positive and less than 1 for any $s\in \lbrack 0,1).$

    For brevity we set $\zeta _{l}^{-}(s):=\zeta _{l,0}^{-}(s)$ and $\zeta
    ^{-}(s):=\zeta _{\infty ,0}^{-}(s).$ Observe that $$\zeta :=\lim_{\min (l,n-l)\rightarrow \infty }\zeta
    _{l}^{-}(f_{0,n-l}^{+}(0))=\zeta ^{-}(q^{+})  \label{aszet}$$exists $\hat{\mathbf{P}}-$ a.s. and, moreover, $\zeta \in (0,1]$ with probability 1.

Main results
============

Now we are ready to formulate the main results of the present paper. Below our two main theorems we offer some intuition as to their interpretation.

\[Tclose\]Suppose that $A1-$ $A3$ hold. Then for any $R\in \mathbb{Z}$, any $t\in (0,1)$ and $s\in (0,1]$

1\) $$\left\{ \mathbf{E}_{\mathbf{k}}\left[ s^{Z_{\tau (nt)+R}}|T=n\right]
\left\vert \tau (n)\geq nt\right. \right\} \overset{d}{\rightarrow }s\left( 
\frac{1-\Theta _{R}}{1-\Theta _{R}s}\right) ^{2},\,n\rightarrow \infty ,$$where$$\Theta _{R}=\left\{ 
\begin{array}{ccc}
\zeta ^{-}\left( f_{0,R}^{+}(0)\right) e^{-S_{R}^{+}} & \text{if} & R\geq 0,
\\ 
&  &  \\ 
\zeta _{\infty ,R}^{-}(0)e^{-S_{R}^{-}} & \text{if} & R<0,\end{array}\right.$$and $\Theta _{R}\in (0,1)$ with probability 1;

2)$$\left\{ \mathbf{E}_{\mathbf{k}}\left[ s^{Z_{\tau (nt,n)+R}}|T=n\right]
\left\vert \tau (n)<nt\right. \right\} \overset{d}{\rightarrow }s\left( 
\frac{1-\theta _{R}}{1-\theta _{R}s}\right) ^{2},\,n\rightarrow \infty ,$$where$$\theta _{R}=\left\{ 
\begin{array}{ccc}
q_{R}^{+} & \text{if} & R\geq 0, \\ 
&  &  \\ 
f_{R,0}^{-}(q^{+}) & \text{if} & R<0,\end{array}\right.$$and $\theta _{R}\in (0,1)$ with probability 1.

The next theorem deals with the distribution of the number of particles at moments $nt$, $0<t<1.$ Let $$O_{m,n}:=\frac{1-f_{0,n}(0)}{1-f_{m,n}(0)}f_{m,n}(0)b_{m}.  \label{DefOO}$$

\[Tfar\]Suppose that Assumptions $A1-$ $A3$ hold. Then, for any 