Asymptotic properties of the branching random walk
<p>The branching random walk is a Galton-Watson process with the additional feature that people have positions. The initial ancestor is at the origin. Let {3<sup>(1)</sup><sub style="position: relative; left: -.8em;">r</sub>} be the po...
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| Formato: | Tesis |
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1976
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| Sumario: | <p>The branching random walk is a Galton-Watson process with the additional feature that people have positions. The initial ancestor is at the origin. Let {3<sup>(1)</sup><sub style="position: relative; left: -.8em;">r</sub>} be the positions on the real line of his children. The people in the nth generation give birth independently of one another and of the preceeding generations to form the (n+1)th generation and the positions of the children of an nth generation person at x has the same distributions as {3<sup>(1)</sup><sub style="position: relative; left: -.8em;">r</sub>+x} . Let {3<sup>(n)</sup><sub style="position: relative; left: -.8em;">r</sub>} be positions of the nth generation people in this process.</p> <p>In the first chapter the convergence of certain martingales associated with this process is examined. A generalization of the Kesten-Stigum theorem for the Galton-Watson process is obtained. The convergence of one of these martingales is shown to be closely related to some known results on the growth rate of age-dependent branching process.</p> <p>If B<sup>(n)</sup> is the position of the person on the extreme left of the nth generation then it is shown in the second chapter that B<sup>(n)</sup>/n &rightarrow;γ for some constant γ when the process survives. Subsequent chapters are generalizations of this result. Thus the same result holds for a multitype process with a finite number of different types and a weaker result holds when there is a countable number of different types. The generalization to the branching random walk on ℝ<sup>P</sup> is also considered. Let <em>C</em><sup>(n)</sup> be the set of points {3<sup>(n)</sup><sub style="position: relative; left: -.8em;">r</sub>/n:r}. It is shown that there is a compact convex set <em>C</em> such that <em>C</em><sup>(n)</sup><sup>Δ</sup><sub style="position: relative; left:-.8em;">&rightarrow;</sub><em>C</em> when the process survives where Δ is a suitable metric on the compact subsets of ℝ<sup>P</sup> .(All of these results are proved under the 'natural' conditions.)</p> |
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