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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2016 • Question 5

Question

Math 6010 • Linear Models Fall 2016 • Question 5

Let \(\{X_n\}_{n=1}^{\infty }\) be i.i.d. random variables, and let \(p>0\) be fixed. Prove that the following are equivalent:

(a) \(\mathbb {E}(|X_1|^p)<\infty \);

(b) \(n^{-1/p}X_n\to 0\) almost surely; and

(c) \(n^{-1/p}\max _{1\leq j\leq n}|X_j|\to 0\) almost surely.

Raw LaTeX

Let \(\{X_n\}_{n=1}^{\infty}\) be i.i.d. random variables, and let
\(p>0\) be fixed. Prove that the following are equivalent:

\textbf{(a)} \(\E(|X_1|^p)<\infty\);

\textbf{(b)} \(n^{-1/p}X_n\to0\) almost surely; and

\textbf{(c)} \(n^{-1/p}\max_{1\leq j\leq n}|X_j|\to0\) almost surely.

Topic Tags

Convergence in Probability and Almost Sure Convergence Laws of Large Numbers Measure-Theoretic Probability Uniform Integrability and Truncation Methods