L¨ohr/Winter Wintersemester 2010/11
Ubungen zur Vorlesung Wahrscheinlichkeitstheorie II ¨
Ubungsblatt 6¨
Optional Sampling
Aufgabe 6.1. Seien Xk, k ∈ N, unabh¨angige, auf { −1,1} gleichverteilte ZV, und Sn :=
Pn
k=1Xk. F¨ur a, b∈Ndefiniere T =Tab := inf{n∈N|Sn=aoder Sn=−b}. Zeige:
(a) T is eine f.s. endliche Stoppzeit.
(b) P(ST =a) = b a+b.
Aufgabe 6.2. SeienXk,Sn,a,bundTab wie in Aufgabe 6.1. SeiTa:= inf{n∈N|Sn=a}.
(a) Berechne E(Tab).
Hinweis: Benutze die Doob-Zerlegung von (Sn2)n∈N
(b) Berechne E(Ta).
Aufgabe 6.3. Finde ein Martingal (Mn)n∈N und eine Folge von StoppzeitenT1 < T2 <· · · mitP(Tn<∞) = 1 f¨ur allenund
E(MTn) n−→ ∞.→∞
Insbesondere ist MTn
n∈Ndann kein Martingal.
Abgabe: Di, 30.11. in der ¨Ubungsstunde Arbeitsgruppenvortr¨age:
Am 30.11.gibt Lorenz Pfeiffroth von der TU M¨unchen einen Vortrag ¨uber Frogs in a random environment on Z
Abstract: The frog model in a fixed environment can be described as follows. Let G be a graph and take one vertex as origin. Initially there is a number of sleeping frogs at each vertex except the origin. At the origin there is one active frog which jumps according to a random walk on G. If an active frog jumps to a vertex where sleeping frogs are, they get awake and move according to the same random walk, independently from everything else. The idea of this model is that every active frog has some information and it shares it with the sleeping frogs for the first time when they meet. Alves, Machado and Popov proved a recurrence criterion if the graph is Zd orTd and the underlying random walk is a symmetric simple random walk.
The first time other underlying random walks were investigated was by Gantert and Schmidt in 2008. The random walk was a simple random walk in Z with drift to the right. In the first part of this talk we consider a more general setting of underlying random walks. I.e. the only
assumption for our random walk is that he is transient to the right. The question, we are interested in, is if the origin is visited infinitely often by active frogs with probability 1 or not. This is not a trivial question in this setting because all random walks in this model won’t eventually visit the negative integers. But intuitively spoken if there are enough frogs on the positive integers, which will be activated surely, the change of visiting the negative integers is increasing and thus also the origin. So we expect if there are enough frogs on the right of the origin the model will be recurrent. We give a necessary and sufficient condition that this will happened. Also we show that our result is a generalization of the model, which Gantert and Schmidt investigate, and present a 0-1 law for this model. Now the question naturally arise is if we take the jumping probability random, can we derive analogue conditions for the recurrence of such a model. The second part of this talk deals with that kind of problem.
We give recurrence criteria for such a model. If we take the starting configuration of sleeping frogs also as random, we derive a 0-1 law too and show that the recurrence of such a model only depends on the distribution of the starting configuration and it does not depend on the distribution of the jumping probability of the underlying random walk. In the last part I sketch the proof of the recurrence criteria for a frog model in a fixed and random environment, respectively.
Hierzu ergeht eine herzliche Einladung. Zeit:16.00 – 17.00. Raum: S05 T03 B72
