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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2005 • Question 1

Question

Math 6010 • Linear Models 2005 • Question 1

Prove that for all submartingales \(\{X_n\}_{n=1}^{\infty }\), \[ \mathbb {P}\left \{\max _{n\le m}|X_n|\ge \lambda \right \} \le \frac {3}{\lambda }\max _{n\le m}\mathbb {E}\{|X_n|\}, \qquad \lambda >0,\; m=1,2,\ldots . \]

Raw LaTeX

Prove that for all submartingales \(\{X_n\}_{n=1}^{\infty}\),
\[
  \Prob\left\{\max_{n\le m}|X_n|\ge \lambda\right\}
  \le \frac{3}{\lambda}\max_{n\le m}\E\{|X_n|\},
  \qquad \lambda>0,\; m=1,2,\ldots .
\]

Topic Tags

Martingales and Supermartingales Uniform Integrability and Truncation Methods