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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1994 • Question 3

Question

Math 6010 • Linear Models 1994 • Question 3

Let \(\{S_n,\mathcal F_n,\,n\ge 1\}\) be a nonnegative martingale with \(\mathbb {E} S_n=1\). Show that for all \(\lambda >0\), \[ \mathbb {P}\{S_n\ge \lambda ,\text { for some } n\ge 1\}\le \frac {1}{\lambda }. \]

Raw LaTeX

Let \(\{S_n,\mathcal F_n,\,n\ge 1\}\) be a nonnegative martingale with
\(\E S_n=1\). Show that for all \(\lambda>0\),
\[
\Prob\{S_n\ge \lambda,\text{ for some } n\ge 1\}\le \frac{1}{\lambda}.
\]

Topic Tags

Exponential Martingales Infinite Products and Multiplicative Martingales Martingales and Supermartingales Uniform Integrability and Truncation Methods