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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2009 • Question 3

Question

Math 6010 • Linear Models Fall 2009 • Question 3

Let \(X_1,X_2,\ldots \) be independent with \[ \mathbb {P}(X_n=n^2-1)=n^{-2}=1-\mathbb {P}(X_n=-1),\qquad n\geq 1. \] Notice that \(\mathbb {E}[X_n]=0\) for all \(n\geq 1\), which might lead one to expect that \(S_n:=X_1+\cdots +X_n\) satisfies \(S_n/n\to 0\) a.s. Show that in fact \(S_n/n\to -1\) a.s.

Raw LaTeX

Let \(X_1,X_2,\ldots\) be independent with
\[
  \Prob(X_n=n^2-1)=n^{-2}=1-\Prob(X_n=-1),\qquad n\geq1.
\]
Notice that \(\E[X_n]=0\) for all \(n\geq1\), which might lead one to expect that
\(S_n:=X_1+\cdots+X_n\) satisfies \(S_n/n\to0\) a.s. Show that in fact
\(S_n/n\to -1\) a.s.

Topic Tags

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