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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2004 • Question 3

Question

Math 6010 • Linear Models 2004 • Question 3

Suppose \(\{X_n\}_{n=1}^{\infty }\) are i.i.d. standard normal random variables. Prove that \[ M_n:=(n+1)^{-1/2}\exp \left (\frac {S_n^2}{2n+2}\right ) \] defines a martingale, where \(S_n=\sum _{i=1}^{n}X_i\).

Raw LaTeX

Suppose \(\{X_n\}_{n=1}^{\infty}\) are i.i.d. standard normal random
variables. Prove that
\[
  M_n:=(n+1)^{-1/2}\exp\left(\frac{S_n^2}{2n+2}\right)
\]
defines a martingale, where \(S_n=\sum_{i=1}^{n}X_i\).

Topic Tags

Central Limit Theorems Exponential Martingales Martingales and Supermartingales Uniform Integrability and Truncation Methods