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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2017 • Question 1

Question

Math 6010 • Linear Models Spring 2017 • Question 1

Consider an array \(\{X_{i,j}\}\) of random variables such that, for every \(n\geq 1\), the collection \(X_{1,n},\ldots ,X_{n,n}\) is i.i.d. with common distribution \[ \mathbb {P}\{X_{1,n}=1\} = \mathbb {P}\{X_{1,n}=-1\} = \frac {1}{2}\left (1-\frac {1}{n^2}\right ); \] and \(\mathbb {P}\{X_{1,n}=n^2\}=n^{-2}\). Let \(S_n:=X_{1,n}+\cdots +X_{n,n}\). Prove that \(S_n/\sqrt {n}\Rightarrow \mathrm {N}(0,\sigma ^2)\) for some \(\sigma \in (0,\infty )\), although \(\mathbb {E}(S_n/\sqrt {n})\to \infty \) as \(n\to \infty \).

Raw LaTeX

Consider an array \(\{X_{i,j}\}\) of random variables such that, for
every \(n\geq1\), the collection \(X_{1,n},\ldots,X_{n,n}\) is i.i.d. with common
distribution
\[
  \Prob\{X_{1,n}=1\}
  =
  \Prob\{X_{1,n}=-1\}
  =
  \frac{1}{2}\left(1-\frac{1}{n^2}\right);
\]
and \(\Prob\{X_{1,n}=n^2\}=n^{-2}\). Let
\(S_n:=X_{1,n}+\cdots+X_{n,n}\). Prove that
\(S_n/\sqrt{n}\Rightarrow \mathrm{N}(0,\sigma^2)\) for some
\(\sigma\in(0,\infty)\), although \(\E(S_n/\sqrt{n})\to\infty\) as
\(n\to\infty\).

Topic Tags

Central Limit Theorems Estimator Distributions Multivariate Normal Distributions Uniform Integrability and Truncation Methods