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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2000 • Question 5

Question

Math 6010 • Linear Models 2000 • Question 5

Let \(X_1,X_2,X_3,\ldots \) be i.i.d. with finite mean and variance, and put \(S_n=X_1+X_2+\cdots +X_n\). Show that \(S_n-n\mathbb {E}[X_1]\) and \((S_n-n\mathbb {E}[X_1])^2-n\operatorname {Var}(X_1)\) are martingales.

Raw LaTeX

Let \(X_1,X_2,X_3,\ldots\) be i.i.d. with finite mean and variance, and
put \(S_n=X_1+X_2+\cdots+X_n\). Show that \(S_n-n\E[X_1]\) and
\((S_n-n\E[X_1])^2-n\Var(X_1)\) are martingales.

Topic Tags

Analysis of Variance Martingales and Supermartingales Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods