Transient phenomena for random walks with nonidential jumps having nonidetically infinite variances

Let $\zeta_1,\zeta_2,\dots$ be independent random variables,
$$ Z_n=\sum_{i=1}^n\zeta_i,\qquad \overline{Z}_n=\max_{k\leq n}Z_k,\qquad Z=\overline{Z}_\infty. $$
It is well known that if $\zeta_i$ are identically distributed, then $Z$ is a proper random variable when ${\mathbf{E}\zeta_i=-a<0}$, and $Z=\infty$ a.s. when $a=0$. The limiting distribution for $\overline{Z}_n$ as $n\to\infty$, $a\to 0$ (in the triangular array scheme) when $\mathbf{E}\zeta_i^2<\infty$ is well studied (see, e.g., [J. F. C. Kingman, J. Roy. Statist. Soc. Ser. B, 24 (1962), pp. 383–392], [Yu. V. Prokhorov, Litov. Math. Sb., 3 (1963), pp. 199–204 (in Russian)], and [A. A. Borovkov, Stochastic Process in Queueing Theory, Springer-Verlag, New York, 1976]).
In the present paper, we study the limiting distribution for $\overline{Z}_n$ as $\mathbf{E}\zeta_i\to 0$, $n\to\infty$, in the case when $\mathbf{E}\zeta_i^2=\infty$ and the summands $\zeta_i$ are nonidentically distributed.