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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2005 • Question 2

Question

Math 6010 • Linear Models 2005 • Question 2

Suppose \(X_1,X_2,\ldots \) are independent with distribution, \[ \mathbb {P}\{X_n=n^\alpha \}=\mathbb {P}\{X_n=-n^{-\alpha }\}=\frac {1}{2}, \qquad n=1,2,\ldots . \] Prove that \(\sum _{i=1}^{n}X_n/n\to 0\) a.s. if \(0<\alpha <\frac {1}{2}\).

Raw LaTeX

Suppose \(X_1,X_2,\ldots\) are independent with distribution,
\[
  \Prob\{X_n=n^\alpha\}=\Prob\{X_n=-n^{-\alpha}\}=\frac{1}{2},
  \qquad n=1,2,\ldots .
\]
Prove that \(\sum_{i=1}^{n}X_n/n\to 0\) a.s. if \(0<\alpha<\frac{1}{2}\).

Topic Tags

Convergence in Probability and Almost Sure Convergence Covariance and Independence Estimator Distributions Uniform Integrability and Truncation Methods