{k}}}^{(j)}}({S_{k-1}})\gt 0\}}}\lt \varepsilon {n^{-3}},k\le \vartheta (n)\hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{{S_{\vartheta (n)}}}}\gt 0\right\}.\end{array}\]
Invoking $\mathbb{P}\{{Z_{{S_{\vartheta (n)}}}}\gt 0\}\ge (1/{C_{6}}){n^{-1/2}}$ for some ${C_{6}}\gt 0$ and large n, we obtain, for such n,
(26)
\[\begin{aligned}{}& \sum \limits_{k\ge 1}\mathbb{P}\Big\{0\lt \frac{1}{{Z_{{S_{k-1}}}}}{\sum \limits_{j=1}^{{Z_{{S_{k-1}}}}}}{1_{\{{\widetilde{Z}_{{d_{k}}}^{(j)}}({S_{k-1}})\gt 0\}}}\lt \frac{\varepsilon }{{n^{3}}},k\le \vartheta (n)\hspace{0.1667em}\Big|\hspace{0.1667em}{Z_{{S_{\vartheta (n)}}}}\gt 0\Big\}\\ {} & \le {C_{6}}\sqrt{n}\sum \limits_{k\ge 1}\mathbb{P}\Big\{0\lt \frac{1}{{Z_{{S_{k-1}}}}}{\sum \limits_{j=1}^{{Z_{{S_{k-1}}}}}}{1_{\{{\widetilde{Z}_{{d_{k}}}^{(j)}}({S_{k-1}})\gt 0\}}}\lt \frac{\varepsilon }{{n^{3}}},k\le \vartheta (n),{Z_{{S_{\vartheta (n)}}}}\gt 0\Big\}\\ {} & ={C_{6}}\sqrt{n}\sum \limits_{k\ge 1}\mathbb{P}\Big\{0\lt \frac{1}{{Z_{{S_{k-1}}}}}{\sum \limits_{j=1}^{{Z_{{S_{k-1}}}}}}{1_{\{{\widetilde{Z}_{{d_{k}}}^{(j)}}({S_{k-1}})\gt 0\}}}\lt \frac{\varepsilon }{{n^{3}}},{S_{k-1}}\le n,{Z_{{S_{\vartheta (n)}}}}\gt 0\Big\}\\ {} & \le {C_{6}}\sqrt{n}\sum \limits_{k\ge 1}\mathbb{P}\Big\{0\lt \frac{1}{{Z_{{S_{k-1}}}}}{\sum \limits_{j=1}^{{Z_{{S_{k-1}}}}}}{1_{\{{\widetilde{Z}_{{d_{k}}}^{(j)}}({S_{k-1}})\gt 0\}}}\lt \frac{\varepsilon }{{n^{3}}},{S_{k-1}}\le n,{Z_{{S_{k-1}}}}\gt 0\Big\}.\end{aligned}\]
\[ {\sum \limits_{j=1}^{{Z_{{S_{k-1}}}}}}{1_{\{{\widetilde{Z}_{{d_{k}}}^{(j)}}({S_{k-1}})\gt 0\}}}\]
has a binomial distribution with parameters $({Z_{{S_{k-1}}}},\widetilde{p})$.In what follows we denote by $\mathrm{Bin}(N,p)$ a random variable with a binomial distribution with N and p interpreted as the number of independent trials and a success probability, respectively. The next lemma provides a uniform in N estimate for $\mathbb{P}\{0\lt {N^{-1}}\mathrm{Bin}(N,p)\le x\}$, which is useful when x is close to zero.
Proof.
Plainly, $\mathbb{P}\{0\lt {N^{-1}}\mathrm{Bin}(N,p)\le x\}=0$ if $x\lt 1/N$. If $x\ge 1/N$, then by Chebyshev’s inequality
□
Using Lemma 3 we estimate the summands in (26) as follows. For $k\ge 1$ and n large enough,
\[\begin{aligned}{}& \hspace{-5.69054pt}\mathbb{P}\left\{0\lt \frac{1}{{Z_{{S_{k-1}}}}}{\sum \limits_{j=1}^{{Z_{{S_{k-1}}}}}}{1_{\{{\widetilde{Z}_{{d_{k}}}^{(j)}}({S_{k-1}})\gt 0\}}}\lt \varepsilon {n^{-3}},{S_{k-1}}\le n,{Z_{{S_{k-1}}}}\gt 0\right\}\\ {} & =\mathbb{P}\left\{0\lt \frac{1}{{Z_{{S_{k-1}}}}}{\sum \limits_{j=1}^{{Z_{{S_{k-1}}}}}}{1_{\{{\widetilde{Z}_{{d_{k}}}^{(j)}}({S_{k-1}})\gt 0\}}}\lt \varepsilon {n^{-3}}\hspace{0.1667em}\Big|\hspace{0.1667em}{S_{k-1}}\le n,{Z_{{S_{k-1}}}}\gt 0\right\}\\ {} & \hspace{199.16928pt}\times \mathbb{P}\{{S_{k-1}}\le n,{Z_{{S_{k-1}}}}\gt 0\}\\ {} & \le \frac{\widetilde{p}(1-\widetilde{p})\varepsilon }{(\widetilde{p}-\varepsilon {n^{-3}})}{n^{-3}}\mathbb{P}\{{S_{k-1}}\le n,{Z_{{S_{k-1}}}}\gt 0\}\\ {} & \le \frac{\widetilde{p}(1-\widetilde{p})\varepsilon }{(\widetilde{p}-\varepsilon {n^{-3}})}{n^{-3}}\mathbb{P}\{{S_{k-1}}\le n\}.\end{aligned}\]
Summarizing, the probability on the left-hand side of (25) is bounded from above by
\[ {C_{6}}{n^{1/2}}\frac{\widetilde{p}(1-\widetilde{p})\varepsilon }{(\widetilde{p}-\varepsilon {n^{-3}})}{n^{-3}}\sum \limits_{k\ge 1}\mathbb{P}\{{S_{k-1}}\le n\}=O({n^{-3/2}}),\hspace{1em}n\to \infty ,\]
thereby finishing the proof of (25) and Theorem 2.
