Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2019 • Question 3

Question

Math 6010 • Linear Models Fall 2019 • Question 3

Let \(X_k\) have characteristic function \(\phi _k\) with \(X_1,X_2,\ldots \), independent random variables. Show that \(\sum _{k=1}^{n}X_k\) converges almost surely \(\Longleftrightarrow \) there exists a neighborhood \(U\) of 0 and a function \(h\) such that \[ \prod _{k=1}^{n}\phi _k(u) \xrightarrow [n\to \infty ]{} h(u)\ne 0, \] for all \(u\in U\). Hint: For the \(\Leftarrow \) direction consider the characteristic function of the partial sums \(\sum _{k=m}^{n}X_k\).

Let \(X_k\) have characteristic function \(\phi_k\) with
\(X_1,X_2,\ldots\), independent random variables. Show that
\(\sum_{k=1}^{n}X_k\) converges almost surely \(\Longleftrightarrow\) there exists
a neighborhood \(U\) of 0 and a function \(h\) such that
\[
  \prod_{k=1}^{n}\phi_k(u)
  \xrightarrow[n\to\infty]{}
  h(u)\ne0,
\]
for all \(u\in U\). \textbf{Hint:} For the \(\Leftarrow\) direction consider the
characteristic function of the partial sums \(\sum_{k=m}^{n}X_k\).

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Topic Tags

Characteristic Functions and Fourier Inversion Convergence in Probability and Almost Sure Convergence Covariance and Independence Uniform Integrability and Truncation Methods