L¨ohr/Winter Wintersemester 2012/13
Ubungen zur Vorlesung ¨ Wahrscheinlichkeitstheorie II
Ubungsblatt 12 ¨
Brownsche Bewegung & Satz von Donsker
Sei (Bt)t≥0 eine Standard Brownsche Bewegung.
Aufgabe 12.1. (3 Punkte)
SeiM := supt∈[0,1]Bt, T:= inf
t∈[0,1]
Bt=M . (a) Zeige oder widerlege:T ist eine Stoppzeit.
(b) Zeige, dass es fast sicher genau eint∈[0,1] mitBt=M gibt.
Hinweis:Verwende Aufgabe 11.4
Aufgabe 12.2 (Konvergenz im Pfadraum vs. fdd-Konvergenz). (4 Punkte) SeiM := supt∈[0,1]|Bt|undXt(n):= BMtn
. BetrachteX(n):= (Xt(n))t∈[0,1]. (a) Seit∈[0,1] fest. Zeige:Xt(n) n→∞−→ 0 f.s.
Insbesondere konvergieren die endlichdimensionalen Verteilungen vonX(n)gegen 0.
(b) Zeige, dass X(n) als Folge C [0,1]
-wertiger Zufallsvariablen nicht in Verteilung gegen 0 konvergiert.
Aufgabe 12.3 (Donsker f¨ur die Brownsche Br¨ucke). (4 Punkte) SeienU1, U2, . . . unabh¨angig und gleichverteilt auf [−1,1], und f¨ur n∈N
Zt(n) := (1−nt)
⌊t⌋
X
k=1
Uk−nt n
X
k=⌊t⌋+1
Uk+ t− ⌊t⌋ U⌊t⌋+1.
Zeige, dass
q3 nZnt(n)
t∈[0,1]
n→∞=⇒ (Xt)t∈[0,1],
wobei (Xt)t≥0 eine Brownsche Br¨ucke und die Konvergenz die schwache Konvergenz auf dem PfadraumC [0,1]
ist.
Hinweis:Verwende Donsker und eine geeignete AbbildungF:C [0,1]
→ C [0,1]
.
Aufgabe 12.4 (Folgerung aus dem Satz von Donsker). (5 Punkte) SeienX1, X2, . . .unabh¨angig, identisch verteilt mit E(X1) = 0 und Var(X1) = 1. Zeige f¨urSn :=
Pn
k=1Xk unda∈R: 1 n#
m≤n
Sm> a√
n n→∞=⇒ λ
t∈[0,1]
Bt> a , wobeiλdas Lebesguemaß auf [0,1] ist.
Hinweis:Zeige λ {t|Bt=a}
= 0 f.s. und wende Donsker auf ψ(f) =λ {t|f(t)> a} an.
Abgabe bis Di, 29.01. am Anfang der ¨Ubungsstunde Klausur: Do, 28.02., 10.15
Arbeitsgruppenvortr¨age:
Am22.01.gibt Martina Baar (Universit¨at Bonn) einen Vortrag ¨uber Stochastic individual-based models of adaptive dynamics
Abstract: In this talk, we study the limit behavior of a model for the Darwinian evolution in an asexual population characterized by a natural birth rate, a death rate due to age or competition and a probability of mutation at each birth event. The model is a stochastic, generic, individual-based model and belongs to the models of adaptive dynamics. We focus on the combination of the three main limits of the theory, large population size, rare mutations and small mutation effects, on the long-term evolution of the population.
More precisely, we consider the following tree limit behaviors: first, only the limit of large population, second, the limit of large population and rare mutations, third the limit of large population and rare mutation and afterwards the one of small mutation. Finally we investigate the limit of large population, rare and small mutation simultaneously in one single step. We obtain for this limit that on a specific time scale coexistence of two traits cannot occur in the population process with monomorphic initial condition. In other words, the population stays essentially single modal centered around a trait, that evolves continuously.
Am29.01.gibt Patric Gl¨ode (Universit¨at Erlangen) einen Vortrag ¨uber
Dynamics of Genealogical Trees for Autocatalytic Branching Processes
Abstract: My talk will feature the dynamics of genealogical trees for autocatalytic branching processes. In such populations each individual dies at a rate depending on the total population size and, upon its death, produces a random number of offspring. I will consider finite as well as infinite populations. Formally, processes take values in the space of ultrametric measure spaces. The dynamics are characterised by means of martingale problems. Key issues are to prove well-posedness for the martingale problems and to find invariance principles linking finite and infinite populations. In fact, infinite populations arise as scaling limits of finite populations in the sense of weak convergence on path space with respect to the (polar) Gromov-weak topology. I will show that there is a close relationship between the genealogies of infinite autocatalytic branching processes and the Fleming-Viot process. I will also mention an abstract uniqueness result for martingale problems of skew product form which is of importance for my own processes but also applies to more general situations.
Hierzu ergeht eine herzliche Einladung. Zeit:Di, 16.00 – 17.00. Raum: WSC-N-U-4.04
