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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2010 • Question 10

Question

Math 6010 • Linear Models Spring 2010 • Question 10

Let \(Z\) be a random variable with all moments finite. Choose \(X\) and \(Y\) appropriately as in the Cauchy–Schwarz inequality or the Hölder inequality to prove that \(f(p):=\ln \mathbb {E}[|Z|^p]\) is convex on \((0,\infty )\).

Raw LaTeX

Let \(Z\) be a random variable with all moments finite. Choose \(X\)
and \(Y\) appropriately as in the Cauchy--Schwarz inequality or the H\"{o}lder
inequality to prove that \(f(p):=\ln\E[|Z|^p]\) is convex on \((0,\infty)\).

Topic Tags

Measure-Theoretic Probability Probability Inequalities and Tail Bounds Random Walks and Maximal Inequalities Uniform Integrability and Truncation Methods