  For a continuous-time catalytic branching random walk (CBRW) on Z, with an
arbitrary finite number of catalysts, we study the asymptotic behavior of
position of the rightmost particle when time tends to infinity. The mild
requirements include the regular variation of the jump distribution tail for
underlying random walk and the well-known L log L condition for the offspring
numbers. In our classification, given in the previous paper, the analysis
refers to supercritical CBRW. The principle result demonstrates that, after a
proper normalization, the maximum of CBRW converges in distribution to a
non-trivial law. An explicit formula is provided for this normalization and
non-linear integral equations are obtained to determine the limiting
distribution function. The novelty consists in establishing the weak
convergence for CBRW with "heavy" tails, in contrast to the known behavior in
case of "light" tails of the random walk jumps. The new tools such as
"many-to-few lemma" and spinal decomposition appear non-efficient here. The
approach developed in the paper combines the techniques of renewal theory,
Laplace transform, non-linear integral equations and large deviations theory
for random sums of random variables.
  Keywords and phrases: catalytic branching random walk, heavy tails, regular
varying tails, spread of population, L log L condition.
