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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2019 • Question 5

Question

Math 6010 • Linear Models Fall 2019 • Question 5

Let \((Z_n)\) be i.i.d. random variables with \(\mathbb {P}(Z_1=1)=\mathbb {P}(Z_1=2)=1/2\). Show that \(\mathbb {E}[Z_1\cdots Z_n]=(1.5)^n\) and yet \(Z_1\cdots Z_n/(1.45)^n\to 0\) almost surely. Identify the set of numbers \(a>0\) for which \(Z_1\cdots Z_n/a^n\to 0\) almost surely.

Raw LaTeX

Let \((Z_n)\) be i.i.d. random variables with
\(\Prob(Z_1=1)=\Prob(Z_1=2)=1/2\). Show that
\(\E[Z_1\cdots Z_n]=(1.5)^n\) and yet
\(Z_1\cdots Z_n/(1.45)^n\to0\) almost surely. Identify the set of numbers \(a>0\)
for which \(Z_1\cdots Z_n/a^n\to0\) almost surely.

Topic Tags

Convergence in Probability and Almost Sure Convergence Laws of Large Numbers Measure-Theoretic Probability Uniform Integrability and Truncation Methods