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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2017 • Question 9

Question

Math 6010 • Linear Models Spring 2017 • Question 9

Let \(X_1,X_2,\ldots \) be i.i.d. with \(\mathbb {P}\{X_1=1\}=p\) and \(\mathbb {P}\{X_1=-1\}=q\), where \(p\notin \{0,\frac {1}{2},1\}\) and \(q=1-p\). Define \(S_n:=X_1+\cdots +X_n\) for all \(n\geq 1\) and consider the stopping time, \(T:=\inf \{n\geq 1:|S_n|=5\}\), where \(\inf \emptyset :=\infty \). Prove that \(T\in L^1(\mathbb {P})\) and compute \(\mathbb {E}(T)\). (HINT: Begin by proving that \(M_n:=(q/p)^{S_n}\) and \(N_n:=S_n-(p-q)n\) are martingales.)

Raw LaTeX

Let \(X_1,X_2,\ldots\) be i.i.d. with
\(\Prob\{X_1=1\}=p\) and \(\Prob\{X_1=-1\}=q\), where
\(p\notin\{0,\frac{1}{2},1\}\) and \(q=1-p\). Define
\(S_n:=X_1+\cdots+X_n\) for all \(n\geq1\) and consider the stopping time,
\(T:=\inf\{n\geq1:|S_n|=5\}\), where \(\inf\emptyset:=\infty\). Prove that
\(T\in L^1(\Prob)\) and compute \(\E(T)\). (HINT: Begin by proving that
\(M_n:=(q/p)^{S_n}\) and \(N_n:=S_n-(p-q)n\) are martingales.)

Topic Tags

Exponential Martingales Filtrations and Stopping Times Martingales and Supermartingales Uniform Integrability and Truncation Methods