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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2017 • Question 3

Question

Math 6010 • Linear Models Spring 2017 • Question 3

Let \(X_1,X_2,\ldots \) be a collection of i.i.d. random variables, and define \(S_n:=X_1+\cdots +X_n\) for all \(n\geq 1\). Suppose \[ \mathbb {P}\{X_1=k\}=\frac {3}{\pi ^2 k^2} \qquad \text {for all } k=\pm 1,\pm 2,\ldots . \] Prove that \(S_n/n\) converges weakly to a non-constant random variable as \(n\to \infty \).

Raw LaTeX

Let \(X_1,X_2,\ldots\) be a collection of i.i.d. random variables,
and define \(S_n:=X_1+\cdots+X_n\) for all \(n\geq1\). Suppose
\[
  \Prob\{X_1=k\}=\frac{3}{\pi^2 k^2}
  \qquad\text{for all } k=\pm1,\pm2,\ldots.
\]
Prove that \(S_n/n\) converges weakly to a non-constant random variable as
\(n\to\infty\).

Topic Tags

Laws of Large Numbers Measure-Theoretic Probability Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits