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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2010 • Question 2

Question

Math 6010 • Linear Models Spring 2010 • Question 2

Suppose \((\Omega ,\mathcal {F},\mathbb {P})\) is a probability space, and \(\mathcal {F}_1\subset \mathcal {F}_2\subset \cdots \) defines a filtration of sigma-algebras of subsets of \(\mathcal {F}\).

(a) State, without proof, Doob’s martingale convergence theorem.

(b) Prove that \(\mathcal {L}:=\lim _{n\to \infty }\mathbb {E}(Z\mid \mathcal {F}_n)\) exists almost surely and in \(L^1(\mathbb {P})\) for all \(Z\in L^1(\mathbb {P})\) [this is Lévy’s martingale convergence theorem]. Identify the limit \(\mathcal {L}\).

Raw LaTeX

Suppose \((\Omega,\mathcal{F},\Prob)\) is a probability space, and
\(\mathcal{F}_1\subset\mathcal{F}_2\subset\cdots\) defines a filtration of
sigma-algebras of subsets of \(\mathcal{F}\).

\textbf{(a)} State, without proof, Doob's martingale convergence theorem.

\textbf{(b)} Prove that \(\mathcal{L}:=\lim_{n\to\infty}\E(Z\mid\mathcal{F}_n)\)
exists almost surely and in \(L^1(\Prob)\) for all \(Z\in L^1(\Prob)\) [this is
L\'{e}vy's martingale convergence theorem]. Identify the limit \(\mathcal{L}\).

Topic Tags

Convergence in Probability and Almost Sure Convergence Filtrations and Stopping Times Martingale Convergence Theorems Martingales and Supermartingales