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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2004 • Question 1

Question

Math 6010 • Linear Models 2004 • Question 1

A sequence \(X_1,X_2,\ldots \) of random variables is said to be uniformly integrable if \[ \lim _{c\to \infty }\sup _{n\ge 1}\mathbb {E}\{|X_n|;\ |X_n|>c\}=0. \] Prove that \(X_n\to X\) in \(L^1(\mathbb {P})\) if and only if both of the following happen: (i) \(X_n\to X\) in probability; and (ii) \(\{X_n\}_{n=1}^{\infty }\) is uniformly integrable.

Raw LaTeX

A sequence \(X_1,X_2,\ldots\) of random variables is said to be
uniformly integrable if
\[
  \lim_{c\to\infty}\sup_{n\ge 1}\E\{|X_n|;\ |X_n|>c\}=0.
\]
Prove that \(X_n\to X\) in \(L^1(\Prob)\) if and only if both of the following happen:
\textbf{(i)} \(X_n\to X\) in probability; and \textbf{(ii)}
\(\{X_n\}_{n=1}^{\infty}\) is uniformly integrable.

Topic Tags

Classical Probability and Inclusion-Exclusion Convergence in Probability and Almost Sure Convergence Measure-Theoretic Probability Uniform Integrability and Truncation Methods