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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1996 • Question 5

Question

Math 6010 • Linear Models 1996 • Question 5

Let \(\{B(t),\, t\ge 0\}\) be a standard Brownian motion. Prove directly, using \[ V_n=\sum _{i=1}^{2^n}\left |B(i/2^n)-B((i-1)/2^n)\right |, \] that \(B(\cdot ,\omega )\) is of unbounded variation on \([0,1]\) for a.e. \(\omega \).

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Raw LaTeX

Let \(\{B(t),\, t\ge 0\}\) be a standard Brownian motion. Prove
directly, using
\[
V_n=\sum_{i=1}^{2^n}\left|B(i/2^n)-B((i-1)/2^n)\right|,
\]
that \(B(\cdot,\omega)\) is of unbounded variation on \([0,1]\) for a.e. \(\omega\).

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Topic Tags

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