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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 \(\{S_n,\mathcal {F}_n,n\geq 1\}\) be a nonnegative martingale with \(ES_n=1\). Show that for all \(\lambda >0\), \[ P\{S_n\geq \lambda ,\text { for some }n\geq 1\}\leq \frac {1}{\lambda }. \]

Raw LaTeX

Let \(\{S_n,\mathcal{F}_n,n\geq 1\}\) be a nonnegative martingale with \(ES_n=1\). Show that for all \(\lambda>0\),
\[
P\{S_n\geq\lambda,\text{ for some }n\geq 1\}\leq \frac{1}{\lambda}.
\]

Topic Tags

Exponential Martingales Filtrations and Stopping Times Martingales and Supermartingales Uniform Integrability and Truncation Methods