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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2006 • Question 2

Question

Math 6010 • Linear Models 2006 • Question 2

Let \(\{X_i\}_{i=1}^{\infty }\) be a sequence of i.i.d. random variables, and define \[ S(r):=\sum _{i=1}^{\infty }(1-r)^iX_i \qquad \text {for all } 0<r<1. \]

(a) Prove that for every fixed \(r\in (0,1)\), \(S(r)\) is a.s. absolutely convergent if and only if \(\mathbb {E}\{\log (1+|X_1|)\}<\infty \).

(b) Suppose that \(\mathbb {E}[X_1]=\mu \) and \(\operatorname {Var}(X_1)=\sigma ^2<\infty \). Then prove that there exists a constant \(c\) such that \(rS(r)\to c\) in probability, as \(r\to 0\). Compute \(c\).

Raw LaTeX

Let \(\{X_i\}_{i=1}^{\infty}\) be a sequence of i.i.d. random variables,
and define
\[
  S(r):=\sum_{i=1}^{\infty}(1-r)^iX_i
  \qquad \text{for all } 0<r<1.
\]

\textbf{(a)} Prove that for every fixed \(r\in(0,1)\), \(S(r)\) is a.s. absolutely
convergent if and only if \(\E\{\log(1+|X_1|)\}<\infty\).

\textbf{(b)} Suppose that \(\E[X_1]=\mu\) and \(\Var(X_1)=\sigma^2<\infty\). Then prove
that there exists a constant \(c\) such that \(rS(r)\to c\) in probability, as \(r\to 0\).
Compute \(c\).

Topic Tags

Convergence in Probability and Almost Sure Convergence Measure-Theoretic Probability Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods