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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2019 • Question 6

Question

Math 6010 • Linear Models Spring 2019 • Question 6

Fix \(\alpha \in (0,1)\). Let \(X_n\) be a sequence of independent random variables with \(\mathbb {P}(X_n=n)=\mathbb {P}(X_n=-n)=n^{-\alpha }/2\) and \(\mathbb {P}(X_n=0)=1-n^{-\alpha }\). Prove that \((X_1+\cdots +X_n)/n^{(3-\alpha )/2}\) converges weakly to a centered normal.

Raw LaTeX

Fix \(\alpha\in(0,1)\). Let \(X_n\) be a sequence of independent random
variables with \(\Prob(X_n=n)=\Prob(X_n=-n)=n^{-\alpha}/2\) and
\(\Prob(X_n=0)=1-n^{-\alpha}\). Prove that
\((X_1+\cdots+X_n)/n^{(3-\alpha)/2}\) converges weakly to a centered normal.

Topic Tags

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