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Some functionals for random walks and critical branching processes in an extremely unfavourable random environment V. A. Vatutina  , C. Dongb, E. E. Dyakonovaa a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia b Xidian University, Xi'an, P. R. China
Bibliographic databases:
    § 1. Introduction and main resultsWe consider the asymptotic behaviour of some functionals specified on the trajectories of a random walk
$$
\begin{equation*}
S_{0}=0, \quad S_{n}=X_{1}+\dots+X_{n}, \qquad n\geqslant 1,
\end{equation*}
\notag
$$
with independent identically distributed increments $X_{i}$, $i=1,2,\dots$ . To describe the conditions we impose on the increments, let
$$
\begin{equation*}
\mathcal{A}:=\{\alpha \in (0,2)\setminus \{1\},\,|\beta|<1\}\cup \{\alpha =1,\,\beta=0\}\cup \{\alpha=2,\,\beta=0\}
\end{equation*}
\notag
$$
be a subset in $\mathbb{R}^{2}$. For $(\alpha,\beta)\in \mathcal{A}$ and a random variable $X$ we write $X\in \mathcal{D}(\alpha,\beta)$ if the distribution of $X$ belongs to the domain of attraction of a stable law with density $g_{\alpha,\beta}(x)$, $x\in (-\infty,+\infty)$, and characteristic function
$$
\begin{equation*}
G_{\alpha,\beta}(w)=\int_{-\infty}^{+\infty}e^{iwx}g_{\alpha,\beta}(x)\,dx =\exp \biggl\{ -c|w|^{\alpha}\biggl(1-i\beta \frac{w}{|w|}\tan\frac{\pi \alpha}{2}\biggr) \biggr\}, \qquad c>0,
\end{equation*}
\notag
$$
and, in addition, $\mathbf{E}X=0$ if this moment exists. This implies, in particular, that there is an increasing sequence of positive numbers with a slowly varying sequence $\ell(1),\ell(2),\dots$, such that, as $n\to\infty$
$$
\begin{equation*}
\biggl\{ \frac{S_{[ nt]}}{a_{n}},\,t\geqslant 0\biggr\} \quad \Longrightarrow\quad \mathcal{Y}=\{ Y_{t},\,t\geqslant 0\},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\mathbf{E}e^{iwY_{t}}=G_{\alpha,\beta}(wt^{1/\alpha}), \qquad t\geqslant 0,
\end{equation*}
\notag
$$
and the symbol $\Longrightarrow$ denotes weak convergence in the space $D[0,\infty)$ of càdlàg functions endowed with the Skorokhod topology. Observe that if $X_{n}\overset{d}{=}X\in \mathcal{D}(\alpha,\beta)$ for all $n\in \mathbb{N}:=\{1,2,\dots\}$, then
$$
\begin{equation*}
\lim_{n\to \infty}\mathbf{P}(S_{n}>0)=:\rho =\mathbf{P}(Y_{1}>0) \in (0,1).
\end{equation*}
\notag
$$
We now list our main restrictions on the properties of the random walk. Condition A1. The random variables $X_{n}$, $n\in \mathbb{N}$, are independent copies of a random variable $X\in \mathcal{D}(\alpha,\beta)$. In addition, the distribution of $X$ is non-lattice. Some of our statements need a stronger assumption. Condition A2. The law of $X$ under $\mathbf{P}$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$, and there exists $n\in\mathbb{N}$ such that the density $f_{n}(x):=\mathbf{P}(S_{n}\in dx)/dx$ of $S_{n}$ is bounded. We set
$$
\begin{equation}
\begin{gathered} \, M_{n}:=\max (S_{1},\dots,S_{n}), \qquad L_{n}:=\min (S_{1},\dots,S_{n}), \notag \\ \tau_{n} :=\min \{ 0\leqslant k\leqslant n\colon S_{k}=\min (0,L_{n})\}, \notag \\ b_{n}:=\frac{1}{a_{n}n}=\frac{1}{n^{1/\alpha +1}\ell(n)}, \end{gathered}
\end{equation}
\tag{1.1}
$$
and introduce a renewal function:
$$
\begin{equation*}
U(x):= \begin{cases} 0, &x<0, \\ \displaystyle 1+\sum_{n=1}^{\infty}\mathbf{P}(S_{n}\geqslant-x,\, M_{n}<0) \\ \displaystyle \qquad\ \ =1+\sum_{n=1}^{\infty}\mathbf{P}(S_{n}\geqslant -x,\, \tau_{n}=n), &x\geqslant 0, \end{cases}
\end{equation*}
\notag
$$
where the last equality follows from the duality principle for random walks (see, for example, [7], Ch. XII, § 2). We use, along with the function $U$, two more renewal functions:
$$
\begin{equation}
\begin{gathered} \, V_0(x):= \begin{cases} 0, &x>0, \\ \displaystyle 1+\sum_{n=1}^{\infty}\mathbf{P}(S_{n}\leqslant -x,\, L_{n}\geqslant 0),&x\leqslant 0, \end{cases} \\ V(x):= \begin{cases} 0,&x\geqslant 0, \\ \displaystyle 1+\sum_{n=1}^{\infty}\mathbf{P}( S_{n}<-x,\, L_{n}\geqslant 0), & x<0. \end{cases} \end{gathered}
\end{equation}
\tag{1.2}
$$
Observe that
$$
\begin{equation}
V_{0}(0)=1+\sum_{n=1}^{\infty}\mathbf{P}(S_{n}=0,\, L_{n}\geqslant 0) =\frac{1}{1-\zeta},
\end{equation}
\tag{1.3}
$$
where
$$
\begin{equation*}
\begin{aligned} \, \zeta &=\mathbf{P}(S_{1}=0) +\sum_{n=2}^{\infty}\mathbf{P}(S_{1}>0,\dots,S_{n-1}>0,\, S_{n}=0) \\ &=\mathbf{P}(S_{1}=0) +\sum_{n=2}^{\infty}\mathbf{P}(S_{1}<0,\dots,S_{n-1}<0,\, S_{n}=0) \in (0,1). \end{aligned}
\end{equation*}
\notag
$$
Here, to justify the transition from the first to the second line it is necessary to use the fact that
$$
\begin{equation*}
\{ S_{n}-S_{n-k},\,k=0,1,\dots,n\} \overset{d}{=}\{S_{k},\,k=0,1,\dots,n\},
\end{equation*}
\notag
$$
which follows from the duality principle. One can check also that if Condition A2 is valid, then and $V(x)=V_{0}(x)$ for all $x\in(-\infty,+\infty)$.In what follows we will consider the random walks starting at time $n=0$ from an arbitrary point $x\in \mathbb{R}$ and denote the corresponding probabilities and expectations by $\mathbf{P}_{x}(\,\cdot\,)$ and $\mathbf{E}_{x}[\,\cdot\,]$. We will also write $\mathbf{P}$ and $\mathbf{E}$ instead of $\mathbf{P}_{0}$ and $\mathbf{E}_{0}$, respectively. Now we formulate the main results of this paper dealing with the properties of random walks. Theorem 1. Let Condition A1 be valid. If $\varphi(n)$, $n\in \mathbb{N}$, is a positive deterministic function such that $\varphi(n)\to+\infty$ as $n\to\infty$ and $\varphi(n)=o(a_{n})$, then for any $\lambda >0$
$$
\begin{equation*}
\begin{aligned} \, \mathbf{E}[ e^{\lambda S_{\tau_{n}}};\, S_{n}\leqslant \varphi(n)] &\sim\theta \, \mathbf{P}(S_{n}\leqslant \varphi(n),\, L_{n}\geqslant 0)\int_{0}^{\infty }e^{-\lambda z}U(z)\,dz \\ &\sim \lambda g_{\alpha,\beta}(0)b_{n}\int_{0}^{\varphi (n)}V(-w)\,dw\int_{0}^{\infty}e^{-\lambda z}U(z)\,dz \end{aligned}
\end{equation*}
\notag
$$
as $n\to\infty$. In the following three statements we analyze the case when $S_{n}$ $\to-\infty$ almost surely as $n\to\infty$. Theorem 2. Let Condition A1 be valid. If $\psi(n)$, $n\in \mathbb{N}$, is a deterministic function such that $\psi(n)\to-\infty$ as $n\to \infty$ and $\psi(n)=o(a_{n})$, then for any $\lambda >0$ and $x\leqslant 0$ as $n\to\infty$.Using the duality principle for random walks and setting $x=0$ in (1.4) we immediately obtain the following result. Corollary 1. If Condition A1 is valid and $\psi(n)$, $n\in \mathbb{N}$, is a deterministic function such that $\psi(n)\to-\infty$ as $n\to\infty$ and $\psi(n)=o(a_{n})$, then for any $\lambda >0$ as $n\to\infty$.The next statement is a natural complement to Corollary 1. Theorem 3. Let Condition A1 be valid. If $\psi(n)$, $n\in \mathbb{N}$, is a deterministic function such that $\psi(n)\to-\infty$ as $n\to \infty$ and $\psi(n)=o(a_{n})$, then for any $\lambda >0$ We now consider the case $S_{n}\leqslant K$ for some fixed $K$. Theorem 4. Let Conditions A1 and A2 be valid. Then for any fixed $K$ and $\lambda >0$
$$
\begin{equation*}
\begin{aligned} \, \lim_{n\to \infty}\frac{\mathbf{E}[ e^{\lambda S_{\tau_{n}}};\, S_{n}\leqslant K]}{b_{n}} &=g_{\alpha,\beta}(0)\int_{-\infty}^{K\wedge 0}e^{\lambda x}\, U(-dx)\int_{0}^{K-x}V(-w)\,dw \\ &\qquad+g_{\alpha,\beta}(0)\int_{-\infty}^{K\wedge 0}e^{\lambda x}U(-x)V_{0}(K-x)\,dx. \end{aligned}
\end{equation*}
\notag
$$
We note that problems related to the ones under consideration here were investigated by several authors. Thus, Hirano [8] analyzed the asymptotic behaviour as $n\to\infty$ of the functional
$$
\begin{equation*}
\mathbf{E}_{x}[ e^{\lambda S_{n}};\, M_{n}\leqslant K]=e^{\lambda x}\mathbf{E}[ e^{\lambda S_{n}};\, M_{n}\leqslant K-x],
\end{equation*}
\notag
$$
assuming that $\mathbf{E}X^{2}<\infty$ and the pair $(K,x)$ is fixed. Hirano’s results were generalized in [2] to the case $X\in \mathcal{D}(\alpha,\beta)$ by showing that, as $n\to\infty$, for $\lambda >0$ and fixed $x\leqslant 0$,
$$
\begin{equation}
\mathbf{E}_{x}[ e^{\lambda S_{n}}; M_{n}<0] \sim g_{\alpha,\beta }(0)b_{n}V(x)\int_{0}^{\infty}e^{-\lambda z}U(w)\,dw,
\end{equation}
\tag{1.6}
$$
while for fixed $x\geqslant 0$
$$
\begin{equation*}
\mathbf{E}_{x}[ e^{-\lambda S_{n}};\, L_{n}\geqslant 0] \sim g_{\alpha,\beta}(0)b_{n}U(x)\int_{0}^{\infty}e^{-\lambda z}V(-w)\,dw.
\end{equation*}
\notag
$$
It follows from (1.6) and the continuity theorem for Laplace transforms that for fixed $y\leqslant 0$
$$
\begin{equation*}
\mathbf{E}[ e^{\lambda S_{n}};\, S_{n}<y,\, M_{n}<0] \sim g_{\alpha,\beta}(0)b_{n}\int_{-y}^{\infty}e^{-\lambda z}U(w)\,dw
\end{equation*}
\notag
$$
as $n\to\infty$, which hints at the form of the asymptotic behaviour of the left-hand side in (1.5). The structure of the remaining sections of this paper looks as follows. In § 2 we formulate a number of known results for random walks conditioned to stay nonnegative or negative. In § 3 we prove Theorem 1. Section 4 is devoted to the proof of Theorem 2. Section 5 contains the proof of Theorem 3. The proof of Theorem 4 is given in § 6. In § 7 we introduce measures $\mathbf{P}_{x}^{+}$ and $\mathbf{P}_{x}^{-}$ generated, respectively, by random walks conditioned to stay nonnegative or negative and use Theorem 4 to investigate the survival probability of a critical branching process evolving in extreme random environment. In what follows we denote by $C,C_{1},C_{2},\dots$, some positive constants that can be different in different formulae or even within one and the same formula. § 2. Auxiliary resultsWe now formulate a number of statements that show the importance of the functions $U,V_{0}$ and $V$. We recall that a positive sequence $\{c_{n},\,n\in \mathbb{N}\}$ (or a real function $c(x)$, $x\geqslant x_{0}$) is said to be regularly varying at infinity with index $\gamma \in \mathbb{R}$, denoted by $c_{n}\in R_{\gamma}$ or $c(x)\in R_{\gamma}$, if $c_{n}= n^{\gamma}l(n)$ ($c(x)= x^{\gamma}l(x)$), where $l(x)$ is a slowly varying function, that is, a positive real function with the property that $l(tx)/l(x)\to1$ as $x\to\infty$ for any fixed $t>0$. It is known (see, for instance, [10] and [11]) that if Condition A1 is valid, then
$$
\begin{equation}
\mathbf{P}(M_{n}<0) \in R_{-\rho}, \qquad U(x)\in R_{\alpha \rho},
\end{equation}
\tag{2.1}
$$
and
$$
\begin{equation}
\mathbf{P}(L_{n}\geqslant 0) \in R_{-(1-\rho)}, \qquad V(-x)\in R_{\alpha (1-\rho)}.
\end{equation}
\tag{2.2}
$$
Some basic inequalities used below in our proofs are contained in the following lemma. Lemma 1 (see [2], Proposition 2.3, and [12], Lemma 2). If Condition A1 is valid, then there is a positive constant $C$ such that, for all $n$ and $x,y\geqslant 0$,
$$
\begin{equation}
\mathbf{P}_{x}(0\leqslant S_{n}<y,\, L_{n}\geqslant 0)\leqslant Cb_{n}U(x)\int_{0}^{y}V(-w)\,dw
\end{equation}
\tag{2.3}
$$
and, for $x,z\leqslant 0$, and therefore We need some equivalence relations established by Doney [6] and rewritten below in our notation. Lemma 2 ([6], Proposition 18). Suppose that $X\in\mathcal{D}(\alpha,\beta)$ and the distribution of $X$ is non-lattice. Then, for each fixed $\Delta >0$ and uniformly in the nonnegative $z$ and $y$ such that $\max (z,y)\in [0,\delta_{n}a_{n}]$, where $\delta_{n}\to0$ as $n\to\infty$.Integration (2.5) over $y\in (0,x)$ leads to the following important conclusion for ${z=0}$ (see also Theorem 4 in [15]). Corollary 2. Under the assumptions of Lemma 2 uniformly in $x\in (0,\delta_{n}a_{n}]$, where $\delta_{n}\to0$ as $n\to\infty$.Combining (2.1) and (2.6) we arrive at the following statement. Corollary 3. Suppose that $X\in \mathcal{D}(\alpha,\beta)$ and the distribution of $X$ is non-lattice. Then for each fixed $\Delta >0$ uniformly in $x\in [0,\delta_{n}a_{n}]$ and $y\in [T_{n},\delta_{n}a_{n}]$, where $T_{n}\to\infty$ and $\delta_{n}\to0$ as $n\to\infty$ in such a way that $T_{n}<\delta_{n}a_{n}$.We also need the following simple observation, which we refer to several times in what follows. Lemma 3. Let $g(x)$, $x\geqslant 0$, be a positive nondecreasing function such that ${g(2x_{0})\geqslant 1}$ for some $x_{0}>0$. Then for any $x\geqslant x_{0}$ and $y\geqslant 0$ If, in addition, then there is a constant $C\in (0,\infty)$ such that for all $y\geqslant 0$ and all sufficiently large $x$.§ 3. Proof of Theorem 1We fix a positive integer $J<n$ and write
$$
\begin{equation}
\mathbf{E}[ e^{\lambda S_{\tau_{n}}};\, S_{n}\leqslant \varphi(n)] =\sum_{j=0}^{J}\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant \varphi(n),\, \tau_{n}=j] +R(J+1,n),
\end{equation}
\tag{3.1}
$$
where
$$
\begin{equation*}
R(J+1,n):=\sum_{j=J+1}^{n}\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant \varphi(n),\, \tau_{n}=j] .
\end{equation*}
\notag
$$
It follows from (2.4) and (2.3) that for all positive integers $n$ and $k$
$$
\begin{equation}
\begin{split} &\mathbf{P}(S_{n}\in [ -k,-k+1),\, M_{n}<0) \\ &\qquad=\mathbf{P}(S_{n}\in [ -k,-k+1),\, \tau_{n}=n)\leqslant Cb_{n}U(k) \end{split}
\end{equation}
\tag{3.2}
$$
and
$$
\begin{equation}
\mathbf{P}(0\leqslant S_{n}<y,\, L_{n}\geqslant 0) \leqslant Cb_{n}\int_{0}^{y}V(-w)\,dw.
\end{equation}
\tag{3.3}
$$
Let $\{S_{n}',\,n\geqslant 0\}$ be an independent probabilistic copy of the random walk $\{S_{n},n\geqslant 0\}$, and let $L_{n}'=\min\{S_{1}',\dots,S_{n}'\}$. Using (3.2) and (3.3) we obtain
$$
\begin{equation}
\begin{aligned} \, &\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant \varphi(n),\, \tau_{n}=j] \notag \\ &\qquad =\mathbf{E}[ e^{\lambda S_{j}}\, \mathbf{P}(S_{n-j}'\leqslant \varphi(n)-S_{j},\, L_{n-j}'\geqslant 0\mid S_{j}); \, \tau_{j}=j] \notag \\ &\qquad \leqslant \sum_{k=1}^{\infty}e^{\lambda (-k+1)}\, \mathbf{P}(S_{j}\in [ -k,-k+1),\, \tau_{j}=j) \, \mathbf{P}(S_{n-j}\leqslant \varphi(n)+k,\, L_{n-j}\geqslant 0) \notag \\ &\qquad \leqslant Cb_{j}\sum_{k=1}^{\infty}e^{\lambda (-k+1)}U(k)\, \mathbf{P}(S_{n-j}\leqslant \varphi(n)+k,\, L_{n-j}\geqslant 0) \notag \\ &\qquad \leqslant Cb_{j}b_{n-j}\sum_{k=1}^{\infty}e^{-\lambda k}U(k)\int_{0}^{\varphi(n)+k}V(-z)\,dz. \end{aligned}
\end{equation}
\tag{3.4}
$$
Note that in view of (1.1) and the properties of regularly varying functions there exists a constant $C\in (0,\infty)$ such that $b_{j}\leqslant Cb_{n}$ for all $n$ and $j\in [n/2,n]$. Since $U(x)$ and $V(-x)$ are renewal functions, there exists a constant $C\in (0,\infty)$ such that for all $x\in \mathbb{R}$
$$
\begin{equation}
U(x)+V(-x)\leqslant C(|x|+1) \quad\text{and}\quad U(x)V(-2x)\leqslant C(|x|+1)^2.
\end{equation}
\tag{3.5}
$$
Therefore,
$$
\begin{equation*}
\sum_{k=1}^{\infty}e^{-\lambda k}U(k)(1+V(-2k))<\mathbf{\infty}
\end{equation*}
\notag
$$
for all $\lambda >0$. Using this estimate and inequality (2.7) for $g(x)=V(-x)$, we conclude that
$$
\begin{equation}
\begin{aligned} \, &\sum_{j=[ n/2] +1}^{n}\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant \varphi(n),\, \tau_{n}=j] \notag \\ &\qquad \leqslant C\sum_{k=1}^{\infty}e^{-\lambda k}U(k)\sum_{j=[ n/2]+1}^{n}b_{j}\, \mathbf{P}(S_{n-j}\leqslant \varphi(n)+k,\, L_{n-j}\geqslant 0) \notag \\ &\qquad \leqslant C_{1}b_{n}\sum_{k=1}^{\infty}e^{-\lambda k}U(k)\sum_{j=[n/2] +1}^{n}\mathbf{P}(S_{n-j}\leqslant \varphi(n)+k,\, L_{n-j}\geqslant 0) \notag \\ &\qquad \leqslant C_{1}b_{n}\sum_{k=1}^{\infty}e^{-\lambda k}U(k)V(-\varphi(n)-k) \notag \\ &\qquad \leqslant C_{2}b_{n}V(-\varphi(n))\biggl(\sum_{k=1}^{\infty}e^{-\lambda k}U(k)(1+V(-2k))\biggr) \notag \\ &\qquad \leqslant C_{3}b_{n}V(-\varphi(n)). \end{aligned}
\end{equation}
\tag{3.6}
$$
We know by (2.2) that $V(-w)\in R_{\alpha (1-\rho)}$. Consequently,
$$
\begin{equation}
\int_{0}^{y}V(-w)\,dw\sim \frac{yV(-y)}{\alpha (1-\rho)+1}\in R_{\alpha(1-\rho)+1}
\end{equation}
\tag{3.7}
$$
as $y\to\infty$ (see [7], Ch. VIII, § 9, Theorem 1). This fact, in combination with (3.6) and Corollary 2, shows that
$$
\begin{equation}
\begin{aligned} \, &\sum_{j=[ n/2] +1}^{n}\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant \varphi(n),\, \tau_{n}=j] \leqslant Cb_{n}V(-\varphi(n)) \notag \\ &\qquad =o\biggl(b_{n}\int_{0}^{\varphi(n)}V(-w)\,dw\biggr) =o(\mathbf{P}(S_{n}\leqslant \varphi(n),\, L_{n}\geqslant 0)) \end{aligned}
\end{equation}
\tag{3.8}
$$
as $n\to\infty$, and that there exists a positive constant $C$ such that
$$
\begin{equation*}
\int_{0}^{2y}V(-w)\,dw\leqslant C\int_{0}^{y}V(-w)\,dw
\end{equation*}
\notag
$$
for all sufficiently large $y$. Combining this estimate with (3.4) we conclude that
$$
\begin{equation}
\begin{aligned} \, &\sum_{J+1\leqslant j\leqslant n/2}\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant \varphi(n),\, \tau_{n}=j] \notag \\ &\qquad \leqslant C\sum_{J\leqslant j\leqslant n/2}b_{j}b_{n-j}\sum_{k=1}^{\infty}e^{-\lambda k}U(k)\int_{0}^{\varphi(n)+k}V(-w)\,dw \notag \\ &\qquad \leqslant C_{1}b_{n}\int_{0}^{2\varphi(n)}V(-w)\,dw\sum_{J\leqslant j\leqslant n/2}b_{j}\biggl(\sum_{k=1}^{\infty}e^{-\lambda k}U(k)\biggr( 1+\int_{0}^{2k}V(-w)\,dw\biggr)\biggr) \notag \\ &\qquad \leqslant C_{2}\sum_{J\leqslant j\leqslant \infty}b_{j} b_{n}\int_{0}^{\varphi(n)}V(-w)\,dw=\varepsilon_{J}\mathbf{P}(S_{n}\leqslant \varphi(n),\, L_{n}\geqslant 0), \end{aligned}
\end{equation}
\tag{3.9}
$$
where $\varepsilon_{J}\to0$ as $J\to\infty$ in view of (1.1) and Corollary 2. Estimates (3.8) and (3.9) imply that
$$
\begin{equation}
\lim_{J\to \infty}\lim_{n\to \infty }\frac{R(J+1,n)}{\mathbf{P}(S_{n}\leqslant \varphi(n),\, L_{n}\geqslant 0)}\leqslant C\lim_{J\to \infty}\varepsilon_{J}=0.
\end{equation}
\tag{3.10}
$$
We now fix $j\in [0,J]$ and write the decomposition
$$
\begin{equation*}
\begin{aligned} \, &\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant \varphi(n),\, \tau_{n}=j] \\ &\qquad=\mathbf{E}\bigl[ e^{\lambda S_{j}};\, S_{n}\leqslant \varphi(n),\, \tau_{n}=j,\, S_{j}>-\sqrt{\varphi(n)}\, \bigr] \\ &\qquad\qquad +\mathbf{E}\bigl[ e^{\lambda S_{j}};\, S_{n}\leqslant \varphi(n),\, \tau_{n}=j,\, S_{j}\leqslant -\sqrt{\varphi(n)}\, \bigr] . \end{aligned}
\end{equation*}
\notag
$$
According to Lemma 5 in [13], for each fixed $j$
$$
\begin{equation*}
\mathbf{E}\bigl[ e^{\lambda S_{j}};\, S_{n}\leqslant \varphi(n),\, \tau_{n}=j,\, S_{j}\leqslant -\sqrt{\varphi(n)}\, \bigr]=o(\mathbf{P}(S_{n-j}\leqslant\varphi(n),\, L_{n-j}\geqslant 0))
\end{equation*}
\notag
$$
as $n\to\infty$. It is clear that
$$
\begin{equation*}
\begin{aligned} \, &\mathbf{E}\bigl[ e^{\lambda S_{j}};\, S_{n}\leqslant \varphi(n),\, \tau_{n}=j,\, S_{j}>-\sqrt{\varphi(n)}\, \bigr] \\ &\qquad =\int_{-\sqrt{\varphi(n)}}^{0}e^{\lambda x}\, \mathbf{P}(S_{j}\in dx,\, \tau_{j}=j) \, \mathbf{P}(S_{n-j}\leqslant \varphi(n)-x,\, L_{n-j}\geqslant 0). \end{aligned}
\end{equation*}
\notag
$$
For $x\in [-\sqrt{\varphi(n)},0]$ and fixed $j$ we have
$$
\begin{equation*}
0\leqslant \varphi(n)-x\leqslant \varphi(n)+\sqrt{\varphi(n)}=o(a_{n})=o(a_{n-j})
\end{equation*}
\notag
$$
as $n\to\infty$. This estimate and Corollary 2 imply that
$$
\begin{equation*}
\mathbf{P}(S_{n-j}\leqslant \varphi(n)-x,\, L_{n-j}\geqslant 0)\sim g_{\alpha,\beta }(0)b_{n-j}\int_{0}^{\varphi(n)-x}V(-w)\,dw
\end{equation*}
\notag
$$
as $n\to\infty$ uniformly in $x\in (-\sqrt{\varphi(n)},0]$. We know that $b_{n}\in R_{1+\alpha ^{-1}}$. Therefore, $b_{n-j}\sim b_{n}$ as $n\to\infty$ for each fixed $j$. Moreover,
$$
\begin{equation*}
\lim_{n\to \infty}\sup_{x\in (-\sqrt{\varphi(n)},0]} \left(\frac{\displaystyle\int_{0}^{\varphi(n)-x}V(-w)\,dw} {\displaystyle\int_{0}^{\varphi(n)}V(-w)\,dw}-1\right)=0,
\end{equation*}
\notag
$$
according to (3.7). Hence we conclude that for each fixed $j$
$$
\begin{equation*}
\begin{aligned} \, \mathbf{P}(S_{n-j}\leqslant \varphi(n)-x,\, L_{n-j}\geqslant 0) &\sim g_{\alpha,\beta}(0)b_{n}\int_{0}^{\varphi(n)}V(-w)\,dw \\ &\sim \mathbf{P}(S_{n}\leqslant \varphi(n),\, L_{n}\geqslant 0) \end{aligned}
\end{equation*}
\notag
$$
as $n\to\infty$ uniformly in $x\in (-\sqrt{\varphi(n)},0]$. Thus,
$$
\begin{equation*}
\begin{aligned} \, &\int_{-\sqrt{\varphi(n)}}^{0}e^{\lambda x}\, \mathbf{P}(S_{j}\in dx,\, \tau_{j}=j) \, \mathbf{P}(S_{n-j}\leqslant \varphi(n)-x,\, L_{n-j}\geqslant 0) \\ &\qquad \sim \mathbf{P}(S_{n}\leqslant \varphi(n),\, L_{n}\geqslant 0)\int_{-\infty}^{0}e^{\lambda x}\, \mathbf{P}(S_{j}\in dx,\, \tau_{j}=j) \end{aligned}
\end{equation*}
\notag
$$
as $n\to\infty$. Using (3.10) and summing the estimates above over $j$ from $0$ to $\infty$ we obtain from (3.1) that
$$
\begin{equation*}
\begin{aligned} \, \mathbf{E}[ e^{\lambda S_{\tau_{n}}};\, S_{n}\leqslant \varphi(n)] &\sim\mathbf{P}(S_{n}\leqslant \varphi(n),\, L_{n}\geqslant 0)\int_{-\infty}^{0}e^{\lambda x}\sum_{j=0}^{\infty}\mathbf{P}(S_{j}\in dx,\, \tau_{j}=j) \\ &=\mathbf{P}(S_{n}\leqslant \varphi(n),\, L_{n}\geqslant 0) \biggl(1+\int_{-\infty}^{0}e^{\lambda x}U(-dx)\biggr) \\ &=\mathbf{P}(S_{n}\leqslant \varphi(n),\, L_{n}\geqslant 0)\lambda \int_{0}^{\infty }e^{-\lambda x}U(x)\,dx \end{aligned}
\end{equation*}
\notag
$$
as $n\to\infty$, where we used the equality $U(0+)=1$ at the last step. We complete the proof of Theorem 1 by combining the equivalence relation obtained with Corollary 2. § 4. Proof of Theorem 2The starting point of our arguments is the representation
$$
\begin{equation*}
\begin{aligned} \, &\mathbf{E}_{x}[ e^{\lambda S_{n}};\, S_{n}\leqslant \psi(n),\, M_{n}<0] =\int_{-\infty}^{\psi(n)}e^{\lambda w}\, \mathbf{P}_{x}(S_{n}\in dw,M_{n}<0) \\ &\qquad=e^{\lambda \psi(n)}\int_{0}^{\infty }e^{-\lambda y}\, \mathbf{P}_{x}(S_{n}\in \psi(n)-dy,\, M_{n}<0) . \end{aligned}
\end{equation*}
\notag
$$
Introducing for $h\in (0,1]$ and $N\in \mathbb{N}$ the notation
$$
\begin{equation*}
T_{1}(n,N,h):= \sum_{k=0}^{[ N/h]}e^{-\lambda kh}\, \mathbf{P}_{x}(S_{n}\in [ \psi(n)-(k+1) h,\, \psi (n)-kh),\, M_{n}<0)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
T_{2}(n,N):= \sum_{r=N}^{\infty}e^{-\lambda r}\, \mathbf{P}_{x}( S_{n}\in [ \psi(n)-(r+1),\, \psi(n)-r),\, M_{n}<0),
\end{equation*}
\notag
$$
we have
$$
\begin{equation*}
\begin{aligned} \, e^{-\lambda h}T_{1}(n,N,h) &\leqslant \int_{0}^{\infty}e^{-\lambda y}\, \mathbf{P}_{x}(S_{n}\in \psi(n)-dy,\, M_{n}<0) \\ &\leqslant T_{1}(n,N,h)+T_{2}(n,N). \end{aligned}
\end{equation*}
\notag
$$
Using (2.4), Lemma 3 and the first inequality in (3.5), we conclude that there exists a positive integer $N_{0}$ such that the estimates
$$
\begin{equation*}
\begin{aligned} \, T_{2}(n,N) &\leqslant Cb_{n}V(x)\sum_{r=N}^{\infty}e^{-\lambda r}U(r+1-\psi(n)) \\ &\leqslant Cb_{n}V(x)U(-\psi(n))\biggl(\sum_{r=N}^{\infty}e^{-\lambda r}(1+U(2r+2))\biggr) \\ &=Cb_{n}V(x)U(-\psi(n))e^{-\lambda N}\biggl(\sum_{r=0}^{\infty }e^{-\lambda r}(1+U(2r+2N+2))\biggr) \\ &\leqslant 2Cb_{n}V(x)U(-\psi(n))e^{-\lambda N}\sum_{r=0}^{\infty}e^{-\lambda r}(r+N+2) \\ &\leqslant C_{1}b_{n}V(x)U(-\psi(n))Ne^{-\lambda N} \end{aligned}
\end{equation*}
\notag
$$
are valid for all $N\geqslant N_{0}$. Further, if Condition A1 is satisfied, then, according to Corollary 3,
$$
\begin{equation}
\begin{aligned} \, &\mathbf{P}_{x}(S_{n}\in [ \psi(n)-(k+1) h,\, \psi(n)-kh),\, M_{n}<0) \notag \\ &\qquad \sim g_{\alpha,\beta}(0)b_{n}V(x)hU((k+1)h-\psi(n)) \end{aligned}
\end{equation}
\tag{4.1}
$$
as $n\to\infty$ uniformly in negative $x=o(a_{n})$ and $\psi(n)-(k+1)h=o(a_{n})$, where $(k+1)h\leqslant 2N$. Moreover, as $n\to\infty$ uniformly in $z=o(\psi(n))$. As a result,
$$
\begin{equation*}
\begin{aligned} \, T_{1}(n,N,h) &=(1+o(1))g_{\alpha,\beta}(0)b_{n}V(x)h\sum_{k=0}^{[ N/h]}e^{-\lambda kh}U(kh-\psi(n)) \\ &=(1+o(1))g_{\alpha,\beta}(0)b_{n}V(x)U(-\psi(n)) h\sum_{k=0}^{[ N/h]}e^{-\lambda kh} \\ &\leqslant (1+\varepsilon)g_{\alpha,\beta}(0)b_{n}V(x)U(-\psi (n)) h(1-e^{-\lambda h})^{-1} \end{aligned}
\end{equation*}
\notag
$$
for any $\varepsilon >0$ and sufficiently large $n\geqslant n_{0}(\varepsilon)$. Thus,
$$
\begin{equation*}
\limsup_{n\to \infty}\frac{\displaystyle\int_{0}^{\infty}e^{-\lambda y}\, \mathbf{P}_{x}(S_{n}\in \psi(n)-dy,\, M_{n}<0)}{b_{n}U(-\psi(n))V(x)}\leqslant (1+\varepsilon)g_{\alpha,\beta}(0)h(1-e^{\lambda h})^{-1}.
\end{equation*}
\notag
$$
Since $\varepsilon >0$ and $h>0$ can be selected arbitrary small, we conclude that
$$
\begin{equation*}
\limsup_{n\to \infty}\frac{\displaystyle\int_{0}^{\infty}e^{-\lambda y}\, \mathbf{P}_{x}(S_{n}\in \psi(n)-dy,\, M_{n}<0)}{b_{n}U(-\psi(n))V(x)}\leqslant \frac{g_{\alpha,\beta}(0)}{\lambda}.
\end{equation*}
\notag
$$
In a similar way we obtain
$$
\begin{equation*}
\begin{aligned} \, e^{-\lambda h}T_{1}(n,N,h) &\geqslant (1-\varepsilon)g_{\alpha,\beta}(0)b_{n}V(x)U(-\psi(n)) h \\ &\qquad \times \biggl((1-e^{-\lambda h})^{-1}-\sum_{r=N}^{\infty}e^{-\lambda r}\biggr) \end{aligned}
\end{equation*}
\notag
$$
for any $\varepsilon >0$ and sufficiently large $n$, which implies that
$$
\begin{equation*}
\begin{aligned} \, &\liminf_{n\to \infty} \frac{\displaystyle\int_{0}^{\infty}e^{-\lambda y}\, \mathbf{P}_{x}(S_{n}\in \psi(n)-dy,\, M_{n}<0)}{b_{n}U(-\psi(n))V(x)} \\ &\qquad \geqslant \liminf_{N\to \infty}\liminf_{h\downarrow 0}\lim_{\varepsilon \downarrow 0}\liminf_{n\to \infty }\frac{e^{-\lambda h}T_{1}(n,N,h)}{b_{n}U(-\psi(n))V(x)}\geqslant \frac{g_{\alpha ,\beta}(0)}{\lambda}. \end{aligned}
\end{equation*}
\notag
$$
It follows that
$$
\begin{equation*}
\lim_{n\to \infty}\frac{\displaystyle\int_{0}^{\infty}e^{-\lambda y}\, \mathbf{P}_{x}(S_{n}\in \psi(n)-dy,\, M_{n}<0)}{b_{n}U(-\psi (n))V(x)}=\frac{g_{\alpha,\beta}(0)}{\lambda},
\end{equation*}
\notag
$$
as required.
§ 5. Proof of Theorem 3We write
$$
\begin{equation*}
\mathbf{E}[ e^{\lambda S_{\tau_{n}}};\, S_{n}\leqslant \psi(n)] =R(0,n-J)+\sum_{j=n-J+1}^{n}\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant \psi (n),\tau_{n}=j],
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
R(0,n-J):=\sum_{j=0}^{n-J}\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant \psi(n),\, \tau_{n}=j] .
\end{equation*}
\notag
$$
Clearly, for each $j\in [0,n]$
$$
\begin{equation}
\begin{aligned} \, &\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant \psi(n),\, \tau_{n}=j] \notag \\ &\qquad =\int_{-\infty}^{\psi(n)}e^{\lambda x}\, \mathbf{P}(S_{j}\in dx,\, \tau_{j}=j) \, \mathbf{P}(S_{n-j}\leqslant \psi(n)-x,\, L_{n-j}\geqslant 0) \notag \\ &\qquad =e^{\lambda \psi(n)}T_{3}(n,j), \end{aligned}
\end{equation}
\tag{5.1}
$$
where
$$
\begin{equation*}
T_{3}(n,j):=\int_{0}^{\infty}e^{-\lambda y}\, \mathbf{P}(S_{j}\in \psi (n)-dy,\, \tau_{j}=j) \, \mathbf{P}(S_{n-j}\leqslant y,\, L_{n-j}\geqslant0).
\end{equation*}
\notag
$$
In view of (2.2) there is a constant $C\in (0,\infty)$ such that
$$
\begin{equation}
e^{-\lambda y}\int_{0}^{y}V(-w)\,dw\leqslant Ce^{-\lambda y/2}
\end{equation}
\tag{5.2}
$$
for all $y\geqslant 0$. Using now the inequalities (2.3), (2.4) and (2.1) it is not difficult to check the validity of the following chain of inequalities:
$$
\begin{equation}
\begin{aligned} \, T_{3}(n,j) &\leqslant Cb_{n-j}\int_{0}^{\infty}e^{-\lambda y}\, \mathbf{P}( S_{j}\in \psi(n)-dy,\, \tau_{j}=j) \int_{0}^{y}V(-w)\,dw \notag \\ &\leqslant Cb_{n-j}\sum_{k=0}^{\infty}e^{-\lambda k}\, \mathbf{P}(S_{j}\,{\in}\, [ \psi(n)\,{-}\,k\,{-}\,1,\, \psi(n)\,{-}\,k),\, \tau_{j}\,{=}\,j) \int_{0}^{k+1}V(-w)\,dw \notag \\ &\leqslant C_{1}b_{n-j}b_{j}\sum_{k=0}^{\infty}e^{-\lambda k/2}U(k+1-\psi(n)) \notag \\ &\leqslant C_{1}b_{n-j}b_{j}U(-2\psi(n))\sum_{k=0}^{\infty}e^{-\lambda k/2}(U(2k+2)+1) \notag \\ &\leqslant C_{2}b_{n-j}b_{j}U(-\psi(n)). \end{aligned}
\end{equation}
\tag{5.3}
$$
In view of (1.1) $b_{n-j}\leqslant Cb_{n}$ for all $j\leqslant n/2$. Therefore,
$$
\begin{equation*}
\begin{aligned} \, &\sum_{j=J}^{n-J}\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant \psi(n),\, \tau _{n}=j] \leqslant C_{2}U(-\psi(n))e^{\lambda \psi (n)}\sum_{j=J}^{n-J}b_{n-j}b_{j} \\ &\qquad \leqslant C_{3}b_{n}U(-\psi(n))e^{\lambda \psi (n)}\sum_{j=J}^{n/2}b_{j}\leqslant \varepsilon_{J}b_{n}U(-\psi(n))e^{\lambda \psi(n)}, \end{aligned}
\end{equation*}
\notag
$$
where $\varepsilon_{J}=C_{3}\sum_{j=J}^{\infty}b_{j}\to0$ as $J\to \infty$. Further, for any fixed $j\in [0,J]$ we have
$$
\begin{equation*}
\begin{aligned} \, T_{3}(n,j) &\leqslant Cb_{n-j}\sum_{k=0}^{\infty}e^{-\lambda k/2}\, \mathbf{P}(S_{j}\in [ \psi(n)-k-1,\, \psi(n)-k),\, \tau_{j}=j) \\ &\leqslant C_{1}b_{n}\, \mathbf{P}(S_{j}\leqslant \psi(n))\sum_{k=0}^{\infty}e^{-\lambda k/2}=o(b_{n}) \end{aligned}
\end{equation*}
\notag
$$
as $n\to\infty$. Thus,
$$
\begin{equation*}
\limsup_{J\to \infty}\limsup_{n\to \infty }\frac{R(0,n-J)}{b_{n}U(-\psi(n)) e^{\lambda \psi(n)}}=0.
\end{equation*}
\notag
$$
Consider now $j=n-t$ for $t\in [0,J]$. In this case, for any $N\in \mathbb{N}$ and $h>0$ we have where
$$
\begin{equation*}
\begin{aligned} \, T_{4}(n,N,h,j) &:=\sum_{k=0}^{[ N/h]}e^{-\lambda kh}\, \mathbf{P}(S_{j}\in [ \psi(n)-(k+1)h,\, \psi(n)-kh),\, \tau_{j}=j) \\ &\qquad\times\mathbf{P}(S_{t}\leqslant (k+1)h,\, L_{t}\geqslant 0) \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, T_{5}(n,N,j) &:=\sum_{r=N}^{\infty}e^{-\lambda r}\, \mathbf{P}(S_{j}\in [ \psi(n)-r-1,\, \psi(n)-r),\, \tau_{j}=j) \\ &\qquad\times\mathbf{P}(S_{t}\leqslant r+1,\, L_{t}\geqslant 0) . \end{aligned}
\end{equation*}
\notag
$$
Using (2.4) for $x=0$ we conclude that
$$
\begin{equation}
\begin{aligned} \, T_{5}(n,N,j) &\leqslant Cb_{j}\sum_{r=N}^{\infty}e^{-\lambda r}U(r+1-\psi (n))\, \mathbf{P}(S_{t}\leqslant r+1,\, L_{t}\geqslant 0) \notag \\ &\leqslant Cb_{j}U(-2\psi(n))\sum_{r=N}^{\infty}e^{-\lambda r}( U(2r+2)+1) \notag \\ &\leqslant C_{1}b_{j}U(-\psi(n))e^{-\lambda N}\sum_{r=0}^{\infty}e^{-\lambda r}(U(2r+2N+2)+1) \notag \\ &\leqslant C_{2}b_{j}U(-\psi(n))e^{-\lambda N}\sum_{r=0}^{\infty}e^{-\lambda r}(r+N+2) \notag \\ &\leqslant C_{3}b_{j}U(-\psi(n))Ne^{-\lambda N}. \end{aligned}
\end{equation}
\tag{5.4}
$$
Further, if $0\leqslant n-j\leqslant J$ and $n\to\infty$, then by (2.6) and the duality principle for random walks
$$
\begin{equation*}
\begin{aligned} \, &\mathbf{P}(S_{j}\in [ \psi(n)-(k+1) h,\, \psi(n)-kh),\, \tau_{j}=j) \\ &\qquad =\mathbf{P}(S_{j}\in [ \psi(n)-(k+1)h,\, \psi(n)-kh),\, M_{j}<0) \\ &\qquad \sim g_{\alpha,\beta}(0)b_{j}\int_{kh-\psi(n)}^{( k+1) h-\psi(n)}U(z)\,dz\sim g_{\alpha,\beta}(0)b_{j}hU(-\psi(n)) \end{aligned}
\end{equation*}
\notag
$$
as $n\to\infty$, uniformly in $0\leqslant (k+1)h\leqslant N$. Using these estimates we see that for each fixed $h>0$
$$
\begin{equation}
T_{4}(n,N,h,j)\sim g_{\alpha,\beta}(0)b_{j}U(-\psi(n))h\sum_{k=0}^{[ N/h]}e^{-\lambda kh}\,\mathbf{P}(S_{t}\leqslant (k+1)h,\,L_{t}\geqslant 0) .
\end{equation}
\tag{5.5}
$$
Some obvious estimates lead to the chains of inequalities
$$
\begin{equation}
\begin{aligned} \, &h\sum_{k=0}^{[ N/h]}e^{-\lambda kh}\, \mathbf{P}(S_{t}\leqslant (k+1) h,\, L_{t}\geqslant 0) \notag \\ &\qquad\leqslant e^{2\lambda h}\sum_{k=0}^{[ N/h] }\int_{(k+1) h}^{(k+2) h}e^{-\lambda z}\, \mathbf{P}(S_{t}\leqslant z,\, L_{t}\geqslant 0) \,dz \notag \\ &\qquad \leqslant e^{2\lambda h}\int_{0}^{\infty}e^{-\lambda z}\, \mathbf{P}(S_{t}\leqslant z,\, L_{t}\geqslant 0) \,dz \end{aligned}
\end{equation}
\tag{5.6}
$$
and
$$
\begin{equation}
\begin{aligned} \, &h\sum_{k=0}^{[ N/h]}e^{-\lambda kh}\, \mathbf{P}(S_{t}\leqslant (k+1) h,\, L_{t}\geqslant 0) \notag \\ &\qquad \geqslant e^{-2\lambda h}\sum_{k=0}^{[ N/h]}he^{-\lambda (k-1)h}\, \mathbf{P}(S_{t}\leqslant kh,\, L_{t}\geqslant 0) \notag \\ &\qquad \geqslant e^{-2\lambda h}\int_{0}^{N-1}e^{-\lambda z}\, \mathbf{P}(S_{t}\leqslant z,\, L_{t}\geqslant 0) \,dz. \end{aligned}
\end{equation}
\tag{5.7}
$$
Combining (5.5)–(5.7) we see that for $j=n-t\in (n-J,n]$
$$
\begin{equation*}
\lim_{h\downarrow 0}\lim_{N\to \infty}\lim_{n\to \infty }\frac{T_{4}(n,N,h,j)}{g_{\alpha,\beta}(0)b_{j}U(-\psi (n))}=\int_{0}^{\infty}e^{-\lambda z}\, \mathbf{P}(S_{t}\leqslant z,\, L_{t}\geqslant 0) \,dz.
\end{equation*}
\notag
$$
Hence, by the equivalence $b_{j}\sim b_{n}$ for $j\in (n-J,n]$, taking (5.4) into account we deduce that
$$
\begin{equation}
\limsup_{n\to \infty}\frac{T_{3}(n,j)}{g_{\alpha,\beta }(0)b_{n}U(-\psi(n))}\leqslant \int_{0}^{\infty}e^{-\lambda z}\mathbf{P}( S_{t}\leqslant z,L_{t}\geqslant 0) \,dz.
\end{equation}
\tag{5.8}
$$
To obtain a similar estimate from below we observe that
$$
\begin{equation*}
\begin{aligned} \, &T_{3}(n,j)\geqslant T_{6}(N,n,h,j) \\ &:=\sum_{k=1}^{[ N/h]}e^{-\lambda (k+1)h}\, \mathbf{P}( S_{j}\in [ \psi(n)-(k+1) h,\, \psi(n)-kh),\, \tau_{j}=j) \, \mathbf{P}(S_{t}\,{\leqslant}\, kh,\, L_{t}\,{\geqslant}\, 0) \\ &\sim g_{\alpha,\beta}(0)b_{j}U(-\psi(n))h\sum_{k=1}^{[ N/h] }e^{-\lambda (k+1)h}\, \mathbf{P}(S_{t}\leqslant kh,\, L_{t}\geqslant 0) \end{aligned}
\end{equation*}
\notag
$$
as $n\to\infty$ and
$$
\begin{equation*}
\begin{aligned} \, &h\sum_{k=1}^{[ N/h]}e^{-\lambda (k+1)h}\, \mathbf{P}( S_{t}\leqslant kh,\, L_{t}\geqslant 0) \\ &\qquad \geqslant e^{-2\lambda h}\sum_{k=1}^{[ N/h]}he^{-\lambda (k-1)h}\, \mathbf{P}(S_{t}\leqslant kh,\, L_{t}\geqslant 0) \\ &\qquad \geqslant e^{-2\lambda h}\int_{0}^{N-1}e^{-\lambda z}\, \mathbf{P}(S_{t}\leqslant z,\, L_{t}\geqslant 0)\, dz. \end{aligned}
\end{equation*}
\notag
$$
Hence it follows that
$$
\begin{equation*}
\begin{aligned} \, &\liminf_{n\to \infty}\frac{T_{3}(n,j)}{g_{\alpha,\beta }(0)b_{j}U(-\psi(n))} \geqslant \lim_{h\downarrow 0}\lim_{N\to \infty }\lim_{n\to \infty}\frac{T_{6}(n,N,h,j)}{g_{\alpha,\beta }(0)b_{j}U(-\psi(n))} \\ &\qquad =\int_{0}^{\infty}e^{-\lambda z}\, \mathbf{P}(S_{t}\leqslant z,\, L_{t}\geqslant 0) \,dz. \end{aligned}
\end{equation*}
\notag
$$
Thus, for each $t=n-j\in [0,J]$
$$
\begin{equation*}
\lim_{n\to \infty}\frac{T_{3}(n,j)}{g_{\alpha,\beta}(0)b_{n}U(-\psi(n))} =\int_{0}^{\infty}e^{-\lambda z}\, \mathbf{P}(S_{t}\leqslant z,\, L_{t}\geqslant 0) \,dz.
\end{equation*}
\notag
$$
Using now (5.1) we conclude that for each fixed $J$
$$
\begin{equation*}
\begin{aligned} \, &\sum_{j=n-J+1}^{n}\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant \psi(n),\, \tau_{n}=j] \\ &\qquad \sim g_{\alpha,\beta}(0)b_{n}U(-\psi(n))e^{\lambda \psi (n)}\int_{0}^{\infty}e^{-\lambda z}\sum_{t=0}^{J}\mathbf{P}(S_{t}\leqslant z,\, L_{t}\geqslant 0)\,dz \end{aligned}
\end{equation*}
\notag
$$
as $n\to \infty $. Letting $J$ tend to infinity we see that
$$
\begin{equation*}
\mathbf{E}[ e^{\lambda S_{\tau_{n}}};\, S_{n}\leqslant \psi(n)] \sim g_{\alpha,\beta}(0)b_{n}U(-\psi(n)) e^{\lambda \psi (n)}\int_{0}^{\infty}e^{-\lambda z}V_{0}(-z)\,dz,
\end{equation*}
\notag
$$
as required.
§ 6. Proof of Theorem 4We begin with the decomposition
$$
\begin{equation*}
\mathbf{E}[ e^{\lambda S_{\tau_{n}}};\, S_{n}\leqslant K] =R(0,J)+R(J+1,n-J)+R(n-J+1,n),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
R(N_{1},N_{2}):=\sum_{j=N_{1}}^{N_{2}}\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant K,\, \tau_{n}=j] .
\end{equation*}
\notag
$$
We show that the term $R(J+1,n-J)$ is negligibly small with respect the other terms in the expectation we are interested in. By Lemma 1 and estimate (5.2)
$$
\begin{equation*}
\begin{aligned} \, &\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant K,\, \tau_{n}=j] \\ &\qquad =\int_{-\infty}^{K\wedge 0}e^{\lambda x}\, \mathbf{P}(S_{j}\in dx,\, \tau_{j}=j) \, \mathbf{P}(S_{n-j}\leqslant K-x,\, L_{n-j}\geqslant 0) \\ &\qquad =e^{\lambda K}\int_{K\vee 0}^{\infty}e^{-\lambda y}\, \mathbf{P}(S_{j}\in K-dy,\, M_{j}<0) \, \mathbf{P}(S_{n-j}\leqslant y,\, L_{n-j}\geqslant 0) \\ &\qquad \leqslant Cb_{n-j}\int_{K\vee 0}^{\infty}e^{-\lambda y}\, \mathbf{P}(S_{j}\in K-dy,\, M_{j}<0) \int_{0}^{y}V(-w)\,dw \\ &\qquad \leqslant Cb_{n-j}\sum_{k\geqslant K\vee 0}e^{-\lambda k}\, \mathbf{P}(S_{j}\in [ K-k-1,\, K-k),\, M_{j}<0) \int_{0}^{k+1}V(-w)\,dw \\ &\qquad \leqslant C_{1}b_{n-j}b_{j}\sum_{k\geqslant K\vee 0}e^{-\lambda k/2}U(k+1-K) \\ &\qquad \leqslant C_{1}b_{n-j}b_{j}U((-2K)\vee 0)\sum_{k\geqslant K\vee 0}e^{-\lambda k/2}(U(2k+2)+1) \leqslant C_{2}b_{n-j}b_{j}. \end{aligned}
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
R(J+1,n-J)\leqslant C\sum_{j=J+1}^{n-J}b_{n-j}b_{j}\leqslant C_{1}b_{n}\sum_{j=J+1}^{n/2}b_{j}\leqslant \varepsilon_{J}b_{n},
\end{equation*}
\notag
$$
where $\varepsilon_{J}=C_{1}\sum_{j=J+1}^{\infty}b_{j}\to0$ as $J\to\infty $. Further, for fixed $j\in [0,J]$ we have
$$
\begin{equation*}
\begin{aligned} \, &\frac{\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant K,\, \tau_{n}=j] }{b_{n}} \\ &\qquad=\int_{-\infty}^{K\wedge 0}e^{\lambda x}\, \mathbf{P}(S_{j}\in dx,\, \tau_{j}=j) \, \frac{\mathbf{P}(S_{n-j}\leqslant K-x,\, L_{n-j}\geqslant 0)}{b_{n}}. \end{aligned}
\end{equation*}
\notag
$$
Note that for any fixed $x\leqslant K\wedge 0$
$$
\begin{equation*}
\mathbf{P}(S_{n-j}\leqslant K-x,\, L_{n-j}\geqslant 0)\sim g_{\alpha,\beta}(0)b_{n-j}\int_{0}^{K-x}V(-w)\,dw
\end{equation*}
\notag
$$
as $n\to\infty$, and by (2.3) there is a constant $C\in (0,\infty)$ such that
$$
\begin{equation*}
\frac{\mathbf{P}(0\leqslant S_{n}<K-x,\, L_{n}\geqslant 0)}{b_{n}}\leqslant C\int_{0}^{K-x}V(-w)\,dw
\end{equation*}
\notag
$$
for all $n$ and all $x\leqslant K\wedge 0$. These relations, in combination with (3.7), show that for any $\lambda >0$
$$
\begin{equation*}
\begin{aligned} \, &\int_{-\infty}^{K\wedge 0}e^{\lambda x}\, \mathbf{P}(S_{j}\in dx,\, \tau_{j}=j) \int_{0}^{K-x}V(-w)\,dw \\ &\qquad \leqslant \int_{-\infty}^{0}e^{\lambda x}\, \mathbf{P}(S_{j}\in dx) \int_{0}^{K-x}V(-w)\,dw \\ &\qquad \leqslant C_{1}\int_{-\infty}^{0}e^{\lambda x}\, \mathbf{P}(S_{j}\in dx) (|x|^{2}+|K|^{2}+1) <\infty . \end{aligned}
\end{equation*}
\notag
$$
Using Corollary 2 and applying the dominated convergence theorem we conclude that, for each $j\in [0,J]$
$$
\begin{equation*}
\begin{aligned} \, &\lim_{n\to \infty}\frac{\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant K,\,\tau_{n}=j]}{b_{n}} \\ &\qquad =g_{\alpha,\beta}(0)\int_{-\infty}^{K\wedge 0}e^{\lambda x}\, \mathbf{P}(S_{j}\in dx,\, \tau_{j}=j) \int_{0}^{K-x}V(-w)\,dw. \end{aligned}
\end{equation*}
\notag
$$
As a result,
$$
\begin{equation}
\begin{aligned} \, &\lim_{J\to \infty}\lim_{n\to \infty}\frac{R(0,J)}{b_{n}} \notag \\ &\qquad=g_{\alpha,\beta}(0)\int_{-\infty}^{K\wedge 0}e^{\lambda x}\sum_{j=0}^{\infty}\mathbf{P}(S_{j}\in dx,\, \tau_{j}=j) \int_{0}^{K-x}V(-w)\,dw \notag \\ &\qquad =g_{\alpha,\beta}(0)\int_{-\infty}^{K\wedge 0}e^{\lambda x}\, U(-dx)\int_{0}^{K-x}V(-w)\,dw<\infty. \end{aligned}
\end{equation}
\tag{6.1}
$$
To evaluate $R(n-J+1,n)$, for $j=n-t\in [n-J+1,n]$ we write the representation
$$
\begin{equation*}
\begin{aligned} \, &\frac{\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant K,\, \tau_{n}=j] }{b_{n}} \\ &\qquad=\int_{-\infty}^{K\wedge 0}e^{\lambda x}\, \frac{\mathbf{P}( S_{j}\in dx,\, \tau_{j}=j)}{b_{n}}\, \mathbf{P}(S_{t}\leqslant K-x,\, L_{t}\geqslant 0) \end{aligned}
\end{equation*}
\notag
$$
and observe that by (2.4) and the duality principle for random walks
$$
\begin{equation}
\begin{aligned} \, &\int_{-\infty}^{-H}e^{\lambda x}\, \frac{\mathbf{P}(S_{j}\in dx,\, \tau _{j}=j)}{b_{n}}\, \mathbf{P}(S_{t}\leqslant K-x,\, L_{t}\geqslant 0) \notag \\ &\qquad \leqslant \mathbf{P}(L_{t}\geqslant 0) \int_{-\infty }^{-H}e^{\lambda x}\, \frac{\mathbf{P}(S_{j}\in dx,\, \tau_{j}=j) }{b_{n}} \notag \\ &\qquad \leqslant \mathbf{P}(L_{t}\geqslant 0) \sum_{k=H}^{\infty}e^{-\lambda k}\, \frac{\mathbf{P}(S_{j}\in [ -k-1,-k),\, \tau_{j}=j)}{b_{n}} \notag \\ &\qquad \leqslant C_{1}\, \mathbf{P}(L_{t}\geqslant 0) \sum_{k=H}^{\infty}e^{-\lambda k}U(k+1), \end{aligned}
\end{equation}
\tag{6.2}
$$
for any $H>0$, and therefore the right-hand side of this relation vanishes as $H\to\infty$. Thus, we are left with the integral
$$
\begin{equation*}
\int_{-H}^{K\wedge 0}e^{\lambda x}\, \frac{\mathbf{P}(S_{j}\in dx,\, \tau _{j}=j)}{b_{n}}\, \mathbf{P}(S_{t}\leqslant K-x,\, L_{t}\geqslant 0) .
\end{equation*}
\notag
$$
To estimate this term we use Theorem 5.1 in [3] (rewritten in our notation) according to which
$$
\begin{equation*}
\frac{\mathbf{P}(S_{j}\in dx,\, \tau_{j}=j)}{b_{n}dx}\sim g_{\alpha,\beta}(0)U(-x)
\end{equation*}
\notag
$$
as $n\to\infty$, uniformly in $x<0$ such that $ x=o(a_{n})$, if Condition A2 is valid. Thus,
$$
\begin{equation}
\begin{aligned} \, &\lim_{n\to \infty}\int_{-H}^{K\wedge 0}e^{\lambda x}\, \frac{\mathbf{P}(S_{j}\in dx,\, \tau_{j}=j)}{b_{n}}\, \mathbf{P}(S_{t}\leqslant K-x,\, L_{t}\geqslant 0) \notag \\ &\qquad =g_{\alpha,\beta}(0)\int_{-H}^{K\wedge 0}e^{\lambda x}U(-x)\, \mathbf{P}(S_{t}\leqslant K-x,\,L_{t}\geqslant 0)\, dx. \end{aligned}
\end{equation}
\tag{6.3}
$$
Combining (6.2) and (6.3) we deduce that for fixed $t=n-j\in [0,J]$
$$
\begin{equation*}
\begin{aligned} \, &\lim_{n\to \infty}\frac{\mathbf{E}[ e^{\lambda S_{j}};\, S_{n}\leqslant K,\, \tau_{n}=j]}{b_{n}} \\ &\qquad=g_{\alpha,\beta}(0)\int_{-\infty}^{K\wedge 0}e^{\lambda x}U(-x)\, \mathbf{P}(S_{t}\leqslant K-x,\, L_{t}\geqslant 0) \,dx. \end{aligned}
\end{equation*}
\notag
$$
Summing with respect to $t$ from $0$ to $\infty$ we obtain
$$
\begin{equation}
\begin{aligned} \, &\lim_{J\to \infty}\lim_{n\to \infty}\frac{R(n-J+1,n)}{b_{n}} \nonumber \\ &\qquad=g_{\alpha,\beta}(0)\int_{-\infty}^{K\wedge 0}e^{\lambda x}U(-x)V_{0}(K-x)\,dx<\infty. \end{aligned}
\end{equation}
\tag{6.4}
$$
Thus,
$$
\begin{equation*}
\begin{aligned} \, &\lim_{n\to \infty}\frac{\mathbf{E}[ e^{\lambda S_{\tau_{n}}};\, S_{n}\leqslant K]}{b_{n}} =g_{\alpha,\beta}(0)\int_{-\infty}^{K\wedge 0}e^{\lambda x}\, U(-dx)\int_{0}^{K-x}V(-w)\,dw \\ &\qquad\qquad+g_{\alpha,\beta}(0)\int_{-\infty}^{K\wedge 0}e^{\lambda x}U(-x)V_{0}(K-x)\,dx, \end{aligned}
\end{equation*}
\notag
$$
as required.
§ 7. Survival probability for branching processes evolving in extremely unfavourable random environmentIn this section we apply the results obtained for random walks to study the asymptotic behaviour of the survival probabilities of the critical branching process evolving in an unfavourable random environment. For a formal description of the problems we are planning to consider we denote by $\mathfrak{F}=\{\mathfrak{f}\}$ the space of all probability measures on $\mathbb{N}_{0}:=\{0,1,2,\dots\}$. For notational reasons, we identify a measure $\mathfrak{f}=\{\mathfrak{f}(\{0\}),\mathfrak{f}(\{1\}),\dots\}\in \mathfrak{F}$ with the corresponding probability generating function
$$
\begin{equation*}
f(s)=\sum_{k=0}^{\infty}\mathfrak{f}(\{ k\})s^{k}, \qquad s\in [0,1],
\end{equation*}
\notag
$$
and make no difference between $\mathfrak{f}$ and $f$. Equipped with the metric of total variation, $\mathfrak{F}=\{\mathfrak{f}\}=\{f\}$ becomes a Polish space. Let
$$
\begin{equation*}
F(s)=\sum_{j=0}^{\infty}F(\{ j\}) s^{j}, \qquad s\in[ 0,1],
\end{equation*}
\notag
$$
be a random variable taking values in $\mathfrak{F}$, and let
$$
\begin{equation*}
F_{n}(s)=\sum_{j=0}^{\infty}F_{n}(\{ j\}) s^{j}, \qquad s\in [ 0,1], \quad n\in \mathbb{N},
\end{equation*}
\notag
$$
be a sequence of independent probabilistic copies of the random variable $F$. The infinite sequence $\mathcal{E}=\{F_{n},\,n\in \mathbb{N}\}$ is called a random environment. A sequence of nonnegative random variables $\mathcal{Z}=\{Z_{n},\, n\in \mathbb{N}_{0}\}$ specified on a probability space $(\Omega, \mathcal{F},\mathbf{P})$ is called a branching process in a random environment (BPRE) if $Z_{0}$ is independent of $\mathcal{E}$ and, given $\mathcal{E}$, the process $\mathcal{Z}$ is a Markov chain with
$$
\begin{equation*}
\mathcal{L}(Z_{n}\mid Z_{n-1}=z_{n-1},\, \mathcal{E}=(f_{1},f_{2},\dots)) =\mathcal{L}(\xi_{n1}+\dots +\xi_{ny_{n-1}})
\end{equation*}
\notag
$$
for all $n\in \mathbb{N}$, $z_{n-1}\in \mathbb{N}_{0}$ and $f_{1},f_{2},\ldots\in \mathfrak{F}$, where $\xi_{n1},\xi_{n2},\dots$ is a sequence of independent identically distributed random variables with distribution $f_{n}$. Thus, $Z_{n-1}$ is the $(n-1)$st generation size of the population of the branching process and $f_{n}$ is the offspring distribution of an individual at generation $n-1$. The sequence
$$
\begin{equation*}
S_{0}=0, \qquad S_{n}=X_{1}+\dots+X_{n}, \quad n\geqslant 1,
\end{equation*}
\notag
$$
where $X_{i}=\log F_{i}'(1)$, $i=1,2,\dots$, is called the associated random walk for the process $\mathcal{Z}$. We assume below that $Z_{0}=1$ and impose the following restrictions on the properties of the BPRE. According to the classification of BPREs (see, for instance, [1] and [9]), Condition B1 means that we consider the critical BPREs. Our second assumption on the environment concerns reproduction laws of particles. Set
$$
\begin{equation*}
\gamma (b):=\frac{\sum_{k=b}^{\infty}k^{2}F(\{ k\}) }{\bigl(\sum_{i=0}^{\infty}iF(\{ i\})\bigr)^{2}}.
\end{equation*}
\notag
$$
Condition B2. There exist $\varepsilon >0$ and $b\in\mathbb{N}$ such that where $\log ^{+}x=\log (x\vee 1)$.It is known (see [1], Theorem 1.1 and Corollary 1.2) that if Conditions B1 and B2 are valid, then there exist a number $\theta \in(0,\infty)$ and a sequence $l(1),l(2),\dots$ varying slowly at infinity such that, as $n\to\infty$,
$$
\begin{equation}
\mathbf{P}(Z_{n}>0) \sim\theta \, \mathbf{P}(L_n\geqslant 0)\sim \theta n^{-(1-\rho)}l(n),
\end{equation}
\tag{7.1}
$$
and for any $x\geqslant 0$
$$
\begin{equation}
\begin{aligned} \, \mathbf{P}(Z_{n}>0,\, S_{n}\leqslant xa_{n}) &=\mathbf{P}(S_{n}\leqslant xa_{n}\mid Z_{n}>0) \, \mathbf{P}(Z_{n}>0) \notag \\ &\sim \mathbf{P}(Y_{1}^{+}\leqslant x)\, \mathbf{P}(Z_{n}>0), \end{aligned}
\end{equation}
\tag{7.2}
$$
where $\mathcal{Y}^{+}=\{Y_{t}^{+},\,0\leqslant t\leqslant 1\}$ denotes the meander of a strictly $\alpha$-stable process $\mathcal{Y}$, so that $\mathcal{Y}^+$ is a strictly $\alpha$-stable Lévy process, which is assumed to be positive on the half-open interval $(0,1]$ (see [4] and [5]). Thus, if a BPRE is critical, then, given $Z_{n}>0$, the random variable $S_{n}$, which is the value of the associated random walk that provides the survival of the population to a distant moment $n$, grows like $a_{n}$ times a random positive multiplier. Since $\mathbf{P}(Y_{1}^{+}\leqslant 0)=0$, it follows from (7.2) that if $\varphi(n)$ satisfies the restriction
$$
\begin{equation}
\limsup_{n\to \infty}\frac{\varphi(n)}{a_{n}}\leqslant 0,
\end{equation}
\tag{7.3}
$$
then
$$
\begin{equation*}
\begin{aligned} \, \mathbf{P}(Z_{n}>0,\, S_{n}\leqslant \varphi(n)) &=\mathbf{P}(S_{n}\leqslant \varphi(n)\mid Z_{n}>0) \mathbf{P}(Z_{n}>0) \\ &=o(\mathbf{P}(Z_{n}>0)) \end{aligned}
\end{equation*}
\notag
$$
as $n\to\infty$. It is natural to consider an environment meeting condition (7.3) as unfavourable for the development of the critical BPRE. An important case of an unfavourable random environment was considered in [12], and [14], where it was in particular shown that if $\varphi(n)\to\infty$ as $n\to\infty$ in such a way that $\varphi(n)=o(a_{n})$, then
$$
\begin{equation*}
\mathbf{P}(Z_{n}>0,\, S_{n}\leqslant \varphi(n)) \sim C b_{n}\int_{0}^{\varphi(n)}V(-w)\,dw, \qquad C \in (0,\infty).
\end{equation*}
\notag
$$
The authors of [12] also investigated the conditional distribution of the number of particles in the process at time $m\leqslant n$ given the event $\{Z_{n}>0,\,S_{n}\leqslant \varphi(n)\}$. In this paper we complement the results of [12] by imposing even more stringent restrictions on the environment. Namely, we assume that at the time $n$ of observation the environment meets the assumption $S_{n}\leqslant K$ for a fixed constant $K$ and call such an environment extremely unfavourable. Our main result looks as follows. Theorem 5. Let Conditions B1 and B2 be valid. Then for any fixed $K$
$$
\begin{equation*}
\lim_{n\to \infty}\frac{\mathbf{P}(Z_{n}>0,\, S_{n}\leqslant K)}{b_{n}}=G_{\mathrm{left}}(K)+G_{\mathrm{right}}(K),
\end{equation*}
\notag
$$
where the constants $G_{\mathrm{left}}(K)\in (0,\infty)$ and $G_{\mathrm{right}}(K)\in(0,\infty)$ are specified below by formulae (7.9) and (7.14), respectively. To prove the theorem we introduce two new probability measures $\mathbf{P}^{+}$ and $\mathbf{P}^{-}$ by using the identities
$$
\begin{equation}
\begin{aligned} \, \mathbf{E}[U(x+X);\, X+x\geqslant 0]&=U(x), \qquad x\geqslant 0 , \\ \mathbf{E}[V(x+X);\, X+x<0]&=V(x), \qquad x\leqslant 0, \end{aligned}
\end{equation}
\tag{7.4}
$$
which are valid for any oscillating random walk (see [9], § 4.4.3). The construction procedure of these measures is standard and is explained for $\mathbf{P}^{+}$ and $\mathbf{P}^{-}$ in detail in [1] and [2] (see also [9], § 5.2). We recall here only some basic definitions related to this construction. Let $\mathcal{F}_{n}$, $n\geqslant 0$, be the $\sigma $-field of events generated by the random variables $F_{1},F_{2},\dots,F_{n}$ and $Z_{0},Z_{1},\dots,Z_{n}$. This $\sigma $-field form a filtration $\mathcal{F}$. We assume that the random walk $\mathcal{S}=\{S_{n},\,n\geqslant0\}$ with the initial value $S_{0}=x$, $x\in \mathbb{R}$, is adapted to the filtration $\mathcal{F}$ and construct for $x\geqslant 0$ probability measures $\mathbf{P}_{x}^{+}$ and expectations $\mathbf{E}_{x}^{+}$ as follows. For every sequence $T_{0},T_{1},\dots$ of random variables with values in some space $\mathcal{T}$ and adopted to $\mathcal{F}$ and for any bounded and measurable function $g\colon\mathcal{T}^{n+1}\to\mathbb{R}$, $n\in \mathbb{N}_{0}$, we set
$$
\begin{equation*}
\mathbf{E}_{x}^{+}[g(T_{0},\dots,T_{n})]:= \frac{1}{U(x)}\, \mathbf{E}_{x}[g(T_{0},\dots,T_{n})U(S_{n});\, L_{n}\geqslant 0].
\end{equation*}
\notag
$$
Similarly, for $x<0$, $V$ gives rise to probability measures $\mathbf{P}_{x}^{-}$ and expectations $\mathbf{E}_{x}^{-}$ characterized for each $n\in \mathbb{N}_{0}$ by the equation
$$
\begin{equation*}
\mathbf{E}_{x}^{-}[g(T_{0},\dots,T_{n})]:= \frac{1}{V(x)}\, \mathbf{E}_{x}[g(T_{0},\dots,T_{n})V(S_{n});\, M_{n}<0].
\end{equation*}
\notag
$$
In virtue of (7.4) these definitions are consistent and in agreement with respect to $n$. For the convenience of the reader we present the following two lemmas, which provide the major steps in the proof of Theorem 5. Lemma 4 ([12], Lemma 4). Assume Condition B1. Let $H_{1},H_{2},\dots$, be a uniformly bounded sequence of real-valued random variables adapted to some filtration $\widetilde{\mathcal{F}}=\{\widetilde{\mathcal{F}}_{k},\,k\in \mathbb{N}\}$, which converges $\mathbf{P}^{+}$-almost surely to a random variable $H_{\infty}$. Suppose that $\varphi(n)$, $n\in\mathbb{N}$, is a real-valued function such that $\inf_{n\in \mathbb{N}}\varphi(n)\geqslant C>0$ and $\varphi(n)=o(a_{n})$ as $n\to\infty$. Then The statement of the following lemma uses the first $n$ elements $F_{1},\dots,F_{n}$ of the random environment $\mathcal{E}=\{F_{k},\,k\in \mathbb{N}\}$. Lemma 5. Let $0<\delta <1$. Let
$$
\begin{equation}
\widehat{W}_{n}=g_{n}(F_{1},\dots,F_{\lfloor \delta n\rfloor },Z_{0},Z_{1},\dots,Z_{\lfloor \delta n\rfloor}), \qquad n\geqslant 1,
\end{equation}
\tag{7.5}
$$
be random variables with values in a Euclidean (or a Polish) space $\mathcal{W}$ such that
$$
\begin{equation}
\widehat{W}_{n}\to \widehat{W}_{\infty} \quad \mathbf{P}_{x}^{+}\textit{-a.s.}
\end{equation}
\tag{7.6}
$$
for some $\mathcal{W}$-valued random variable $\widehat{W}_{\infty}$ and for all $x\geqslant 0$. Also let $B_{n}=h_{n}(F_{1},\dots,F_{\lfloor\delta n\rfloor})$, $n\geqslant 1$, be random variables with values in a Euclidean (or a Polish) space $\mathcal{B}$ such that
$$
\begin{equation*}
B_{n}\to B_{\infty} \quad \mathbf{P}^{-}\textit{-a.s.}
\end{equation*}
\notag
$$
for some $\mathcal{B}$-valued random variable $B_{\infty}$. Set
$$
\begin{equation*}
\widetilde{B}_{n}:=h_{n}(F_{n},\dots,F_{n-\lfloor \delta n\rfloor +1}).
\end{equation*}
\notag
$$
Then for any bounded continuous function $\Psi\colon\mathcal{W}\times \mathcal{B}\times \mathbb{R}\to\mathbb{R}$ and for $\lambda >0$
$$
\begin{equation*}
\begin{aligned} \, & \frac{\mathbf{E}[\Psi (\widehat{W}_{n}, \widetilde{B}_{n},S_{n})e^{\lambda S_{n}};\, \tau_{n}=n]}{\mathbf{E}[e^{\lambda S_{n}};\, \tau_{n}=n]} \\ &\qquad \to K_{1}\iiint \Psi (w,b,-z)\, \mathbf{P}_{z}^{+}(\widehat{W}_{\infty}\in dw) \, \mathbf{P}^{-}(B_{\infty}\in db)U(z)e^{-\lambda z}\,dz \end{aligned}
\end{equation*}
\notag
$$
as $n\to\infty$, where Proof. First of all, note that the statement of the lemma coincides almost literally with the statement of Theorem 2.8 in [2]. The only difference is that [2] deals, instead of the sequence $\{\widehat{W}_{n},\,n\geqslant 1\}$, with the sequence
$$
\begin{equation*}
W_{n}=g_{n}(F_{1},\dots,F_{\lfloor \delta n\rfloor}), \qquad n\geqslant 1,
\end{equation*}
\notag
$$
meeting the condition
$$
\begin{equation*}
W_n\to W_{\infty} \quad \mathbf{P}_{x}^{+}\text{-a.s.}
\end{equation*}
\notag
$$
for some $\mathcal{W}$-valued random variable $W_{\infty}$ and all $x\geqslant 0$. In particular, it was shown in [2] that for all continuous bounded functions $\varphi_{1}(u)$, $\varphi_{2}(b)$ and $\varphi_{3}(z)$
$$
\begin{equation*}
\begin{aligned} \, &\frac{\mathbf{E}[\varphi_1(W_n)\varphi_2(\widetilde{B}_n)\varphi_3(S_n)e^{\lambda S_{n}};\, \tau _{n}=n]}{\mathbf{E}[e^{\lambda S_{n}};\, \tau_{n}=n]} \\ &\qquad \to K_{1}\iiint \varphi_1(u)\varphi_2(b)\varphi_3(-z)\, \mathbf{P}_{z}^{+}(W_{\infty}\in du) \, \mathbf{P}^{-}(B_{\infty}\in db) U(z)e^{-\lambda z}\,dz \end{aligned}
\end{equation*}
\notag
$$
as $n\to\infty$. Now let $\widehat{\varphi}_{1}(w),\,w\in (-\infty,+\infty)$ be a continuous function such that $|\widehat{\varphi}_{1}(w)|\leqslant C$ for all $w\in (-\infty,+\infty)$, and let
$$
\begin{equation*}
W_{n}=W_n(F_{1},\dots,F_{\lfloor \delta n\rfloor})=\mathbf{E}[ \widehat{\varphi}_1(\widehat{W}_n)\mid F_{1},\dots,F_{\lfloor \delta n\rfloor}].
\end{equation*}
\notag
$$
It follows from (7.6) and the dominated convergence theorem that
We introduce the function
$$
\begin{equation*}
\varphi_{1}(u)= \begin{cases} 0 & \text{for } u<-2C, \\ -2C-u & \text{for } -2C\leqslant u<-C, \\ u & \text{for } -C\leqslant u\leqslant C, \\ 2C-u & \text{for } C<u<2C, \\ 0 &\text{for } 2C\leqslant u. \end{cases}
\end{equation*}
\notag
$$
In accordance with the definition of $\varphi_{1}$, we have
$$
\begin{equation}
\begin{aligned} \, \notag &\int_{-\infty}^{+\infty} \varphi_1(u)\, \mathbf{P}_{z}^{+}(W_{\infty}\in du) =\int_{-C}^C u\, \mathbf{P}_{z}^{+}(W_{\infty}\in du)=\mathbf{E}^+_z[W_{\infty}] \\ &\qquad=\mathbf{E}_z^+\bigl[\mathbf{E}[\widehat{\varphi}_1(\widehat{W}_{\infty}) \mid \mathcal{E}]\bigr]=\mathbf{E}_z^+[\widehat{\varphi}_1(\widehat{W}_{\infty})] =\int_{-\infty}^{+\infty} \widehat{\varphi}_1(w)\mathbf{P}_{z}^{+}( \widehat{W}_{\infty}\in dw). \end{aligned}
\end{equation}
\tag{7.7}
$$
Since the random variables $\widetilde{B}_n$, $S_n$ and $\tau_n$ depend on the state of the environment rather than on the sequence $Z_0,Z_1,\dots,Z_n$, it follows that
$$
\begin{equation*}
\begin{aligned} \, &\mathbf{E}[\widehat{\varphi}_1(\widehat{W}_n)\varphi_2(\widetilde{B}_n)\varphi_3(S_n)e^{\lambda S_{n}};\, \tau_{n}=n] \\ &\qquad=\mathbf{E}[\varphi_1(W_n)\varphi_2(\widetilde{B}_n)\varphi_3(S_n)e^{\lambda S_{n}};\, \tau_{n}=n]. \end{aligned}
\end{equation*}
\notag
$$
This equality and (7.7) show that for any continuous bounded functions $\widehat{\varphi}_{1}(w)$, $\varphi_{2}(b)$ and $\varphi_{3}(z)$
$$
\begin{equation*}
\begin{aligned} \, &\frac{\mathbf{E}[\widehat{\varphi}_1(\widehat{W}_n)\varphi_2(\widetilde{B}_n) \varphi_3(S_n)e^{\lambda S_{n}};\, \tau_{n}=n]}{\mathbf{E}[e^{\lambda S_{n}};\, \tau_{n}=n]} \\ &\qquad=\frac{\mathbf{E}[\varphi_1(W_n)\varphi_2(\widetilde{B}_n) \varphi_3(S_n)e^{\lambda S_{n}};\, \tau_{n}=n]}{\mathbf{E}[e^{\lambda S_{n}};\, \tau_{n}=n]} \\ &\qquad\to K_{1}\iiint \varphi_1(u)\varphi_2(b)\varphi_3(-z)\, \mathbf{P}_{z}^{+}( W_{\infty}\in dw)\, \mathbf{P}^{-}(B_{\infty}\in db)U(z)e^{-\lambda z}\,dz \\ &\qquad\to K_{1}\iiint \widehat{\varphi}_1(w)\varphi_2(b)\varphi_3(-z)\, \mathbf{P}_{z}^{+}( \widehat{W}_{\infty}\in dw) \, \mathbf{P}^{-}(B_{\infty}\in db) U(z)e^{-\lambda z}\,dz \end{aligned}
\end{equation*}
\notag
$$
as $n\to\infty$.
To complete the proof of the lemma, it suffices to note that each continuous bounded function $\Psi(w,b,z)$ of three variables on a compact set can be approximated by linear combinations of products of functions of the form $\widehat{\varphi}_{1}(w)\varphi_{2}(b)\varphi_{3}(z)$ as accurately as needed and to observe that the elements of the sequences $\{\widehat{W}_{n},\,n\geqslant 1\}$, $\{\widetilde{B}_{n},n\geqslant 1\}$ and the random variables $\widehat{W}_{\infty}$ and $B_{\infty}$ are bounded, while the integral $\displaystyle\int_J^{\infty}U(z)e^{-\lambda z}\,dz$ can be made arbitrary small by the choice of the parameter $J$. Lemma 5 is proved. Proof of Theorem 5. First observe that
$$
\begin{equation*}
\begin{aligned} \, \mathbf{P}(Z_{n}>0\mid \mathcal{E}) &\leqslant \min_{0\leqslant k\leqslant n}\mathbf{P}(Z_{k}>0\mid \mathcal{E}) \\ &\mathbf{\leqslant}\min_{0\leqslant k\leqslant n}\mathbf{E}[ Z_{k}\mid \mathcal{E}] =\min_{0\leqslant k\leqslant n}e^{S_{k}}=e^{S_{\tau_{n}}}. \end{aligned}
\end{equation*}
\notag
$$
This estimate, in combination with the arguments used in the proof of Theorem 4, implies that
$$
\begin{equation}
\begin{aligned} \, &\limsup_{J\to \infty}\limsup_{n\to \infty}\frac{\mathbf{P}(Z_{n}>0,\, S_{n}\leqslant K,\, \tau_{n}\in [J+1,n-J])}{b_{n}} \notag \\ &\qquad=\limsup_{J\to \infty}\limsup_{n\to \infty}\frac{\mathbf{E}[ \mathbf{P}(Z_{n}>0\mid \mathcal{E});\, S_{n}\leqslant K,\, \tau_{n}\in [ J+1,n-J] ]}{b_{n}} \notag \\ &\qquad \leqslant \limsup_{J\to \infty}\limsup_{n\to \infty} \frac{\mathbf{E}[ e^{S_{\tau_{n}}};\, S_{n}\leqslant K,\, \tau_{n}\in [J+1,n-J] ]}{b_{n}}=0. \end{aligned}
\end{equation}
\tag{7.8}
$$
Thus, it remains to analyze the case when We fix sufficiently large positive integers $J$ and $N>|K|$ and, for $j\in [0,J]$, consider the chain of estimates
$$
\begin{equation*}
\begin{aligned} \, &\mathbf{P}(Z_{n}>0,\, S_{n}\leqslant K,\, \tau_{n}=j, \, S_{j}<-N) \\ &\qquad \leqslant \mathbf{E}[ e^{S_{j}};\, S_{n}\leqslant K, \, \tau_{n}=j,\, S_{j}<-N] \\ &\qquad =\int_{-\infty}^{-N}e^{x}\, \mathbf{P}(S_{j}\in dx,\, \tau_{j}=j) \, \mathbf{P}(S_{n-j}\leqslant K-x, \, L_{n-j}\geqslant 0) \\ &\qquad \leqslant \int_{-\infty}^{-N}e^{x}\, \mathbf{P}(S_{j}\in dx,\, \tau_{j}=j) \, \mathbf{P}(S_{n-j}\leqslant -2x, \, L_{n-j}\geqslant 0) . \end{aligned}
\end{equation*}
\notag
$$
In view of (3.3) and (3.5), for any $\varepsilon >0$ there is a sufficiently large $N=N(\varepsilon)$ such that
$$
\begin{equation*}
\begin{aligned} \, &\int_{-\infty}^{-N}e^{x} \, \mathbf{P}(S_{j}\in dx,\tau_{j}=j) \, \mathbf{P}(S_{n-j}\leqslant -2x,\, L_{n-j}\geqslant 0) \\ &\qquad \leqslant Cb_{n-j}\int_{-\infty}^{-N}e^{x}\, \mathbf{P}(S_{j}\in dx,\, \tau_{j}=j) \int_{0}^{-2x}V(-w)\,dw \\ &\qquad \leqslant C_{1}b_{n-j}\int_{-\infty}^{-N}e^{x} (|x|^{2}+1) \, \mathbf{P}(S_{j}\in dx) \leqslant \varepsilon b_{n-j} \end{aligned}
\end{equation*}
\notag
$$
for all $n-j\geqslant 0$. Thus, for all sufficiently large $N$.
Further, for any $Q\in \mathbb{N}$ and $N>K$, from (3.3) and (2.2) we obtain
$$
\begin{equation*}
\begin{aligned} \, &\mathbf{P}(Z_{n}>0,\, S_{n}\leqslant K,\, \tau_{n}=j,\, S_{j}\geqslant -N,\, Z_{j}>Q) \\ &\qquad \leqslant \mathbf{P}(S_{n}\leqslant K,\, \tau_{n}=j,\, S_{j}\geqslant -N,\, Z_{j}>Q) \\ &\qquad =\int_{-N}^{K\wedge 0}\mathbf{P}(S_{j}\in dx,\, \tau_{j}=j,\, Z_{j}>Q) \, \mathbf{P}(S_{n-j}\leqslant K-x,\, L_{n-j}\geqslant 0) \\ &\qquad \leqslant \int_{-N}^{0}\mathbf{P}(S_{j}\in dx,\, \tau_{j}=j,\, Z_{j}>Q) \, \mathbf{P}(S_{n-j}\leqslant 2N,\, L_{n-j}\geqslant 0) \\ &\qquad \leqslant \mathbf{P}(\tau_{j}=j,\, Z_{j}>Q) \, \mathbf{P}(S_{n-j}\leqslant 2N,\, L_{n-j}\geqslant 0) \\ &\qquad \leqslant Cb_{n-j}\, \mathbf{P}(\tau_{j}=j,\, Z_{j}>Q) \int_{0}^{2N}V(-w)\,dw \\ &\qquad \leqslant C_{1}b_{n-j}\, \mathbf{P}(\tau_{j}=j,\, Z_{j}>Q) N^{2}. \end{aligned}
\end{equation*}
\notag
$$
Clearly,
$$
\begin{equation*}
\lim_{Q\to \infty}\mathbf{P}(\tau_{j}=j,\, Z_{j}>Q) \leqslant \lim_{Q\to \infty}Q^{-1}\, \mathbf{E}[e^{S_{j}};\, \tau_{j}=j]=0.
\end{equation*}
\notag
$$
Therefore, we can select a sufficiently large $Q$ such that
$$
\begin{equation*}
\mathbf{P}(\tau_{j}=j,\, Z_{j}>Q) N^{2}\leqslant \varepsilon,
\end{equation*}
\notag
$$
and obtain the estimate
$$
\begin{equation*}
\mathbf{P}(Z_{n}>0,\, S_{n}\leqslant K,\, \tau_{n}=j,\, Z_{j}>Q) \leqslant C_{1}\varepsilon b_{n-j}.
\end{equation*}
\notag
$$
Now consider the probability
$$
\begin{equation*}
\mathbf{P}(Z_{n}>0,\, S_{n}\leqslant K,\, \tau_{n}=j,\, S_{j}\geqslant -N,\, Z_{j}\leqslant Q).
\end{equation*}
\notag
$$
We introduce the event
$$
\begin{equation*}
A_{\mathrm{u.s}}:=\{ Z_{n}>0\text{ for all }n\geqslant 0\}
\end{equation*}
\notag
$$
and for $0\leqslant s\leqslant 1$ define the iterations
$$
\begin{equation*}
F_{k,n}(s):=F_{k+1}(F_{k+2}(\dots F_{n}(s)\dots))\quad \text{if}\ 0\leqslant k<n\ \text{and}\ F_{n,n}(s):=s.
\end{equation*}
\notag
$$
Since the limit
$$
\begin{equation*}
\lim_{n\to \infty}\mathbf{P}(Z_{n}>0\mid \mathcal{E},\, Z_{0}=q)=\lim_{n\to \infty}(1-F_{0,n}^{q}(0))=:1-F_{0,\infty}^{q}(0)
\end{equation*}
\notag
$$
exists $\mathbf{P}^{+}$-almost surely by the monotonicity of the extinction probability of branching processes, it follows from Lemma 4 that
$$
\begin{equation*}
\mathbf{E}[ 1-F_{0,n}^{q}(0)\mid S_{n}\leqslant y_{n},\, L_{n}\geqslant 0] \to \mathbf{E}^{+}[ 1-F_{0,\infty}^{q}(0)] =\mathbf{P}_{q}^{+}(A_{\mathrm{u.s}})
\end{equation*}
\notag
$$
as $n\to\infty$ if $y_{n}=o(a_{n})$. In addition, $\mathbf{P}_{q}^{+}(A_{\mathrm{u.s}})>0$ for any $q=1,2,\dots$, according to Proposition 3.1 in [1].
Now Corollary 2 shows that for fixed $q\leqslant Q$ and $x\in [-N,K]$
$$
\begin{equation*}
\begin{aligned} \, & \mathbf{P}(Z_{n-j}>0,\, S_{n-j}\leqslant K-x,\, L_{n-j}\geqslant 0,\, Z_{0}=q) \\ &\qquad =\mathbf{E}[ 1-F_{0,n-j}^{q}(0);\, S_{n-j}\leqslant K-x,\, L_{n-j}\geqslant 0] \\ &\qquad \sim \mathbf{P}_{q}^{+}(A_{\mathrm{u.s}}) \, \mathbf{P}(S_{n-j}\leqslant K-x,\, L_{n-j}\geqslant 0) \\ &\qquad \sim g_{\alpha,\beta}(0)\mathbf{P}_{q}^{+}(A_{\mathrm{u.s}})b_{n-j}\int_{0}^{K-x}V(-w)\,dw \end{aligned}
\end{equation*}
\notag
$$
as $n-j\to\infty $. Hence, using (3.3) once again and applying the dominated convergence theorem we conclude that
$$
\begin{equation*}
\begin{aligned} \, &\frac{\mathbf{P}(Z_{n}>0,\, S_{n}\leqslant K,\, \tau_{n}=j,\, S_{j}\geqslant -N,\, Z_{j}=q)}{b_{n-j}} \\ &\quad =\int_{-N}^{K\wedge 0}\mathbf{P}(S_{j}\in dx,\, \tau_{j}=j,\, Z_{j}=q) \, \frac{\mathbf{E}[ 1-F_{0,n-j}^{q}(0);\, S_{n-j}\leqslant K-x,\, L_{n-j}\geqslant 0]}{b_{n-j}} \\ &\quad \sim g_{\alpha,\beta}(0)\, \mathbf{P}_{q}^{+}(A_{\mathrm{u.s}}) \int_{-N}^{K\wedge 0}\mathbf{P}(S_{j}\in dx,\, \tau_{j}=j,\, Z_{j}=q) \int_{0}^{K-x}V(-w)\,dw \end{aligned}
\end{equation*}
\notag
$$
as $n-j\to\infty$.
Combining all these estimates, letting $Q$ and $N$ tend to infinity and using (1.1), we see that for any fixed $j\in [0,J]$
$$
\begin{equation*}
\begin{aligned} \, m_{j} &:=\lim_{n\to \infty}\frac{\mathbf{P}(Z_{n}>0,\, S_{n}\leqslant K,\, \tau_{n}=j)}{b_{n}} \\ &=g_{\alpha,\beta}(0)\int_{-\infty}^{K\wedge 0}\mathbf{E}\biggl[ \mathbf{P}_{Z_{j}}^{+}(A_{\mathrm{u.s}}) \, \mathbf{P}(S_{j}\in dx,\, \tau_{j}=j) \int_{0}^{K-x}V(-w)\,dw\biggr] . \end{aligned}
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation}
G_{\mathrm{left}}(K):=\lim_{J\to \infty}\lim_{n\to \infty} \frac{\mathbf{P}(Z_{n}>0,\, S_{n}\leqslant K,\, \tau_{n}\in [ 0,J])}{b_{n}}=\sum_{j=0}^{\infty}m_{j}.
\end{equation}
\tag{7.9}
$$
Clearly,
$$
\begin{equation*}
\begin{aligned} \, G_{\mathrm{left}}(K) &\geqslant m_{1} \\ &\geqslant g_{\alpha,\beta}(0)\, \mathbf{P}^{+}(A_{\mathrm{u.s}}\mid Z_{0}=1)\int_{-\infty}^{K\wedge 0}\mathbf{P}(S_{1}\in dx) \int_{0}^{K-x}V(-w)\,dw >0. \end{aligned}
\end{equation*}
\notag
$$
The inequality $G_{\mathrm{left}}(K)<\infty$ follows from (6.1) and the estimate
$$
\begin{equation*}
\begin{aligned} \, \mathbf{P}(Z_{n}>0,\, S_{n}\leqslant K,\, \tau_{n}\in [ 0,J]) &=\mathbf{E}[ 1-F_{0,n}(0);\, S_{n}\leqslant K,\, \tau_{n}\in [ 0,J]] \\ &\leqslant \mathbf{E}[ e^{S_{\tau_{n}}};\, S_{n}\leqslant K,\, \tau_{n}\in [0,J]]=R(0,J). \end{aligned}
\end{equation*}
\notag
$$
Now assume that $n-j=t\in [0,J]$. Using the independence of the tuples $F_{1},\dots,F_{j}$ and $F_{j+1},\dots,F_{n}$ we write
$$
\begin{equation*}
\begin{aligned} \, &\mathbf{P}(Z_{n}>0,\, S_{n}\leqslant K,\, \tau_{n}=j)=\mathbf{E}[1-F_{0,n}(0);\, S_{n}\leqslant K,\, \tau_{n}=j] \\ &\quad =\mathbf{E}[ 1-F_{0,j}(F_{j,n}(0));\, S_{j}\leqslant K-(S_{n}-S_{j}),\, \tau_{n}=j] \\ &\quad =\int_{K}^{\infty}\int_{0}^{1}\mathbf{P}(F_{0,t}(0)\in ds,\, S_{t}\in dx,\, L_{t}\geqslant 0) \, \mathbf{E}[1-F_{0,j}(s);\, S_{j}\leqslant K-x,\, \tau_{j}=j] . \end{aligned}
\end{equation*}
\notag
$$
Our aim is to investigate the asymptotic behaviour of the quantity
$$
\begin{equation*}
\mathbf{E}[ 1-F_{0,j}(s);\, S_{j}\leqslant K-x,\, \tau_{j}=j]
\end{equation*}
\notag
$$
as $j\to\infty$ for fixed $s\in [0,1]$ and $x\geqslant K$.
Clearly,
$$
\begin{equation*}
\begin{aligned} \, \mathbf{E}[ 1-s^{Z_{j}}\mid \mathcal{E}] &=\mathbf{E}\bigl[ \mathbf{E}[ 1-s^{Z_{j}}\mid Z_{[j/2]}] \mathcal{E}\bigr]=\mathbf{E}\bigl[ 1-(F_{[ j/2],j}(s)) ^{Z_{[j/2]}}\bigm|\mathcal{E}\bigr] \\ &=\mathbf{E}\bigl[ 1-(F_{[ j/2],j}(s))^{\exp\{ S_{[ j/2]}-S_{j}\} Z_{[j/2]} \exp \{ -S_{[j/2]}\} \exp \{ S_{j}\}}\bigm| \mathcal{E}\bigr]. \end{aligned}
\end{equation*}
\notag
$$
For $w\geqslant 0$, $0\leqslant b\leqslant 1$ and $z\in \mathbb{R}$, setting $0^{0}=1$ we define the function
$$
\begin{equation*}
\Psi_{K-x}(w,b,z)=\bigl(1-b^{w\exp \{ z\}}\bigr)e^{-z}I\{ z\leqslant K-x\},
\end{equation*}
\notag
$$
and extend $\Psi_{K-x}$ to the other values of $w$, $b$ and $z$ as a bounded smooth function. In doing so, points of discontinuity at $(0,0,z)$ and $(w,b,K-x)$ are unavoidable. Our aim is to apply Lemma 5 to $\Psi_{K-x}$. However it is possible to apply this lemma only to bounded continuous functions. We show that in the case of BPREs this difficulty can be bypassed.
With the notation introduced in view, we write
$$
\begin{equation}
\begin{aligned} \, &\mathbf{E}[ 1-F_{0,j}(s);\, S_{j}\leqslant K-x,\, \tau_{j}=j] \nonumber \\ &\qquad=\mathbf{E}\bigl[ \Psi_{K-x}(\widehat{W}_{j},\widetilde{B}_{j}(s),S_{j})e^{S_{j}};\, \tau_{j}=j\bigr], \end{aligned}
\end{equation}
\tag{7.10}
$$
where
$$
\begin{equation*}
\widehat{W}_{j}:=Z_{[j/2]}e^{-S_{[ j/2]}} \quad\text{and}\quad \widetilde{B}_{j}(s):=(F_{[ j/2],j}(s)) ^{\exp \{ S_{[ j/2]}-S_{j}\}}.
\end{equation*}
\notag
$$
Since the sequence $\{\widehat{W}_{j},\,j\geqslant 1\}$ is a nonnegative martingale we have
$$
\begin{equation*}
\widehat{W}_{j}=Z_{[j/2]}e^{-S_{[ j/2]}}\to \widehat{W}_{\infty} \quad \mathbf{P}_{z}^{+}\text{-a.s.}
\end{equation*}
\notag
$$
as $j\to\infty $, where $\widehat{W}_{\infty}$ is a nonnegative random variable. Moreover, it follows from Proposition 3.1 in [1] that for any $z\geqslant 0$. (Estimate (7.11) was proved in [1] for $\mathbf{P}_{0}^{+}$ only. To show the validity of this inequality for all $z>0$ it is sufficient to note that the initial value of the associated random walk has no influence on the reproduction laws and to repeat literally the corresponding arguments from [1], using $S_{\ast}+z$ instead of $S_{\ast}$.)
Further, according to Lemma 3.2 in [2] the sequence
$$
\begin{equation*}
B_{j}(s):=(F_{[ j/2],0}(s))^{\exp \{ -S_{[j/2]}\}}, \qquad j=1,2,\dots,
\end{equation*}
\notag
$$
is nondecreasing and $B_{j}(s)\geqslant s^{\exp \{-S_{0}\}}$. Thus,
$$
\begin{equation*}
B_{j}(s)\to B_{\infty}(s)\geqslant s^{\exp \{ -S_{0}\}}=s \quad \mathbf{P}^{-}\text{-a.s.}
\end{equation*}
\notag
$$
as $j\to\infty$. In view of the equality
$$
\begin{equation*}
\mathbf{P}(\widetilde{B}_{j}(s)\geqslant s)=\mathbf{P}(B_{j}(s)\geqslant s)=1
\end{equation*}
\notag
$$
the second random variable in $\Psi_{K-x}(\widehat{W}_{j},\widetilde{B}_{j}(s),S_{j})$ is separated from zero with probability $1$.
Thus, aiming to apply Lemma 5 we can restrict $\Psi_{K-x}$ to the domain $(w,b,z\neq K-x)$ for $b>0$, where it is continuous. Finally, the additional discontinuity at $z=K-x$ has probability $0$ with respect to the measure
$$
\begin{equation*}
\mu (dz) :=K_{1}e^{-z}U(z)\, \mathbf I\{ z\geqslant 0\}\, dz,
\end{equation*}
\notag
$$
where $\displaystyle K_{1}^{-1}:=\int_{0}^{\infty}e^{-z}U(z)\,dz$.
As a result, we can apply Lemma 5 to conclude that, as $j\to\infty$,
$$
\begin{equation}
\frac{\mathbf{E}[ \Psi_{K-x}(\widehat{W}_{j},\widetilde{B}_{j}(s),S_{j})e^{S_{j}};\, \tau_{j}=j]}{\mathbf{E}[ e^{S_{j}};\, \tau_{j}=j]}\to h(s,K-x),
\end{equation}
\tag{7.12}
$$
where
$$
\begin{equation*}
\begin{aligned} \, &h(s,K-x) \\ &\qquad:=K_{1}\iiint \Psi_{K-x}(w,b,-z)\, \mathbf{P}_{z}^{+}(\widehat{W}_{\infty}\in dw) \, \mathbf{P}^{-}(B_{\infty}(s)\in db)\, e^{-z}U(z)\,dz \end{aligned}
\end{equation*}
\notag
$$
is a bounded function.
It was shown in the proof of Lemma 3.4 in [2] that $B_{\infty}(s)<1$ $\mathbf{P}^{-}$-almost surely under conditions B1 and B2. Altogether, this implies that the right-hand side of (7.12) is positive. Using now the inequality
$$
\begin{equation*}
\mathbf{E}[ 1-F_{0,j}(s);\, S_{j}\leqslant K-x,\, \tau_{j}=j] \leqslant \mathbf{E}[ e^{S_{j}};\, \tau_{j}=j]
\end{equation*}
\notag
$$
and the dominated convergence theorem we conclude that
$$
\begin{equation}
\begin{aligned} \, &\lim_{j\to \infty}\int_{K}^{\infty}\int_{0}^{1}\mathbf{P}(F_{0,t}(0)\in ds,\, S_{t}\in dx,\, L_{t}\geqslant 0)\, \frac{\mathbf{E}[1-F_{0,j}(s);\, S_{j}\leqslant K-x,\, \tau_{j}=j]}{\mathbf{E}[e^{S_{j}};\, \tau_{j}=j]} \notag \\ &\qquad =\int_{K}^{\infty}\int_{0}^{1}\mathbf{P}(F_{0,t}(0)\in ds,\, S_{t}\in dx,\, L_{t}\geqslant 0) h(s,K-x). \end{aligned}
\end{equation}
\tag{7.13}
$$
According to (1.6),
$$
\begin{equation*}
\mathbf{E}[ e^{S_{j}};\, \tau_{j}=j]=\mathbf{E}[e^{S_{j}};\, M_{j}<0] \sim g_{\alpha,\beta}(0)b_{j}\int_{0}^{\infty}e^{-w}U(w)\,dw
\end{equation*}
\notag
$$
as $j\to\infty $. Using this fact and summing (7.13) over $t$ from $0$ to $\infty$ we obtain
$$
\begin{equation}
\begin{aligned} \, &G_{\mathrm{right}}(K):=\lim_{J\to \infty}\lim_{n\to \infty}\frac{\mathbf{P}(Z_{n}>0;\, S_{n}\leqslant K,\, \tau_{n}\in [ n-J+1,n])}{b_{n}} \notag \\ &\qquad =g_{\alpha,\beta}(0)\int_{0}^{\infty}e^{-w}U(w)\,dw \notag \\ &\qquad\qquad \times \int_{K}^{\infty}\int_{0}^{1}\sum_{t=0}^{\infty} \mathbf{P}(F_{0,t}(0)\in ds,\, S_{t}\in dx,\, L_{t}\geqslant 0) h(s,K-x)>0. \end{aligned}
\end{equation}
\tag{7.14}
$$
The inequality $G_{\mathrm{right}}(K)<\infty$ follows from (6.4) and the estimate
$$
\begin{equation*}
\begin{aligned} \, &\mathbf{P}(Z_{n}>0,\, S_{n}\leqslant K,\, \tau_{n}\in [ n-J+1,n]) \\ &\qquad =\mathbf{E}[ 1-F_{0,n}(0);\, S_{n}\leqslant K,\, \tau_{n}\in [n-J+1,n]] \\ &\qquad \leqslant \mathbf{E}[ e^{S_{\tau_{n}}};\, S_{n}\leqslant K,\, \tau_{n}\in [ n-J+1,n]]=R(n-J+1,n). \end{aligned}
\end{equation*}
\notag
$$
as required. AcknowledgementThe authors are grateful to the reviewer, whose constructive comments allowed us to improve the presentation of the results of the paper. 
Citation in format AMSBIB
\Bibitem{VatDonDya24} |
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