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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2013 • Question 8

Question

Math 6010 • Linear Models Fall 2013 • Question 8

Let \(\{W(t)\}_{t\geq 0}\) be a standard Brownian motion, and define \[ U(t):=e^{-t/2}W(e^t)\qquad \text {for all }t\geq 0. \]

(a) Prove that \(\{U(t)\}_{t\geq 0}\) is Gaussian process; compute its mean and covariance functions.

(b) Prove that \[ \limsup _{t\to \infty }\frac {U(t)}{\sqrt {2\ln t}} = -\liminf _{t\to \infty }\frac {U(t)}{\sqrt {2\ln t}} =1 \qquad \text {a.s.} \]

Raw LaTeX

Let \(\{W(t)\}_{t\geq0}\) be a standard Brownian motion, and define
\[
  U(t):=e^{-t/2}W(e^t)\qquad\text{for all }t\geq0.
\]

\textbf{(a)} Prove that \(\{U(t)\}_{t\geq0}\) is Gaussian process; compute its
mean and covariance functions.

\textbf{(b)} Prove that
\[
  \limsup_{t\to\infty}\frac{U(t)}{\sqrt{2\ln t}}
  =
  -\liminf_{t\to\infty}\frac{U(t)}{\sqrt{2\ln t}}
  =1 \qquad\text{a.s.}
\]

Topic Tags

Brownian Motion and Stochastic Processes Convergence in Probability and Almost Sure Convergence Covariance and Independence Weak Convergence and Distributional Limits