## On the supremum of martingale connected with branching random walk.(Ukrainian, English)Zbl 1150.60429

Teor. Jmovirn. Mat. Stat. 74, 44-51 (2006); translation in Theory Probab. Math. Stat. 74, 49-57 (2007).
Let $$S(\cdot)$$ be a point process on the straight line with points $$\infty>A_1\geq A_2\geq\cdots$$. Let us suppose that mother of some population is situated at the origin of real axis. Let us denote by $$S^{(n)}$$, $$n=1,2,\dots$$ the point process, which describes the location of individuals of $$n$$-th generation at the real axis. Individual, which is $$j_{m}$$ offspring of $$j_{m-1}$$ offspring $$\dots$$ $$j_1$$ offspring of mother is characterized by finite sequence of integers $$u=(j_1,\dots,j_{m})$$. The location of individual $$u$$ is denoted by $$A_{u}$$. Let there exist $$\gamma>0$$ such that $$m(\gamma):=E[\sum_{| u|=1}e^{\gamma A_{u}}]<\infty$$. The sequence $$W_{n}:={1\over m^{n}(\gamma)}\sum_{| u|=n}e^{\gamma A_{u}}$$, $$n=1,2,\dots$$ is the martingale connected with considered branching random walk. The authors study behaviour of $$P\{\sup_{u}W_{u}>x\}$$, as $$x\to\infty$$.

### MSC:

 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60G42 Martingales with discrete parameter
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