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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2010 • Question 9

Question

Math 6010 • Linear Models Spring 2010 • Question 9

Let \(X_1,X_2,\ldots \) be i.i.d. with mean \(\mu \). Define \(S_n:=X_1+\cdots +X_n\) for all \(n\geq 1\). For fixed \(n\geq 2\), define \[ M_1:=\frac {S_n}{n},\qquad M_2:=\frac {S_{n-1}}{n-1},\qquad \ldots \qquad M_n:=\frac {X_1}{1}. \] (a) Show that \(\mathbb {E}[X_k\mid S_n]=S_n/n\) for \(1\leq k\leq n\).

(b) Show that \(M_1,M_2,\ldots ,M_n\) is a martingale.

Raw LaTeX

Let \(X_1,X_2,\ldots\) be i.i.d. with mean \(\mu\). Define
\(S_n:=X_1+\cdots+X_n\) for all \(n\geq1\). For fixed \(n\geq2\), define
\[
  M_1:=\frac{S_n}{n},\qquad
  M_2:=\frac{S_{n-1}}{n-1},\qquad \ldots \qquad
  M_n:=\frac{X_1}{1}.
\]
\textbf{(a)} Show that \(\E[X_k\mid S_n]=S_n/n\) for \(1\leq k\leq n\).

\textbf{(b)} Show that \(M_1,M_2,\ldots,M_n\) is a martingale.

Topic Tags

Exponential Martingales Infinite Products and Multiplicative Martingales Martingales and Supermartingales Uniform Integrability and Truncation Methods