  Motivated by seminal paper of Kozlov et al.(1975) we consider in this paper a
branching process with a geometric offspring distribution parametrized by
random success probability $A$ and immigration equals $1$ in each generation.
In contrast to above mentioned article, we assume that environment is
heavy-tailed, that is $\log A^{-1} (1-A)$ is regularly varying with a parameter
$\alpha>1$, that is that ${\bf P} \Big( \log A^{-1} (1-A) > x \Big) =
x^{-\alpha} L(x)$ for a slowly varying function $L$. We will prove that
although the offspring distribution is light-tailed, the environment itself can
produce extremely heavy tails of distribution of the population at $n$-th
generation which gets even heavier with $n$ increasing. Precisely, in this
work, we prove that asymptotic tail ${\bf P}(Z_l \ge m)$ of $l$-th population
$Z_l$ is of order $ \Big(\log^{(l)} m \Big)^{-\alpha} L \Big(\log^{(l)} m
\Big)$ for large $m$, where $\log^{(l)} m = \log \ldots \log m$. The proof is
mainly based on Tauberian theorem. Using this result we also analyze the
asymptotic behaviour of the first passage time $T_n$ of the state $n \in
\mathbb{Z}$ by the walker in a neighborhood random walk in random environment
created by independent copies $(A_i : i \in \mathbb{Z})$ of $(0,1)$-valued
random variable $A$. This version differs from the final version as it contains
an alternative proof for the tail behavior for generation sizes which is not
very sharp (lacks constant) but completely avoids arguments based on Tauberian
theorem. This proof may be of an independent interest.
