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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2016 • Question 10

Question

Math 6010 • Linear Models Fall 2016 • Question 10

Suppose \(\mathbb {P}\{X_1=0\}=\mathbb {P}\{X_1=1\}=1/2\) and for all integers \(n\geq 1\), \[ \mathbb {P}(X_{n+1}=1\mid \mathcal {F}_n) = 1-\mathbb {P}(X_{n+1}=0\mid \mathcal {F}_n) = \frac {S_n}{n}, \] where \(S_n:=X_1+\cdots +X_n\) and \(\mathcal {F}_n:=\sigma (\{X_1,\ldots ,X_n\})\). Prove that \(n^{-1}S_n=X_1\) almost surely for all \(n\geq 1\).

Raw LaTeX

Suppose \(\Prob\{X_1=0\}=\Prob\{X_1=1\}=1/2\) and for all integers
\(n\geq1\),
\[
  \Prob(X_{n+1}=1\mid\mathcal{F}_n)
  =
  1-\Prob(X_{n+1}=0\mid\mathcal{F}_n)
  =
  \frac{S_n}{n},
\]
where \(S_n:=X_1+\cdots+X_n\) and
\(\mathcal{F}_n:=\sigma(\{X_1,\ldots,X_n\})\). Prove that
\(n^{-1}S_n=X_1\) almost surely for all \(n\geq1\).

Topic Tags

Convergence in Probability and Almost Sure Convergence Filtrations and Stopping Times Uniform Integrability and Truncation Methods