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Asymptotic behaviour of the survival probability of almost critical branching processes in a random environment V. V. Kharlamov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
Russian version:
Matematicheskii Sbornik, 2024, Volume 215, Number 1, DOI: https://doi.org/10.4213/sm9923 
Bibliographic databases:
    § 1. IntroductionBranching processes in random environment (BPREs) are a generalization of well-known Galton–Watson branching processes. Unlike the latter, the distribution of the number of offspring in each generation in a BPRE depends on some random factor, called an environment. Like in the case of Galton–Watson branching processes, the subcritical, critical and supercritical cases can be considered for BPREs. There are results concerning the survival probability for all three types of BPREs in the case when the environment consists of independent and identically distributed random elements. In this paper we consider a critical BPRE. The asymptotic behaviour of the survival probability of the process in the case when the generating function of the distribution for the number of offspring of a single particle is linear fractional was deduced in [1]. With no assumption on the explicit form of the distribution for the number of offspring of a single particle, some results were obtained in [2] and generalized in [3]. A sequence $\Xi=\{\xi_i,\,i \in \mathbb{N}\}$ of independent and identically distributed random elements with values in a measurable space $(Y, \mathcal{G})$ is called a random environment. We denote by $(\Omega, \mathcal{F}, \mathsf{P}^{\Xi})$ the probability space on which a random environment $\Xi$ is defined. We consider a family of generating functions Fix $\omega \in \Omega$.
The process $\mathcal{Z}=\{Z_k,\,k \geqslant 0\}$ is called a branching process in a random environment $\Xi$. For each fixed $\omega \in \Omega$ we introduce a probability measure $\mathsf{P}_{\omega}$ on the space of sequences $\mathbb{N}_0^{\infty}$, $\mathbb{N}_0 :=\{0\} \cup \mathbb{N}$. For any $G \subset \mathbb{N}_0^{\infty}$ and $V \in \mathcal{F}$ we set
$$
\begin{equation*}
\mathsf{P}(\mathcal{Z} \in G, \Xi \in V) :=\int_{V}\mathsf{P}_{\omega}(\mathcal{Z} \in G)\, \mathsf{P}^{\Xi}(d \omega)
\end{equation*}
\notag
$$
and extend the probability measure $\mathsf{P}$ to $(\mathbb{N}_0^{\infty} \times \Omega, \sigma(2^{\mathbb{N}_0^{\infty}} \times \mathcal{F}))$. Note that
$$
\begin{equation*}
\mathsf{P}(\mathcal{Z} \in 2^{\mathbb{N}_0^{\infty}},\,\Xi \in V)=\mathsf{P}^{\Xi}(V)
\end{equation*}
\notag
$$
for any $V \in \mathcal{F}$. We set
$$
\begin{equation*}
X_i :=\log F_{i-1}'(1), \qquad S_0:=0 \quad\text{and} \quad S_k :=X_1 + \dots + X_k, \quad i, k \in \mathbb{N}.
\end{equation*}
\notag
$$
By the definition of $\Xi$ the random variables $X_i$, $i \geqslant 1$, are independent and identically distributed. The sequence of random variables $\{S_k,\,k \in \mathbb{N}\}$ is called the associated random walk. We introduce an operator $T$ on the space of generating functions that maps a generating function $f$ to the quantity Theorem 1 (see [4], Theorem 5.1). Under Assumptions 1 and 2, where $c_{-}=\sum_{k=1}^{\infty} k^{-1} (\mathsf{P}(S_k<0)-1/2)$ and $\Upsilon$ is some positive constant. Remark 1. Theorem 1 follows from the results obtained in [3] and [4], but this fact requires a separate substantiation, which is presented in § 3. In this paper we consider the triangular array scheme of branching processes $\{Z_{k,n},\, {k \!\leqslant\! n}\}$ in the random environment $\Xi$. Our goal is to find conditions on $Z_{k,n}$ under which the probabilities $\mathsf{P}(Z_n>0)$ and $\mathsf{P}(Z_{n,n}>0)$ are equivalent as $n \to \infty$. The paper is structured as follows. In § 2 the main result is presented. In § 3 a theorem on survival probability for a BPRE is proved. In § 4 several assertions used to substantiate the main result are established. § 2. The main resultWe consider a family of generating functions and fix $\omega \in \Omega$.
The set of random variables $\{Z_{k,n},\,0 \leqslant k \leqslant n,\,Z_{0,n}=1\}$ is said to be a perturbed branching process in the random environment $\Xi$ (PBPRE). We state the following assumption on the smallness of perturbations. Remark 2. It follows from Assumption 3 that for any $\omega \in \Omega$, $k \in \mathbb{N}$ and $G \subset \mathbb{N}_0^k$. We denote the deviations of the logarithms of the first moments of the generating functions by
$$
\begin{equation*}
a_{i,n}(\omega) :=\log F_{i-1,n;\omega}'(1)-\log F_{i-1;\omega}'(1); \quad\text{let}\quad b_{k,n}(\omega) :=\sum_{i=1}^k a_{i,n}(\omega).
\end{equation*}
\notag
$$
Assumption 4. For some $\delta_2 \in (0, 1/2)$ and $C_2>0$ the sequence of events where $\theta(k) :=C_2 k^{1/2-\delta_2}$, $k \in \mathbb{N}$ and $\theta(0) :=0$, satisfies The main result in this paper is as follows. § 3. Proof of Theorem 1In what follows we use the notation $K,K_1,\dots$ for positive constants, which are generally different in distinct assertions. We prove the theorem on the basis of the proof of Theorem 5.1 in [4]. For an arbitrary nonnegative integer-valued random variable $\zeta$ and a relevant generating function $f$ we use the notation
$$
\begin{equation*}
f[c] :=\mathsf{P}(\zeta=c)\quad\text{and} \quad \varkappa(f; c) :=\frac{\sum_{y=c}^{\infty} y^2 f[y]}{(f'(1))^2}, \qquad c \in \mathbb{N}_0.
\end{equation*}
\notag
$$
Assumption 6 (see [4], Assumption C). There exists $c \in \mathbb{N}_0$ such that In the proof of Theorem 5.1 in [4], Assumption C is used only to prove Lemma 5.5. We prove this lemma in the case when Assumptions 1 and 2 hold. The statement of Lemma 5.5 involves a certain measure $\mathsf{P}^+$. We will not dwell on the details of its construction: it was thoroughly described in [4], § 5. By the definition of the measure $\mathsf{P}^+$,
$$
\begin{equation}
\mathsf{E}^+ Y_n=\mathsf{E}\bigl(Y_n U(S_n);\,L_n \geqslant 0\bigr)
\end{equation}
\tag{3.1}
$$
for a $\sigma(S_1, \dots, S_n)$-measurable nonnegative random variable $Y_n$, where
$$
\begin{equation*}
U(x) :=\mathsf{I}\{x \geqslant 0\} + \sum_{n=1}^{\infty} \mathsf{P}\bigl(S_n \geqslant -x,\,\max\{S_1, \dots, S_n\}<0\bigr)
\end{equation*}
\notag
$$
and The function $U(x)$ has a number of useful properties, which were described in [4], § 4.4.3. In particular, $U(x)$ is a renewal function; hence
$$
\begin{equation}
U(x + y) \leqslant U(x) + U(y), \qquad x, y \geqslant 0.
\end{equation}
\tag{3.2}
$$
Owing to Lemma 4.3 in [4], we have
$$
\begin{equation}
U(x) \sim \frac{x \sqrt{2}}{\sigma}\, e^{c_{-}}, \qquad x \to \infty.
\end{equation}
\tag{3.3}
$$
By virtue of (3.3) and Assumption 1, the quantity $\mathsf{E} U(X)^2$ is finite. Lemma 1 (see [4], Theorem 4.6). Let $L_{k,n} :=\min\{S_k, \dots, S_{n}\}\,{-}\,S_k$. Then, under Assumption 1, for each $k \in \mathbb{N}_0$. In addition, there exists a constant $K$ such that for all $0 \leqslant k \leqslant n$. Lemma 2. Assume that Assumptions 1 and 2 hold. Then the series converges $\mathsf{P}^+$-almost surely. Proof. In was deduced in the proof of Lemma 5.5 in [4] that by Assumption 1, for any $\delta \in (0, 1/2)$ there exists a set $\Omega''=\Omega''(\delta)$, $\mathsf{P}^+(\Omega'')=1$, such that for any $\omega \in \Omega''$ the inequality holds for all $j \in \mathbb{N}$ and some positive function $D_1(\omega)$. We choose $\delta :=\delta_1/2>0$ and fix $\omega \in \Omega''$.
To estimate the sum (3.6) we need an estimate for the probability
$$
\begin{equation*}
\mathsf{P}^+\bigl(T(F_j)>x\bigr), \qquad x \geqslant 1.
\end{equation*}
\notag
$$
By virtue of the independence of $(S_j, L_j)$ and $(T(F_j), X_{j+1})$, in view of (3.1) and (3.2) we have
$$
\begin{equation}
\begin{aligned} \, \notag &\mathsf{P}^+(T(F_j)>x) =\mathsf{E}\bigl(\mathsf{I}\{T(F_j)>x\}\, U(S_{j+1}) \, \mathsf{I}\{L_{j+1} \geqslant 0\}\bigr) \\ \notag &\qquad\leqslant \mathsf{E}\bigl((U(S_j) + U(X_{j+1}))\, \mathsf{I}\{T(F_j)>x\} \, \mathsf{I}\{L_j \geqslant 0\}\bigr) \\ \notag &\qquad=\mathsf{E}\bigl(U(S_j)\, \mathsf{I}\{L_j \geqslant 0\}\bigr)\, \mathsf{P}\big(T(F_j)>x\bigr) + \mathsf{E}\bigl(U(X_{j+1}) \, \mathsf{I}\{T(F_j)>x\}\bigr)\mathsf{P}(L_j \geqslant 0) \\ &\qquad=\mathsf{P}\bigl(T(F_j)>x\bigr)+ \mathsf{E}\bigl(U(X_{j+1}) \, \mathsf{I}\{T(F_j)>x\}\bigr)\mathsf{P}(L_j \geqslant 0). \end{aligned}
\end{equation}
\tag{3.8}
$$
Using the Cauchy–Schwarz–Bunyakovsky inequality, Assumption 1 and relation (3.3), we obtain
$$
\begin{equation}
\begin{aligned} \, \notag \mathsf{E}\bigl(U(X_{j+1}) \mathsf{I}\{T(F_j)>x\}\bigr) &\leqslant \sqrt{\mathsf{E} U(X_{j+1})^2\, \mathsf{E}\, \mathsf{I}^2\{T(F_j)>x\}} \\ &=\sqrt{\mathsf{E} U(X_{j+1})^2\, \mathsf{P}\bigl(T(F_j)>x\bigr)} \leqslant K_1 \sqrt{\mathsf{P}\bigl(T(F_j)>x\bigr)}. \end{aligned}
\end{equation}
\tag{3.9}
$$
Combining the estimates (3.8) and (3.9) and using Lemma 1 we infer that
$$
\begin{equation}
\sum_{j=1}^{\infty} \mathsf{P}^+(T(F_j)>x_j) \leqslant \sum_{j=1}^{\infty} h_j^0 + K_2 \sum_{j=1}^{\infty} \sqrt{\frac{h_j^0}{j}}
\end{equation}
\tag{3.10}
$$
for $x=x_j=\exp\{j^{1/2-\delta_1}\}$. The second term on the right-hand side of (3.10) is finite by Assumption 2. We prove the finiteness of the first term on the right-hand side of (3.10).
Note that
$$
\begin{equation}
\sum_{j=1}^{\infty} h_j^0 =\sum_{j=1}^{\infty} \sqrt{j h_j^0}\, \sqrt{\frac{h_j^0}{j}} \leqslant \sup\Bigl\{\sqrt{j h_j^0}\Bigm| j \in \mathbb{N}\Bigr\} \sum_{j=1}^{\infty} \sqrt{\frac{h_j^0}{j}}.
\end{equation}
\tag{3.11}
$$
If the first factor on the right-hand side of (3.11) is finite, then the first term on the right-hand side of (3.10) is too. Assume the contrary. Then there exists an increasing sequence $\{n_k \in \mathbb{N} \mid k \in \mathbb{N}\}$ such that $h_{n_k}^0>1/n_k$. The sequence $h_j^0$ does not increase, which yields
$$
\begin{equation}
\sum_{j=[n_k/2]}^{n_k} \sqrt{\frac{h_j^0}{j}} \geqslant \sum_{j=[n_k/2]}^{n_k} \sqrt{\frac{2 h_{n_k}^0}{n_k}} \geqslant \sum_{j=[n_k/2]}^{n_k} \frac{\sqrt{2}}{n_k} \geqslant \frac{1}{\sqrt{2}}.
\end{equation}
\tag{3.12}
$$
Since $n_k \geqslant k$ for all $k \in \mathbb{N}$ the relation (3.12) contradicts Assumption 2 in view of Cauchy’s criterion.
By virtue of the convergence of the two series on the right-hand side of (3.10) and the Borel–Cantelli lemma, there exists a set $\Omega'''$, $\mathsf{P}^+(\Omega''')=1$, such that for any $\omega \in \Omega'''$ there exists a positive function $D_2(\omega)$ such that
$$
\begin{equation}
T(F_j) \leqslant D_2(\omega) \exp\{j^{1/2-\delta_1}\}.
\end{equation}
\tag{3.13}
$$
Using (3.13) we obtain
$$
\begin{equation*}
\sum_{j=1}^{\infty} T(F_j) \exp\{-S_{j-1}\} \leqslant D_2(\omega) \sum_{j=1}^{\infty} \exp\{j^{1/2-\delta_1} - D_1(\omega) (j-1)^{1/2-\delta_1/2}\} < \infty
\end{equation*}
\notag
$$
for $\omega \in \Omega' :=\Omega'' \cap \Omega'''$.
Lemma 2 is proved. Proof of Theorem 1. By Lemma 2 and Corollary 5.7 in [4] there is a set $\Omega' \subset \Omega$, $\mathsf{P}^+(\Omega')=1$, such that for $\omega \in \Omega'$. It follows that
$$
\begin{equation}
\mathsf{P}^+\biggl(\bigcap_{n=1}^{\infty} \{Z_n>0\}\biggr)>0.
\end{equation}
\tag{3.14}
$$
This relation replaces Lemma 5.8 in [4], which is used there in the proof of Theorem 5.1. The other assertions in [4] involved in the proof of Theorem 5.1, namely Lemmas 5.2 and 5.9 and Theorem 4.6, use only Assumption 1.
Theorem 1 is proved. § 4. Auxiliary assertions and the proof of the main result Lemma 3. Let Proof. We prove only the first part of (4.1). The second part is a special case of the first for $F_{i,n} \equiv F_i$.
Let
$$
\begin{equation*}
\tau_n :=\min\bigl\{\arg \min\{S_k \mid k \in \{0, \dots, n\}\}\bigr\}
\end{equation*}
\notag
$$
denote the first moment of time at which the random walk attains its minimum on the set $\{0, \dots, n\}$. Then
$$
\begin{equation}
\mathsf{P}(Z_{n,n}>0) =\sum_{k=0}^n \sum_{l=1}^{+\infty} \mathsf{P}(Z_{n,n}>0,\, Z_{k,n}=l,\, \tau_n=k).
\end{equation}
\tag{4.2}
$$
The event $\{\tau_n=k\}$ coincides with the event
$$
\begin{equation*}
\{S_i>S_k \ \forall\, i \in \{0, \dots, k-1\},\,L_{k,n} \geqslant 0\} =J_k \cap \{L_{k,n} \geqslant 0\}.
\end{equation*}
\notag
$$
By the independence of $(S_1, \dots, S_k)$ and $(L_{k,n}, Z_{n,n})$ under the condition $Z_{k,n}=l$, we have
$$
\begin{equation}
\begin{aligned} \, \notag &\mathsf{P}(Z_{n,n}>0,\, Z_{k,n}=l,\, \tau_n=k) \\ &\qquad =\mathsf{P}(\{Z_{k,n}=l\} \cap J_k)\, \mathsf{P}(Z_{n,n}>0, \, L_{k,n} \geqslant 0 \mid Z_{k,n}=l) \end{aligned}
\end{equation}
\tag{4.3}
$$
for arbitrary natural $k$ and $l$. Using (4.2) and (4.3) we arrive at the required equality (4.1).
The lemma is proved. To prove Theorem 2 we divide the double sum in the representation from Lemma 3 into the two parts corresponding to the subsets of indices $k \leqslant m$ and $k>m$, respectively. We show that the second part is negligible in comparison with the first for large $m$. For this purpose we need the following estimate. Lemma 4. Let $\widehat{L}_k$ denote $\min\{S_i + \theta(i) \mid 0 \leqslant i \leqslant k\}$. Then, under Assumption 1, for any $\delta \in (0, 1/2)$ there exists a constant $K$ such that for all $k \in \mathbb{N}$. Proof. We have The inequality holds.
We estimate the quantity $p_2$. We define recursively a random variable $\nu_i$. Set ${\nu_0=0}$ and assume that $\nu_i$ is defined for $i \geqslant 0$; then
$$
\begin{equation}
\nu_{i+1} :=\min\{j>\nu_i\mid S_j-S_{\nu_i} + \theta(j-\nu_i)<0\} \in \mathbb{N} \cup \{+\infty\}.
\end{equation}
\tag{4.7}
$$
We set $\rho_k :=\max\{r \geqslant 0 \mid \nu_r \leqslant k\}$.
Then
$$
\begin{equation}
p_2 \leqslant \mathsf{P}\bigl(\widehat{L}_k >-\log k\bigr) \leqslant \mathsf{P}\bigl(\rho_k \leqslant k^{\delta/2}-1\bigr)+ \mathsf{P}\bigl(\widehat{L}_k >-\log k,\, \rho_k>k^{\delta/2}-1\bigr).
\end{equation}
\tag{4.8}
$$
We estimate the first term on the right-hand side of (4.8). We set $\Delta_0 \,{:=}\, 0$ and $\Delta_i :=\nu_i-\nu_{i-1}$ for $i>0$ and note that the $\Delta_i$ are independent, identically distributed random variables for $i>0$. Since the events $\{\rho_k \leqslant r\}$ and $\{\nu_{r+1}>k\}$ coincide for an arbitrary $r$, Markov’s inequality and the subadditivity of the function $g(x) :=x^{1/2-\delta/2}$, $x \geqslant 0$, imply the relation
$$
\begin{equation}
\begin{aligned} \, \notag \mathsf{P}(\rho_k \leqslant r)=\mathsf{P}(\nu_{r+1}>k) &=\mathsf{P}\biggl(\biggl(\sum_{i=1}^{r+1} \Delta_i\biggr)^{1/2-\delta/2} > k^{1/2-\delta/2}\biggr) \\ &\leqslant \frac{\mathsf{E} \bigl(\sum_{i=1}^{r+1} \Delta_i\bigr)^{1/2-\delta/2} }{k^{1/2-\delta/2}} \leqslant \frac{(r+1) \, \mathsf{E} \Delta_1^{1/2-\delta/2}}{k^{1/2-\delta/2}}. \end{aligned}
\end{equation}
\tag{4.9}
$$
It follows from Assumption 1 that $g(i)=o(\mathsf{D} S_i)$ as $i \to \infty$. Owing to [5], Theorem 1,
$$
\begin{equation*}
\mathsf{P}(\Delta_1 \geqslant i) \leqslant \frac{K_1}{i^{1/2-\delta/4}}
\end{equation*}
\notag
$$
for $i \geqslant 1$. Then
$$
\begin{equation}
\begin{aligned} \, \notag \mathsf{E} \Delta_1^{1/2-\delta/2} &=\sum_{i=1}^{\infty} (i^{1/2-\delta/2}-(i-1)^{1/2-\delta/2}) \, \mathsf{P}(\Delta_1 \geqslant i) \\ &\leqslant K_1 + \sum_{i=1}^{\infty} \frac{K_2}{i^{1+\delta/4}} =: K_3<\infty. \end{aligned}
\end{equation}
\tag{4.10}
$$
By estimates (4.9) and (4.10) we have
$$
\begin{equation}
\mathsf{P}(\rho_k \leqslant k^{\delta/2}-1) \leqslant \frac{K_3 k^{\delta/2}}{k^{1/2-\delta/2}} =\frac{K_3}{k^{1/2-\delta}}.
\end{equation}
\tag{4.11}
$$
Now we estimate the second term on the right-hand side of (4.8). For any $i>0$, by the subadditivity of $\theta(i)$ we have
$$
\begin{equation}
\begin{aligned} \, \notag S_{\nu_i} + \theta(\nu_i)- (S_{\nu_{i-1}} + \theta(\nu_{i-1})) &=S_{\nu_i}-S_{\nu_{i-1}} + \theta(\nu_i)-\theta(\nu_{i-1}) \\ &\leqslant S_{\nu_i}-S_{\nu_{i-1}} + \theta(\nu_i-\nu_{i-1})=: \eta_i. \end{aligned}
\end{equation}
\tag{4.12}
$$
Note that the $\eta_i$, $i>0$, are independent and identically distributed negative random variables. It follows that
$$
\begin{equation}
\begin{aligned} \, \notag \widehat{L}_k &=\min\{S_{\nu_i} + \theta(\nu_i)\mid 0 \leqslant i \leqslant \rho_k\} =S_{\nu_{\rho_k}} + \theta(\nu_{\rho_k}) \\ &=\sum_{i=1}^{\rho_k} \bigl(S_{\nu_i} + \theta(\nu_i) - (S_{\nu_{i-1}} + \theta(\nu_{i-1}))\bigr) \leqslant \sum_{i=1}^{\rho_k} \eta_i. \end{aligned}
\end{equation}
\tag{4.13}
$$
By virtue of (4.13), Markov’s inequality, and since the $\eta_i$ are nonnegative, the second term on the right-hand side of (4.8) is estimated as follows:
$$
\begin{equation}
\begin{aligned} \, \notag &\mathsf{P}(\widehat{L}_k >-\log k,\, \rho_k>k^{\delta/2}-1) \\ &\qquad\leqslant \mathsf{P}\biggl(\sum_{i=1}^{\rho_k} \eta_i >-\log k,\, \rho_k>k^{\delta/2}-1\biggr) \leqslant \mathsf{P}\biggl(\sum_{i=1}^{[k^{\delta/2}-1]} \eta_i >-\log k\biggr) \notag \\ &\qquad=\mathsf{P}\biggl(\exp\biggl\{\sum_{i=1}^{[k^{\delta/2}-1]} \eta_i\biggr\} > \frac{1}{k}\biggr) \leqslant k (\mathsf{E} e^{\eta_1})^{[k^{\delta/2}-1]}. \end{aligned}
\end{equation}
\tag{4.14}
$$
Note that $\mathsf{E} e^{\eta_1}<1$, which yields
$$
\begin{equation}
k (\mathsf{E} e^{\eta_1})^{[k^{\delta/2}-1]} \leqslant \frac{K_4}{k}, \qquad k \in \mathbb{N}.
\end{equation}
\tag{4.15}
$$
The estimates (4.5), (4.6), (4.8), (4.11), (4.14) and (4.15) imply the required inequality
$$
\begin{equation*}
\mathsf{E} \exp\{\widehat{L}_k\} \leqslant \frac{1}{k} + \frac{K_3}{k^{1/2-\delta}} + \frac{K_4}{k} \leqslant \frac{K}{k^{1/2-\delta}}.
\end{equation*}
\notag
$$
Lemma 4 is proved. Lemma 5. Let Proof. We set
$$
\begin{equation}
q_1 :=\mathsf{P}\bigl(\{Z_{k,n}>0,\,S_k \leqslant -k^{1/2-\delta_2/2}\} \cap Q_{k,n} \cap J_k\bigr)
\end{equation}
\tag{4.16}
$$
and
$$
\begin{equation*}
q_2 :=\mathsf{P}\bigl(\{Z_{k,n}>0,\,S_k>-k^{1/2-\delta_2/2}\} \cap Q_{k,n} \cap J_k\bigr).
\end{equation*}
\notag
$$
By the definition of the event $Q_{k,n}$ the first quantity is estimated as follows:
$$
\begin{equation}
\begin{aligned} \, \notag q_1 &\leqslant \mathsf{E}\bigl(Z_{k,n};\, \{S_k \leqslant -k^{1/2-\delta_2/2}\} \cap Q_{k,n} \cap J_k\bigr) \\ \notag &\leqslant \mathsf{E}\bigl(\exp\{S_k + \theta(k)\};\, S_k \leqslant -k^{1/2-\delta_2/2}\bigr) \\ &\leqslant \exp\{k^{1/2-\delta_2}(-k^{\delta_2/2} + k^{\delta_2-1/2} \theta(k))\} =\exp\{k^{1/2-\delta_2} (-k^{\delta_2/2} + C_2)\}. \end{aligned}
\end{equation}
\tag{4.17}
$$
We introduce the notation
$$
\begin{equation*}
\widehat{\tau}_k=\min\biggl\{\arg \min\biggl\{S_i + \theta(i)\biggm| i \leqslant \biggl[\frac k3\biggr]\biggr\}\biggr\}.
\end{equation*}
\notag
$$
Let $\omega \in Q_{k,n}$. On the strength of the fact that the event $\{Z_{k,n}>0\}$ is embedded in $\{Z_{\widehat{\tau}_k,n}>0\}$ and $Z_{\widehat{\tau}_k,n}$ is integer valued, we have
$$
\begin{equation}
\mathsf{P}_{\omega}(Z_{k,n}>0) \leqslant \mathsf{P}_{\omega}(Z_{\widehat{\tau}_k,n}>0) \leqslant \mathsf{E}_{\omega} Z_{\widehat{\tau}_k,n}.
\end{equation}
\tag{4.18}
$$
Owing to the properties of the branching process and the choice of $\omega$, we have
$$
\begin{equation}
\mathsf{E}_{\omega} Z_{\widehat{\tau}_k,n} =\exp\{S_{\widehat{\tau}_k}(\omega) \,{+}\, b_{\widehat{\tau}_k,n}(\omega)\} \leqslant \exp\{S_{\widehat{\tau}_k}(\omega) \,{+}\, \theta(\widehat{\tau}_k)\} =\exp\{\widehat{L}_{[k/3]}(\omega)\}.
\end{equation}
\tag{4.19}
$$
By virtue of relations (4.18) and (4.19),
$$
\begin{equation}
\begin{aligned} \, \notag q_2 &\leqslant \mathsf{E}\bigl(\exp\{\widehat{L}_{[k/3]}\};\, \{S_k>-k^{1/2-\delta_2/2}\} \cap J_k\bigr) \\ \notag &\leqslant \mathsf{E}\biggl(\exp\{\widehat{L}_{[k/3]}\};\, S_i>S_k \ \forall\, i \in \biggl\{\biggl[\frac{2k}{3}\biggr], \dots, k-1\biggr\},\, S_k \in (-k^{1/2-\delta_2/2}, 0)\biggr) \\ &=\mathsf{E} Y \, \mathsf{I}\{S_k \in (-k^{1/2-\delta_2/2}, 0)\}, \end{aligned}
\end{equation}
\tag{4.20}
$$
where
We consider the sigma algebra
$$
\begin{equation*}
\mathcal{H}_k :=\sigma(X_1, \dots, X_{[k/3]}, X_{[2 k/3] + 1}, \dots, X_k).
\end{equation*}
\notag
$$
Since the random variable $Y$ is $\mathcal{H}_k$-measurable, we have
$$
\begin{equation}
\begin{aligned} \, \notag \mathsf{E} Y \, \mathsf{I}\{S_k \in (-k^{1/2-\delta_2/2}, 0)\} &=\mathsf{E} \bigl(\mathsf{E}(Y \mathsf{I}\{S_k \in (-k^{1/2-\delta_2/2}, 0)\} \mid \mathcal{H}_k)\bigr) \\ &=\mathsf{E}(Y\, \mathsf{P}\bigl(S_k \in (-k^{1/2-\delta_2/2}, 0) \mid \mathcal{H}_k)\bigr). \end{aligned}
\end{equation}
\tag{4.21}
$$
Note that
$$
\begin{equation}
\begin{aligned} \, \notag &\mathsf{P}\bigl(S_k \in (-k^{1/2-\delta_2/2}, 0)\bigm| \mathcal{H}_k\bigr) \\ \notag &\qquad =\mathsf{P}\bigl(S_{[2 k/3]}-S_{[k/3]} + (S_{[k/3]} + S_k-S_{[2 k/3]}) \in (-k^{1/2-\delta_2/2}, 0) \bigm| \mathcal{H}_k\bigr) \\ &\qquad \leqslant \sup_{x \in \mathbb{R}} \mathsf{P}\bigl(S_{[2 k/3]}-S_{[k/3]} + x \in (0, k^{1/2-\delta_2/2}) \bigm| \mathcal{H}_k\bigr). \end{aligned}
\end{equation}
\tag{4.22}
$$
Since the random variable $S_{[2 k/3]}-S_{[k/3]}$ is independent of $\mathcal{H}_k$, the following identity holds $\mathsf{P}$-almost surely:
$$
\begin{equation}
\begin{aligned} \, \notag &\sup_{x \in \mathbb{R}} \mathsf{P}\bigl(S_{[2 k/3]}-S_{[k/3]} + x \in (0, k^{1/2-\delta_2/2})\bigm| \mathcal{H}_k\bigr) \\ &\qquad =\sup_{x \in \mathbb{R}}\mathsf{P}\bigl(S_{[2 k/3]-[k/3]} + x\in (0, k^{1/2-\delta_2/2})\bigr). \end{aligned}
\end{equation}
\tag{4.23}
$$
By (4.21)–(4.23) we have
$$
\begin{equation}
\begin{aligned} \, \notag &\mathsf{E} Y \, \mathsf{I}\{S_k \in (-k^{1/2-\delta_2/2}, 0)\} \\ &\qquad \leqslant \mathsf{E} Y\sup_{x \in \mathbb{R}}\mathsf{P}\bigl(S_{[2 k/3]-[k/3]} + x\in (0, k^{1/2-\delta_2/2})\bigr)\leqslant q_{2,1} q_{2,2} q_{2,3}, \end{aligned}
\end{equation}
\tag{4.24}
$$
where
$$
\begin{equation*}
\begin{gathered} \, q_{2,1} :=\mathsf{E} \exp\{\widehat{L}_k\}, \\ q_{2,2} :=\sup_{x \in \mathbb{R}}\mathsf{P}\bigl(S_{[2k/3]}-S_{[k/3]}\in (-x-k^{1/2-\delta_2/2}, -x)\bigr) \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
q_{2,3} :=\mathsf{P}\biggl(S_i>S_k \ \forall\, i \in \biggl\{\biggl[\frac{2k}3\biggr], \dots, k-1\biggr\}\biggr).
\end{equation*}
\notag
$$
We obtain upper estimates for each factor on the right-hand side of (4.24). Owing to Lemma 4, for $\delta=3 \delta_2/8$ we have
$$
\begin{equation}
q_{2,1} \leqslant \frac{K_1}{k^{1/2-3 \delta_2/8}}.
\end{equation}
\tag{4.25}
$$
It follows from the concentration inequality (see [6], Ch. III, Theorem 9) that there exists a positive constant $K_2$ such that
$$
\begin{equation}
q_{2,2} \leqslant \frac{K_2 k^{1/2-\delta_2/2}}{\sqrt{k}} =\frac{K_2}{k^{\delta_2/2}}.
\end{equation}
\tag{4.26}
$$
By Lemma 1 we have Inequalities (4.20) and (4.24)–(4.27) yield the estimate
$$
\begin{equation}
q_2 \leqslant q_{2,1} q_{2,2} q_{2,3} \leqslant \frac{K_4}{k^{1+\delta_2/8}}.
\end{equation}
\tag{4.28}
$$
Finally, estimates (4.17) and (4.28) for $q_1$ and $q_2$ imply the inequality
$$
\begin{equation*}
\begin{aligned} \, \mathsf{P}(\{Z_{k,n}>0\} \cap Q_{k,n} \cap J_k) &=q_1 + q_2 \\ &\leqslant \exp\{k^{1/2-\delta_2} (-k^{\delta_2/2} + C_2)\} + \frac{K_4}{k^{1+\delta_2/8}}\leqslant \frac{K}{k^{1+\delta_2/8}}. \end{aligned}
\end{equation*}
\notag
$$
Lemma 5 is proved. Proof. Owing to the proof of Lemma 3, we have
$$
\begin{equation}
\begin{aligned} \, \notag &\sum_{k=m+1}^n \sum_{l=1}^{\infty} A_{k,l,n} B_{n-k,l,n} =\mathsf{P}(Z_{n,n}>0,\,\tau_n>m) \\ &\qquad =\mathsf{P}(\{Z_{n,n}>0,\, \tau_n>m\} \cap Q_n) + \mathsf{P}(\{Z_{n,n}>0,\,\tau_n>m\} \cap \overline{Q}_n). \end{aligned}
\end{equation}
\tag{4.30}
$$
Note that
$$
\begin{equation}
\sqrt{n}\, \mathsf{P}(\{Z_{n,n}>0,\,\tau_n>m\} \cap \overline{Q}_n) \leqslant \sqrt{n}\, \mathsf{P}(\overline{Q}_n) \leqslant \widetilde{\alpha}_n
\end{equation}
\tag{4.31}
$$
by Assumption 4, where $\{\widetilde{\alpha}_n,\,n \in \mathbb{N}\}$ is a sequence of positive numbers decreasing to zero.
On the other hand, by virtue of the relation
$$
\begin{equation*}
\{\tau_n>m\}=\bigcup_{k=m+1}^n \{\tau_n=k\} =\bigcup_{k=m+1}^n J_k \cap \{L_{k,n} \geqslant 0\}
\end{equation*}
\notag
$$
we have the estimate
$$
\begin{equation}
\begin{aligned} \, \notag &\mathsf{P}(\{Z_{n,n}>0,\,\tau_n>m\} \cap Q_n) \\ \notag &\qquad \leqslant \sum_{k=m+1}^n \mathsf{P}(\{Z_{k,n}>0,\,L_{k,n} \geqslant 0\} \cap J_k \cap Q_{k,n}) \\ &\qquad =\sum_{k=m+1}^n \mathsf{P}(\{Z_{k,n}>0\}\cap J_k \cap Q_{k,n})\, \mathsf{P}(L_{k,n} \geqslant 0) \end{aligned}
\end{equation}
\tag{4.32}
$$
for the first term on the right-hand side of (4.30).
Using (4.30)–(4.32) and Lemmas 1 and 5 we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\sum_{k=m+1}^n \sum_{l=1}^{\infty} A_{k,l,n} B_{n-k,l,n} \\ \notag &\qquad\leqslant \mathsf{P}(\overline{Q}_n) + K_1 \sum_{k=m+1}^n \frac{1}{k^{1 + \delta_2/8}} \frac{K_2}{\max\{\sqrt{n-k}, K_2\}} \\ \notag &\qquad \leqslant \frac{\widetilde{\alpha}_n}{\sqrt{n}} + \sum_{k=m+1}^{n-[n/2]-1} \frac{K_3}{k^{1 + \delta_2/8} \sqrt{n-k}} + \sum_{k=n-[n/2]}^{n-1} \frac{K_3}{k^{1 + \delta_2/8} \sqrt{n-k}} + \frac{K_3}{n^{1 + \delta_2/8}} \\ &\qquad \leqslant \frac{\widetilde{\alpha}_m}{\sqrt{n}} + \frac{K_4}{\sqrt{n} m^{\delta_2/8}} + \frac{K_4}{n^{1/2 + \delta_2/8}} \int_{1/2}^1 \frac{d x}{x^{1 + \delta_2/8} \sqrt{1-x}} + \frac{K_4}{\sqrt{n} m^{\delta_2/8}} \leqslant \frac{\alpha_m}{\sqrt{n}}, \end{aligned}
\end{equation}
\tag{4.33}
$$
where
$$
\begin{equation*}
\alpha_m=K_5 \biggl(\widetilde{\alpha}_m+ \frac{3}{m^{\delta_2/8}}\biggr).
\end{equation*}
\notag
$$
Lemma 6 is proved. Owing to Lemma 6, we have reduced the investigation of the asymptotic behaviour of the sequence $\sum_{k=0}^n \sum_{l=1}^{\infty} A_{k,n,l} B_{n-k,l,n}$ to examining the sums $\sum_{k=0}^m \sum_{l=1}^{\infty} A_{k,n,l} B_{n-k,l,n}$. Now we constrain the values of $l$. Lemma 7. Under Assumptions 1 and 3, for fixed $m$ there exists a sequence of positive numbers $\{\beta_M= \beta_M(m),\,M \in \mathbb{N}\}$ such that $\beta_M \to 0$ as $M \to \infty$ and Proof. We prove that for each $k \in \mathbb{N}$ there exists a sequence of positive numbers $\{\beta_M^{(k)},\,M \in \mathbb{N}\}$ tending to zero as $M \to \infty$ such that
$$
\begin{equation}
\sup_{n\colon n \geqslant k} \mathsf{P}(Z_{k,n}>M)\leqslant \beta_M^{(k)}.
\end{equation}
\tag{4.35}
$$
Assume that the left-hand side of (4.35) does not tend to zero for some $k$. Then there are a positive number $\varepsilon$ and an increasing sequence of natural numbers $\{M_r,\,r \in \mathbb{N}\}$ such that
$$
\begin{equation}
\sup_{n\colon n \geqslant k} \mathsf{P}(Z_{k,n}>M_r)> \varepsilon.
\end{equation}
\tag{4.36}
$$
By virtue of (4.36), for each $r \in \mathbb{N}$ there is $n_r$ such that
If $\sup\{n_r \mid r \in \mathbb{N}\}<\infty$, then there exists a natural number that is infinitely often repeated in the sequence $\{n_r,\,r \in \mathbb{N}\}$. We denote this number by $n$. By the definition of $n$ there is an increasing sequence of natural numbers $\widetilde{M}_r$ such that
$$
\begin{equation}
\mathsf{P}(Z_{k,n}>\widetilde{M}_r) \geqslant \varepsilon.
\end{equation}
\tag{4.37}
$$
However, relation (4.37) contradicts the continuity of the probability measure.
If $\sup\{n_r \mid r \in \mathbb{N}\}=\infty$, then there exist increasing sequences of natural numbers $\{\widetilde{n}_r,\,r \in \mathbb{N}\}$ and $\{\widetilde{M}_r,\,r \in \mathbb{N}\}$ such that
$$
\begin{equation}
\mathsf{P}(Z_{k,\widetilde{n}_r}>\widetilde{M}_r) \geqslant \varepsilon.
\end{equation}
\tag{4.38}
$$
We choose $M>0$ so that We choose $R>0$ so that $\widetilde{M}_r>M$ for $r>R$. Then, owing to (4.38),
$$
\begin{equation}
\mathsf{P}(Z_{k,\widetilde{n}_r}>M) \geqslant \mathsf{P}(Z_{k,\widetilde{n}_r}>\widetilde{M}_r) \geqslant \varepsilon
\end{equation}
\tag{4.40}
$$
for all $r>R$. By virtue of Assumption 3 and Remark 2, the left-hand side of (4.40) tends to $\mathsf{P}(Z_k>M)$ as $r \to \infty$, which yields the estimate $\mathsf{P}(Z_k>M) \geqslant \varepsilon$. However, this contradicts (4.39).
In all above cases we arrive at a contradiction; hence (4.35) is true. By the inequality $B_{n-k,l,n} \leqslant \mathsf{P}(L_{k,n} \geqslant 0)$, Lemma 1 and (4.35) we have
$$
\begin{equation}
\begin{aligned} \, \notag \sum_{k=0}^m \sum_{l=M+1}^{\infty} A_{k,l,n} B_{n-k,l,n} &\leqslant \sum_{k=0}^m \mathsf{P}(Z_{k,n}>M) \frac{K_1}{\max\{\sqrt{n-k}, K_1\}} \\ &\leqslant \sum_{k=0}^m \beta_M^{(k)} \min\biggl\{1, \frac{K_1}{\sqrt{n-k}}\biggr\}. \end{aligned}
\end{equation}
\tag{4.41}
$$
Relation (4.41) makes it possible to deduce the required estimate (4.34).
Lemma 7 is proved. Using Lemmas 6 and 7 we can reduce the problem to the examination of a finite number of combinations $(k, l)$. In this case we can use Assumption 3 to switch from $F_{i-1,n}$ to $F_{i-1}$ for $i \leqslant k$. Lemma 8. Under Assumptions 1 and 3, for fixed $m$ and for any $\varepsilon>0$ there exist $M=M(\varepsilon)$ and $N=N(\varepsilon)$ such that for each $n>N$. Proof. We fix $\varepsilon>0$. By Lemma 7, there exist positive numbers $M$ and $ N_1$ such that
$$
\begin{equation*}
\sum_{k=0}^m \sum_{l=M+1}^{+\infty} A_{k,l,n} B_{n-k,l,n} \leqslant \frac{\varepsilon}{2 \sqrt{n}}
\end{equation*}
\notag
$$
for any $n>N_1 \geqslant 2 m$. Then
$$
\begin{equation}
\begin{aligned} \, \notag &\biggl|\sum_{k=0}^m \sum_{l=1}^{+\infty} A_{k,l,n} B_{n-k,l,n} - \sum_{k=0}^m \sum_{l=1}^M A_{k,l} B_{n-k,l,n}\biggr| \\ &\qquad \leqslant \sum_{k=0}^m \sum_{l=1}^M |A_{k,l,n}-A_{k,l}| B_{n-k,l,n} + \frac{\varepsilon}{2 \sqrt{n}}. \end{aligned}
\end{equation}
\tag{4.43}
$$
Applying Lemma 1 we obtain the estimate for $n>N_1$.
It follows from Assumption 3 and Remark 2 that for $k \leqslant m$ $Z_{k,n}$ converges to $Z_k$ in distribution as $n \to \infty$ for all $(\xi_1, \dots, \xi_k)$. By Lebesgue’s dominated convergence theorem we have
$$
\begin{equation*}
A_{k,l,n}=\mathsf{E}\bigl(\mathsf{I}_{J_k} \mathsf{P}(Z_{k,n}=l \mid \xi_1, \dots, \xi_k)\bigr) \to \mathsf{E}\bigl(\mathsf{I}_{J_k} \mathsf{P}(Z_k=l \mid \xi_1, \dots, \xi_k)\bigr)=A_{k,l}
\end{equation*}
\notag
$$
as $n \to \infty$. Thus, there exists $N \geqslant N_1$ such that
$$
\begin{equation}
|A_{k,l,n}-A_{k,l}| \leqslant \frac{\varepsilon}{4 K_1 M (m+1)}
\end{equation}
\tag{4.45}
$$
for $n>N$, $k \leqslant m$ and $l \leqslant M$. We conclude from (4.43)–(4.45) that
$$
\begin{equation}
\begin{aligned} \, \notag &\biggl|\sum_{k=0}^m \sum_{l=1}^{+\infty} A_{k,l,n} B_{n-k,l,n} - \sum_{k=0}^m \sum_{l=1}^M A_{k,l,n}^0 B_{n-k,l,n}\biggr| \\ &\qquad \leqslant \sum_{k=0}^m \sum_{l=1}^M \frac{\varepsilon}{2 M (m+1) \sqrt{n}} + \frac{\varepsilon}{2 \sqrt{n}} =\frac{\varepsilon}{\sqrt{n}}. \end{aligned}
\end{equation}
\tag{4.46}
$$
Lemma 8 is proved. It follows from Lemmas 6–8 that the main contribution to the asymptotic behaviour of the survival probability of a PBPRE is made by the sum $\sum_{k=0}^m \sum_{l=1}^M A_{k,l} B_{n-k,l,n}$. It remains to establish a result on the asymptotic behaviour of $B_{n-k,l,n}$. Since the number of terms in the sum under consideration is finite, there is no question of uniform asymptotics with respect to $k$ and $l$. In what follows we need an expression for survival probability in the case when the environment is fixed. Lemma 9 (see [4], Propositions 1.3 and 1.4). The survival probability of the PBPRE $Z_{k,n}$ satisfies To prove the next assertion we must pass to the limit under the condition that some part of the trajectory of the sequence $S_i$, $i \geqslant 0$, is nonnegative. A tool making it possible to take the limit is the measure $\mathsf{P}^+$. The properties of this measure are described in § 3. Lemma 10. Under Assumptions 1 and 5, for any $k \in \mathbb{N}_0$ there is a set $\Omega'$, $\mathsf{P}^+(\Omega')=1$, such that for all $\omega\in \Omega'$ and $\varepsilon>0$ there exists a parameter ${R=R(\omega, \varepsilon)}$ such that for any $r>R$ and $n>k + r$. Proof. Our argument is like in the proof of Lemma 2.
By virtue of Assumption 1, relation (3.7) for $\delta :=\min\{\delta_2, \delta_3\}/2$ holds for some $\Omega''=\Omega''(\delta)$ such that $\mathsf{P}^+(\Omega'')=1$. To estimate the sum (4.49) we need an estimate for the probability
$$
\begin{equation*}
\mathsf{P}^+(\widehat{F}_j>x) =\mathsf{P}^+\Bigl(\sup_{n\colon n>j} T(F_{j,n})> x\Bigr).
\end{equation*}
\notag
$$
Owing to the independence of $(S_j, L_j)$ and $(\widehat{F}_j, X_{j+1})$, in view of (3.1) and (3.2) we have
$$
\begin{equation}
\begin{aligned} \, \notag \mathsf{P}^+(\widehat{F}_j>x)&=\mathsf{E}\bigl(\mathsf{I}\{\widehat{F}_j>x\}\, U(S_{j+1}) \, \mathsf{I}\{L_{j+1} \geqslant 0\}\bigr) \\ \notag &\leqslant \mathsf{E}\bigl((U(S_j) + U(X_{j+1}))\, \mathsf{I}\{\widehat{F}_j>x\} \, \mathsf{I}\{L_j \geqslant 0\}\bigr) \\ \notag &=\mathsf{E}\bigl(U(S_j) \, \mathsf{I}\{L_j \geqslant 0\}\bigr)\, \mathsf{P}(\widehat{F}_j>x) + \mathsf{E}\bigl(U(X_{j+1})\, \mathsf{I}\{\widehat{F}_j>x\}\bigr)\, \mathsf{P}(L_j \geqslant 0) \\ &=\mathsf{P}(\widehat{F}_j>x)+ \mathsf{E}\bigl(U(X_{j+1})\, \mathsf{I}\{\widehat{F}_j>x\}\bigr)\, \mathsf{P}(L_j \geqslant 0). \end{aligned}
\end{equation}
\tag{4.50}
$$
Using the Cauchy–Schwarz–Bunyakovsky inequality, Assumption 1 and (3.3) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag \mathsf{E}(U(X_{j+1}) \mathsf{I}\{\widehat{F}_j>x\}) &\leqslant \sqrt{\mathsf{E} U(X_{j+1})^2\, \mathsf{E}\, \mathsf{I}^2\{\widehat{F}_j>x\}} \\ &=\sqrt{\mathsf{E} U(X_{j+1})^2\, \mathsf{P}(\widehat{F}_j>x)} \leqslant K_1 \sqrt{\mathsf{P}(\widehat{F}_j>x)} \end{aligned}
\end{equation}
\tag{4.51}
$$
for $x \geqslant 1$.
From estimates (4.50) and (4.51), Lemma 1 and Assumption 5 we infer the inequality
$$
\begin{equation}
\sum_{j=0}^{\infty}\mathsf{P}^+(\widehat{F}_j>x_j)\leqslant 1 + \sum_{j=1}^{\infty} h_j + K_2 \sum_{j=1}^{\infty}\sqrt{\frac{h_j}{j}}<\infty
\end{equation}
\tag{4.52}
$$
for $x=x_j=\exp\{j^{1/2-\delta_3}\}$. The convergence of the series (4.52) and the Borel–Cantelli lemma imply the existence of a set $\Omega'''$, $\mathsf{P}^+(\Omega''')=1$, such that for any $\omega \in \Omega'''$ there is a positive function $D_3(\omega)$ such that
$$
\begin{equation}
T(F_{j,n;\omega}) \leqslant T(\widehat{F}_{j;\omega}) \leqslant D_3(\omega) \exp\{j^{1/2-\delta_3}\}.
\end{equation}
\tag{4.53}
$$
Using estimate (4.53), for $\omega \in \Omega' :=\Omega'' \cap \Omega'''$ we infer that
$$
\begin{equation}
\begin{aligned} \, \notag &\mathsf{I}_{Q_n}(\omega) \sum_{i=r}^{n-k}T(F_{i+k,n;\omega}) \exp\biggl\{- \sum_{j=k+1}^{i+k}(X_j(\omega) + a_{j,n}(\omega))\biggr\} \\ \notag &\qquad \leqslant \mathsf{I}_{Q_n}(\omega) \exp\{S_k(\omega) + b_{k,n}(\omega)\} \sum_{i=r}^{n-k} T(F_{i+k,n;\omega}) \exp\{-S_{i+k}(\omega)-b_{i+k,n}(\omega)\} \\ \notag &\qquad \leqslant D_3(\omega) \exp\{S_k(\omega) + C_2 k^{1/2-\delta_2}\} \sum_{i=r}^{n-k} \exp\{(i+k)^{1/2-\delta_3}\} \\ &\qquad\qquad \times \exp\{- D_1(\omega) (i+k)^{1/2-\delta} + C_2 (i+k)^{1/2-\delta_2}\}. \end{aligned}
\end{equation}
\tag{4.54}
$$
Since $\delta<\delta_2$ and $\delta<\delta_3$, the terms of the series on the right-hand side of (4.54) are exponentially small. It follows that for all $\omega \in \Omega'$ and $\varepsilon>0$ there exists $R=R(\omega, \varepsilon)$ such that the left-hand side of inequality (4.49) is at most $\varepsilon$ for $r>R$ and $n>k+r$.
Lemma 10 is proved. Proof. Note that
$$
\begin{equation}
B_{n-k,l,n}=\mathsf{E}\bigl(f_l(\widetilde{\pi}_{k,n}) \bigm| L_{k,n} \geqslant 0\bigr)\, \mathsf{P}(L_{k,n} \geqslant 0),
\end{equation}
\tag{4.56}
$$
where In a similar way,
We prove that the difference of the first factors on the right-hand sides of (4.56) and (4.57) tends to zero. Note that
$$
\begin{equation}
\begin{aligned} \, \notag \frac{|B_{n-k,l,n}-B_{n-k,l}|}{\mathsf{P}(L_{k,n} \geqslant 0)} &\leqslant \mathsf{E}\bigl(|f_l(\widetilde{\pi}_{k,n}) - f_l(\widetilde{\pi}_{k,n}^{\,0})| \bigm| L_{k,n} \geqslant 0\bigr) \\ \notag &\leqslant l\, \mathsf{E}\bigl(|\widetilde{\pi}_{k,n} - \widetilde{\pi}_{k,n}^{\,0}| \bigm| L_{k,n} \geqslant 0\bigr) \\ &\leqslant l\, \mathsf{E}\bigl(\mathsf{I}_{Q_n} |\widetilde{\pi}_{k,n} - \widetilde{\pi}_{k,n}^{\,0}| \bigm| L_{k,n} \geqslant 0\bigr) + l \, \mathsf{P}(\overline{Q}_n \mid L_{k,n} \geqslant 0). \end{aligned}
\end{equation}
\tag{4.58}
$$
By Lemma 1 and Assumption 4 the second term on the right-hand side of (4.58) admits the estimate
$$
\begin{equation}
l \, \mathsf{P}(\overline{Q}_n \mid L_{k,n} \geqslant 0) \leqslant l \frac{\mathsf{P}(\overline{Q}_n)}{\mathsf{P}(L_{k,n} \geqslant 0)}\to 0, \qquad n \to \infty.
\end{equation}
\tag{4.59}
$$
We estimate the first term on the right in (4.58). Fix $\varepsilon>0$. By virtue of Lemmas 9 and 10 there exists a set $\Omega' \in \Omega, \mathsf{P}^+(\Omega')=1$ such that for any $\omega \in \Omega'$ there exists $R=R(\omega, \varepsilon)$ such that
$$
\begin{equation}
\mathsf{I}_{Q_n}(\omega) \sum_{i=r+1}^{n-k} d_{i+k,k+r,n}(\omega) \exp\biggl\{-\sum_{j=k+1}^{i+k}(X_j(\omega) + a_{j,n}(\omega))\biggr\} \leqslant \frac{\varepsilon}{3}
\end{equation}
\tag{4.60}
$$
for all $r>R$ and $n>k + r$.
We fix $\omega \in \Omega'$. Note that
$$
\begin{equation}
\widetilde{\pi}_{k,n} =\mathsf{P}_{\omega}(Z_{n,n}>0 \mid Z_{k,n}=1) \leqslant \mathsf{P}_{\omega}(Z_{k+r,n}>0 \mid Z_{k,n}=1).
\end{equation}
\tag{4.61}
$$
Owing to Lemma 9 and relation (4.60), we have
$$
\begin{equation}
\begin{aligned} \, \notag \mathsf{I}_{Q_n}(\omega) \widetilde{\pi}_{k,n}(\omega) &=\mathsf{I}_{Q_n}(\omega)\, \mathsf{P}_{\omega}(Z_{n,n}>0 \mid Z_{k,n}=1) \\ \notag &=\frac{\mathsf{I}_{Q_n}(\omega)}{\sum_{i=0}^{n-k} d_{i+k,k+r,n}(\omega) \exp\bigl\{-\sum_{j=k+1}^{i+k} (X_j(\omega) + a_{j,n}(\omega))\bigr\}} \\ &\geqslant \frac{\mathsf{I}_{Q_n}(\omega)}{\sum_{i=0}^r d_{i+k,k+r,n}(\omega) \exp\bigl\{-\sum_{j=k+1}^{i+k} (X_j(\omega) + a_{j,n}(\omega))\bigr\} + \varepsilon/3}. \end{aligned}
\end{equation}
\tag{4.62}
$$
If $\omega \notin Q_n$, then the last expression in (4.62) is zero. Otherwise, the denominator is at least $1$, which yields
$$
\begin{equation}
\begin{aligned} \, \notag &\frac{\mathsf{I}_{Q_n}(\omega)}{\sum_{i=0}^r d_{i+k,k+r,n}(\omega) \exp\bigl\{-\sum_{j=k+1}^{i+k} (X_j(\omega) + a_{j,n}(\omega))\bigr\} + \varepsilon/3} \\ \notag &\qquad \geqslant \biggl(1-\frac{\varepsilon}{3}\biggr) \frac{\mathsf{I}_{Q_n}(\omega)}{\sum_{i=0}^r d_{i+k,k+r,n}(\omega) \exp\bigl\{-\sum_{j=k+1}^{i+k} (X_j(\omega) + a_{j,n}(\omega))\bigr\}} \\ &\qquad =\biggl(1-\frac{\varepsilon}{3}\biggr)\, \mathsf{I}_{Q_n}(\omega) \, \mathsf{P}_{\omega}(Z_{k+r,n}>0 \mid Z_{k,n}=1). \end{aligned}
\end{equation}
\tag{4.63}
$$
It follows from (4.61)–(4.63) that
$$
\begin{equation}
\mathsf{I}_{Q_n}(\omega) \bigl|\widetilde{\pi}_{k,n}(\omega) - \mathsf{P}_{\omega}(Z_{k+r,n}>0 \mid Z_{k,n}=1)\bigr| \leqslant \frac{\varepsilon}{3}.
\end{equation}
\tag{4.64}
$$
In view of the convergence of the sequence $\widetilde{\pi}_{k,n}^{\,0}(\omega)$ as $n \to \infty$, there exists $r>R$ such that
$$
\begin{equation}
|\widetilde{\pi}_{k,k+r}^{\,0}(\omega) - \widetilde{\pi}_{k,n}^{\,0}(\omega)| \leqslant \frac{\varepsilon}{3}
\end{equation}
\tag{4.65}
$$
for each $n>k+r$.
Assumption 3 and Remark 2 imply that there exists $N>k + r$ such that
$$
\begin{equation}
\bigl|\mathsf{P}_{\omega}(Z_{k+r,n}>0 \mid Z_{k,n}=1) - \widetilde{\pi}_{k,k+r}^{\,0}(\omega)\bigr| \leqslant \frac{\varepsilon}{3}
\end{equation}
\tag{4.66}
$$
for each $n>N$.
It follows from (4.64)–(4.66) that
$$
\begin{equation}
\begin{aligned} \, \notag &\mathsf{I}_{Q_n}(\omega) |\widetilde{\pi}_{k,n}(\omega)- \widetilde{\pi}_{k,n}^{\,0}(\omega)| \\ \notag &\quad\leqslant \mathsf{I}_{Q_n}(\omega)\bigl|\widetilde{\pi}_{k,n}(\omega)- \mathsf{P}_{\omega}(Z_{k+r,n}>0 \mid Z_{k,n}=1)\bigr| \\ &\quad\qquad + \bigl|\mathsf{P}_{\omega}(Z_{k+r,n}>0 \mid Z_{k,n}=1) - \widetilde{\pi}_{k,k+r}^{\,0}(\omega)\bigr| + |\widetilde{\pi}_{k,k+r}^{\,0}(\omega)-\widetilde{\pi}_{k,n}^{\,0}(\omega)| \leqslant \varepsilon. \end{aligned}
\end{equation}
\tag{4.67}
$$
Since $\varepsilon$ is arbitrary, we conclude that the sequence of random variables ${\mathsf{I}_{Q_n} |\widetilde{\pi}_{k,n}-\widetilde{\pi}_{k,n}^{\,0}|}$ converges to $0$ $\mathsf{P}^+$-almost surely as $n \to \infty$. As this sequence is uniformly bounded, it follows from [4], Lemma 5.2, that
$$
\begin{equation}
\mathsf{E}\bigl(\mathsf{I}_{Q_n} |\widetilde{\pi}_{k,n}-\widetilde{\pi}_{k,n}^{\,0}| \bigm| L_{k,n} \geqslant 0\bigr) \to 0, \qquad n \to \infty.
\end{equation}
\tag{4.68}
$$
Lemma 1 and relations (4.58), (4.59) and (4.68) yield the required assertion (4.55).
Lemma 11 is proved. Proof. Fix $\omega \in \Omega$ and $j \in \mathbb{N}_0$. By Assumption 3 and Theorem 3.1.1 in [7] there exist a probability space $(\widehat{\Omega}_{\omega}, \widehat{\mathcal{F}}_{\omega}, \widehat{\mathsf{P}}_{\omega})$ and random variables $\widehat{Y}_{\omega, n}$, $j<n$, and $\widehat{Y}_{\omega}$ defined on it such that $F_{j;\omega}$ is the generating function for $\widehat{Y}_{\omega}$, $F_{j,n;\omega}$ is the generating function for $\widehat{Y}_{\omega,n}$, and $\widehat{Y}_{\omega,n} \to \widehat{Y}_{\omega}$ as $n \to \infty$ $\widehat{\mathsf{P}}_{\omega}$-almost surely. We denote the mean on this space by $\widehat{\mathsf{E}}_{\omega}$. By Fatou’s lemma,
$$
\begin{equation}
\begin{aligned} \, \notag F_{j;\omega}''(1) &=\widehat{\mathsf{E}}_{\omega}\bigl(\widehat{Y}_{\omega}(\widehat{Y}_{\omega}-1)\bigr) \\ &\leqslant \liminf_{n \to \infty} \widehat{\mathsf{E}}_{\omega} \bigl(\widehat{Y}_{\omega,n} (\widehat{Y}_{\omega,n}-1)\bigr) =\liminf_{n \to \infty} F_{j,n;\omega}''(1). \end{aligned}
\end{equation}
\tag{4.69}
$$
We consider $\omega \in Q_n$, $n \in \mathbb{N}$. For an arbitrary $j<n$ we have
$$
\begin{equation}
\begin{aligned} \, \notag |a_{j+1,n}(\omega)| &=|b_{j+1,n}(\omega)-b_{j,n}(\omega)| \\ &\leqslant |b_{j+1,n}(\omega)| + |b_{j,n}(\omega)| \leqslant 2 C_2 (j+1)^{1/2-\delta_2}. \end{aligned}
\end{equation}
\tag{4.70}
$$
Owing to the definition of $a_{j+1,n}$,
$$
\begin{equation}
\frac{F_{j,n;\omega}'(1)}{F_{j;\omega}'(1)} =\exp\{a_{j+1,n}(\omega)\} \leqslant \exp\{2 C_2 (j+1)^{1/2-\delta_2}\}.
\end{equation}
\tag{4.71}
$$
Inequalities (4.70) and (4.71) yield
$$
\begin{equation}
\frac{\mathsf{I}_{Q_n}(\omega)}{(F_{j;\omega}'(1))^2} \leqslant \exp\{4 C_2 (j+1)^{1/2-\delta_2}\} \frac{\mathsf{I}_{Q_n}(\omega)}{(F_{j,n;\omega}'(1))^2}
\end{equation}
\tag{4.72}
$$
for arbitrary $\omega \in \Omega$ and $n \in \mathbb{N}$. Note that, by virtue of Assumption 4, the sequence $\mathsf{I}_{Q_n}$ converges to $1$ in probability as $n \to \infty$. By Riesz’s theorem, there is a subsequence $\mathsf{I}_{Q_{n_k}}$ that converges to $1$ $\mathsf{P}$-almost surely as $k \to \infty$. It follows that
$$
\begin{equation}
\limsup_{n \to \infty} \mathsf{I}_{Q_n} \geqslant \limsup_{k \to \infty} \mathsf{I}_{Q_{n_k}} =1
\end{equation}
\tag{4.73}
$$
$\mathsf{P}$-almost surely. Using (4.69), (4.72) and (4.73), we obtain
$$
\begin{equation}
\begin{aligned} \, \notag T(F_j) &\leqslant \limsup_{n \to \infty} \mathsf{I}_{Q_n} \frac{F_j''(1)}{(F_j'(1))^2} \\ \notag &\leqslant \limsup_{n \to \infty} \exp\{4 C_2 (j+1)^{1/2-\delta_2}\} \frac{\mathsf{I}_{Q_n}}{(F_{j,n}'(1))^2} \liminf_{n \to \infty} F_{j,n}''(1) \\ \notag &\leqslant \exp\{4 C_2 (j+1)^{1/2-\delta_2}\} \limsup_{n \to \infty} \frac{\mathsf{I}_{Q_n} F_{j,n}''(1)}{(F_{j,n}'(1))^2} \\ & \leqslant \exp\{4 C_2 (j+1)^{1/2-\delta_2}\} \limsup_{n \to \infty} T(F_{j,n}) \end{aligned}
\end{equation}
\tag{4.74}
$$
$\mathsf{P}$-almost surely. By (4.74),
$$
\begin{equation}
\begin{aligned} \, \notag &\mathsf{P}\bigl(T(F_j) > \exp\{4 C_2 (j+1)^{1/2-\delta_2} + j^{1/2-\delta_3}\}\bigr) \\ &\qquad\leqslant \mathsf{P}\Bigl(\limsup_{n \to \infty} T(F_{j,n}) > \exp\{j^{1/2-\delta_3}\}\Bigr) \leqslant \mathsf{P}\bigl(\widehat{F}_j > \exp\{j^{1/2-\delta_3}\}\bigr). \end{aligned}
\end{equation}
\tag{4.75}
$$
For some $N \in \mathbb{N}$ we have
$$
\begin{equation}
j^{1/2-\delta_1} \geqslant 4 C_2 (j+1)^{1/2-\delta_2} + j^{1/2-\delta_3}, \quad\text{where } \delta_1 :=\frac{\min\{\delta_2, \delta_3\}}{2} \in \biggl(0, \frac12\biggr),
\end{equation}
\tag{4.76}
$$
for all $j>N$. From Assumption 5 and relations (4.75) and (4.76) we conclude that
$$
\begin{equation}
\begin{aligned} \, \notag \sum_{j=1}^{\infty} \sqrt{\frac{h_j^0}{j}} &=\sum_{j=1}^{\infty}\sqrt{\frac{\mathsf{P}(T(F_j)> \exp\{j^{1/2-\delta_1}\})}{j}} \\ \notag &\leqslant N + \sum_{j=N+1}^{\infty}\sqrt{\frac{\mathsf{P}(T(F_j) > \exp\{4 C_2 (j+1)^{1/2-\delta_2}+ j^{1/2-\delta_3}\})}{j}} \\ &\leqslant N + \sum_{j=N+1}^{\infty}\sqrt{\frac{\mathsf{P}(\widehat{F}_j > \exp\{j^{1/2-\delta_3}\})}{j}}< \infty. \end{aligned}
\end{equation}
\tag{4.77}
$$
Lemma 12 is proved. We have proved all necessary assertions and can switch to the main result. Proof of Theorem 2. Note that Lemmas 6 and 8 apply to $F_{i,n} \equiv F_i$.
Fix $\varepsilon>0$. By Lemma 6 there exists $m$ such that
$$
\begin{equation}
\sqrt{n}\, \biggl|\mathsf{P}(Z_{n,n}>0) - \sum_{k=0}^m \sum_{l=1}^{\infty} A_{k,l,n} B_{n-k,l,n}\biggr| \leqslant \frac{\varepsilon}{6}
\end{equation}
\tag{4.78}
$$
and
$$
\begin{equation}
\sqrt{n}\, \biggl|\mathsf{P}(Z_n>0) - \sum_{k=0}^m \sum_{l=1}^{\infty} A_{k,l} B_{n-k,l}\biggr| \leqslant \frac{\varepsilon}{4}
\end{equation}
\tag{4.79}
$$
for all $n>m$. Owing to Lemma 8 and the relations (4.78) and (4.79), there exist $M, N_1>m$ such that
$$
\begin{equation}
\sqrt{n}\, \biggl|\mathsf{P}(Z_{n,n}>0) - \sum_{k=0}^m \sum_{l=1}^M A_{k,l} B_{n-k,l,n}\biggr| \leqslant \frac{\varepsilon}{3}
\end{equation}
\tag{4.80}
$$
and
$$
\begin{equation}
\sqrt{n}\, \biggl|\mathsf{P}(Z_n>0) - \sum_{k=0}^m \sum_{l=1}^M A_{k,l} B_{n-k,l}\biggr| \leqslant \frac{\varepsilon}{2}
\end{equation}
\tag{4.81}
$$
for each $n>N_1$. By Lemma 11 there exists $N>N_1$ such that
$$
\begin{equation}
\sqrt{n}\, \biggl|\sum_{k=0}^m \sum_{l=1}^M A_{k,l} B_{n-k,l,n} - \sum_{k=0}^m \sum_{l=1}^M A_{k,l} B_{n-k,l}\biggr| \leqslant \frac{\varepsilon}{6}
\end{equation}
\tag{4.82}
$$
for any $n>N$. It follows from (4.80)–(4.82) for $n>N$ that
$$
\begin{equation}
\begin{aligned} \, &\sqrt{n}\, |\mathsf{P}(Z_{n,n}>0)-\mathsf{P}(Z_n>0)| \nonumber \\ &\qquad \leqslant \sqrt{n}\, \biggl|\mathsf{P}(Z_{n,n}>0) - \sum_{k=0}^m \sum_{l=1}^M A_{k,l} B_{n-k,l,n}\biggr| \nonumber \\ &\qquad\qquad + \sqrt{n}\, \biggl|\sum_{k=0}^m \sum_{l=1}^M A_{k,l} B_{n-k,l,n} - \sum_{k=0}^m \sum_{l=1}^M A_{k,l} B_{n-k,l}\biggr| \nonumber \\ &\qquad\qquad + \sqrt{n}\, \biggl|\sum_{k=0}^m \sum_{l=1}^M A_{k,l} B_{n-k,l} - \mathsf{P}(Z_n>0)\biggr| \leqslant \varepsilon. \end{aligned}
\end{equation}
\tag{4.83}
$$
Since $\varepsilon>0$ is arbitrary, we have
$$
\begin{equation}
\sqrt{n}\, |\mathsf{P}(Z_{n,n}>0)-\mathsf{P}(Z_n>0)| \to 0, \qquad n \to \infty.
\end{equation}
\tag{4.84}
$$
Lemma 12 implies that Assumption 2 is fulfilled. By Assumptions 1 and 2, Theorem 1 is true for the BPRE $\{Z_n,\,n \geqslant 0\}$, which implies the convergence
$$
\begin{equation}
\sqrt{n}\, \mathsf{P}(Z_n>0) \to \Upsilon \frac{e^{c_{-}}}{\sqrt{\pi}}, \qquad n \to \infty.
\end{equation}
\tag{4.85}
$$
Relations (4.84) and (4.85) yield the required asymptotics (2.1).
Theorem 2 is proved. AcknowledgementsThe author is grateful to A. V. Shklyaev for his permanent support of this research. The author is grateful to anonymous referees for their comments, which made it possible to improve the presentation considerably. 
Citation in format AMSBIB
\Bibitem{Kha24} |
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