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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2008 • Question 2

Question

Math 6010 • Linear Models Fall 2008 • Question 2

Let \(X_2,X_3,\ldots \) be independent random variables such that \[ \mathbb {P}(X_n=n)=\mathbb {P}(X_n=-n)=\frac {1}{2n\log n}, \qquad \mathbb {P}(X_n=0)=1-\frac {1}{n\log n}. \] Show that this sequence obeys the weak law but not the strong law of large numbers, in the sense that \(n^{-1}\sum _{i=1}^n X_i\) converges to 0 in probability but not almost surely.

Raw LaTeX

Let \(X_2,X_3,\ldots\) be independent random variables such that
\[
  \Prob(X_n=n)=\Prob(X_n=-n)=\frac{1}{2n\log n},
  \qquad
  \Prob(X_n=0)=1-\frac{1}{n\log n}.
\]
Show that this sequence obeys the weak law but not the strong law of large
numbers, in the sense that
\(n^{-1}\sum_{i=1}^n X_i\) converges to 0 in probability but not almost surely.

Topic Tags

Convergence in Probability and Almost Sure Convergence Covariance and Independence Laws of Large Numbers Measure-Theoretic Probability