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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6040 • Probability

Spring 2022 • Question 6

Question

Math 6040 • Probability Spring 2022 • Question 6

Give an example of a sequence of independent random variables \(\{Z_{k}\}_{k \in \mathbb {N}}\) satisfying \(\mathbb {E}[Z_{k}] = 0\), \(\operatorname {Var}(Z_{k}) = 1\), for each \(k\), and such that \((Z_{1} + \cdots + Z_{n})/\sqrt {n}\) does not converge in distribution to a standard normal. Justify any claims you make.

Raw LaTeX

Give an example of a sequence of independent random variables \(\{Z_{k}\}_{k \in \mathbb{N}}\) satisfying
\(\mathbb{E}[Z_{k}] = 0\), \(\operatorname{Var}(Z_{k}) = 1\), for each \(k\), and such that \((Z_{1} + \cdots + Z_{n})/\sqrt{n}\) does not
converge in distribution to a standard normal. Justify any claims you make.

Topic Tags

Central Limit Theorems Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits