Martingale Convergence Theorem

L¨ohr/Winter Winter term 2015/16

Exercises to the lecture Probability Theory II

Exercise sheet 9

Martingale Convergence Theorem

Exercise 9.1 (Moran-model). (4 Points)

LetN, m∈Nwithm < N. We consider a population ofN individuals, out of which initially (in generation 0) m have the (genetic) typea, while the rest has type A. In every timestep (generation) one of the individuals (chosen uniformly at random) dies, and at the same time, a new one is born. Given all previous steps, the probability that the new individual has type a is equal to the relative frequency of typeain the previous generation. Let

X_n := relative frequency ofain the population of then^th generation andT := inf{n∈N|X_n∈ {0,1} }.

(a) Show that (X_n)_n∈N₀ is a martingale, which converges a.s. and in L¹. (b) Show that P {T <∞ }

= 1.

(c) Calculate P {X_T = 1} .

Exercise 9.2. (4 Points)

Let (Fn)_n∈N be a filtration on a probability space (Ω,A,P). Show that for A ∈ F :=

σ S

n∈NF_n

there is a sequence A_n∈ F_n with

n→∞lim P(A_n△A) = 0.

Here,A_n△A= (A_n\A)∪(A\A_n).

Hint: Consider X_n:=P(A| F_n) and use the martingale convergence theorem.

Exercise 9.3. (4 Points)

(a) Let a∈R₊ and (Xn)n∈N be a martingale withX_n+1 ≥X_n−a. Define B :=

lim sup

n→∞

X_n=∞ and C :=

lim inf

n→∞ X_n=−∞ . Show: P(B\C) = 0.

(b) Let X_n,n∈N, be independent withP {X_n=−1}

= _n+1ⁿ andP {X_n=n}

= _n+1¹ . Show that for S_n:=P_n

k=1X_k: P {lim sup

n→∞

S_n=∞ }

= P {lim inf

n→∞ S_n=−∞ }

= 1.

Hint: Use (a) and Borel-Cantelli.

Please turn

Exercise 9.4(Counter examples). (4 Points) (a) Find a martingale (Mn)n∈N withM_n−→

n→∞

∞ a.s.

(b) Find an L¹-bounded martingale (Xn)n∈N which is not uniformly integrable.

(c) Letp >1. Find anL^p-bounded process (Xn)n∈N, such that |Xn|^p

n∈Nis not uniformly integrable.

Due Wed, 13.01. at the beginning of the exercise session

Probability Seminar:

15.12.: Alexander Cox (University of Bath)

Maximising functions of the average of a martingale with given terminal law 22.12.: Sigurd Assing (University of Warwick) .

Tue, 16:15 – 17:15 in WSC-S-U-3.03