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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2016 • Question 3

Question

Math 6010 • Linear Models Spring 2016 • Question 3

(a) Find the distribution of \(\int _0^1 B_t\,dt\), where \((B_t)\) is Brownian motion.

(b) Given an i.i.d. sequence \(X_1,X_2,\ldots \) with mean 0 and variance 1, what does Donsker’s invariance principle tell us about this sequence in connection with the distribution of \(\int _0^1 B_t\,dt\)?

Raw LaTeX

\textbf{(a)} Find the distribution of \(\int_0^1 B_t\,dt\), where
\((B_t)\) is Brownian motion.

\textbf{(b)} Given an i.i.d. sequence \(X_1,X_2,\ldots\) with mean 0 and
variance 1, what does Donsker's invariance principle tell us about this
sequence in connection with the distribution of \(\int_0^1 B_t\,dt\)?

Topic Tags

Analysis of Variance Brownian Motion and Stochastic Processes Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits