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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2014 • Question 4

Question

Math 6010 • Linear Models Fall 2014 • Question 4

Suppose \(\{X_j\}_{j=0}^{\infty }\) is a martingale such that \(X_0=0\) and \(|X_{i+1}-X_i|\leq 1\) for all \(i\geq 0\). Prove that \(\lim _{n\to \infty }n^{-\delta }X_n=0\), a.s. for every \(\delta >1/2\).

Raw LaTeX

Suppose \(\{X_j\}_{j=0}^{\infty}\) is a martingale such that
\(X_0=0\) and \(|X_{i+1}-X_i|\leq1\) for all \(i\geq0\). Prove that
\(\lim_{n\to\infty}n^{-\delta}X_n=0\), a.s. for every \(\delta>1/2\).

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Topic Tags

Convergence in Probability and Almost Sure Convergence Exponential Martingales Martingales and Supermartingales Uniform Integrability and Truncation Methods