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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2015 • Question 4

Question

Math 6010 • Linear Models Spring 2015 • Question 4

Let \(X_1,X_2,\ldots \) be independent and identically distributed random variables with \(\mathbb {E}\log (1+|X_1|)<\infty \). Show that \[ \sum _{i=1}^{\infty }\prod _{j=1}^{i}X_j \] is finite with probability one if and only if \(\mathbb {E}\log |X_1|<0\).

Raw LaTeX

Let \(X_1,X_2,\ldots\) be independent and identically distributed
random variables with \(\E\log(1+|X_1|)<\infty\). Show that
\[
  \sum_{i=1}^{\infty}\prod_{j=1}^{i}X_j
\]
is finite with probability one if and only if \(\E\log|X_1|<0\).

Topic Tags

Classical Probability and Inclusion-Exclusion Covariance and Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods