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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2017 • Question 7

Question

Math 6010 • Linear Models Fall 2017 • Question 7

Suppose \(X\) is a discrete, non-negative random variable with mass function \(p\). Prove that \[ \lim _{n\to \infty }\left (\mathbb {E}[X^n]\right )^{1/n} = \sup \{x\in \mathbb {R}:p(x)>0\}. \]

Raw LaTeX

Suppose \(X\) is a discrete, non-negative random variable with mass
function \(p\). Prove that
\[
  \lim_{n\to\infty}\left(\E[X^n]\right)^{1/n}
  =
  \sup\{x\in\mathbb{R}:p(x)>0\}.
\]

Topic Tags

Measure-Theoretic Probability Uniform Integrability and Truncation Methods