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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2006 • Question 3

Question

Math 6010 • Linear Models 2006 • Question 3

Let \(\{X_n\}_{n=1}^{\infty }\) denote a martingale such that \(X_1\in L^2(\mathbb {P})\), and \[ \sum _{n=1}^{\infty }\mathbb {E}\{(X_{n+1}-X_n)^2\}<\infty . \] Prove that \(\lim _{n\to \infty }X_n\) exists a.s. and in \(L^2(\mathbb {P})\).

Raw LaTeX

Let \(\{X_n\}_{n=1}^{\infty}\) denote a martingale such that
\(X_1\in L^2(\Prob)\), and
\[
  \sum_{n=1}^{\infty}\E\{(X_{n+1}-X_n)^2\}<\infty.
\]
Prove that \(\lim_{n\to\infty}X_n\) exists a.s. and in \(L^2(\Prob)\).

Topic Tags

Convergence in Probability and Almost Sure Convergence L2 Convergence and Covariance Bounds Martingales and Supermartingales Uniform Integrability and Truncation Methods