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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2009 • Question 7

Question

Math 6010 • Linear Models Spring 2009 • Question 7

Choose and fix a sequence of real numbers \(a_1,a_2,\ldots \) such that: (i) \(0<a_k<1\) for all \(k\geq 1\); and (ii) \(\sum _{k=1}^{\infty }a_k^2=\infty \). Let \(X_1,X_2,\ldots \) be independent random variables, where each \(X_k\) is distributed uniformly on \((-a_k,a_k)\). Prove that \(n^{-1/2}(X_1+\cdots +X_n)\) converges weakly to \(N(0,\sigma ^2)\) and compute \(\sigma \).

Raw LaTeX

Choose and fix a sequence of real numbers \(a_1,a_2,\ldots\) such
that: \textbf{(i)} \(0<a_k<1\) for all \(k\geq1\); and \textbf{(ii)}
\(\sum_{k=1}^{\infty}a_k^2=\infty\). Let \(X_1,X_2,\ldots\) be independent random
variables, where each \(X_k\) is distributed uniformly on \((-a_k,a_k)\). Prove
that \(n^{-1/2}(X_1+\cdots+X_n)\) converges weakly to \(N(0,\sigma^2)\) and compute
\(\sigma\).

Topic Tags

Covariance and Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits