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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2013 • Question 4

Question

Math 6010 • Linear Models Fall 2013 • Question 4

Suppose \(\mu \) is a probability measure on the Borel subsets of \((0,1)\) such that \[ \int |\ln (x)|\,\mu (dx)<\infty . \] Prove that \[ I(n):=\left (\int x^{1/n}\,\mu (dx)\right )^n \to \exp \left (\int \ln (x)\,\mu (dx)\right ) \qquad \text {as } n\to \infty . \]

[Hint: \(I(n)=\mathbb {E}\left (\prod _{j=1}^{n}X_j^{1/n}\right )\), where \(X_1,\ldots ,X_n\) are i.i.d.]

Raw LaTeX

Suppose \(\mu\) is a probability measure on the Borel subsets of
\((0,1)\) such that
\[
  \int |\ln(x)|\,\mu(dx)<\infty.
\]
Prove that
\[
  I(n):=\left(\int x^{1/n}\,\mu(dx)\right)^n
  \to
  \exp\left(\int \ln(x)\,\mu(dx)\right)
  \qquad\text{as } n\to\infty.
\]

[Hint: \(I(n)=\E\left(\prod_{j=1}^{n}X_j^{1/n}\right)\), where
\(X_1,\ldots,X_n\) are i.i.d.]

Topic Tags

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