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

Iksanov, O.; Negadajlov, P.

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 \(x\to\infty\).

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

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

Keywords:

supremum of martingale; branching random walk; point process