We study the distribution of the maximal displacement of particle positions for the whole time of the existence of population in the model of critical and subcritical catalytic branching random walk on Z. In particular, we prove that in the case of simple symmetric random walk on Z, the distribution of the maximal displacement has a "heavy"tail', decreasing as a function of the power 1/2 or 1 when the branching process is critical or subcritical, respectively. These statements describe the e ects which had not arisen before in related studies on the maximal displacement of critical and subcritical branching random walks on Z.
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