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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2014 • Question 1

Question

Math 6010 • Linear Models Fall 2014 • Question 1

Let \(X_1,X_2,\ldots \) be independent random variables with 2 finite moments, and \[ \sum _{j=1}^{\infty }\operatorname {Var}(X_j)<\infty . \] Prove that \[ \lim _{n\to \infty }\sum _{j=1}^{n}(X_j-\mathbb {E} X_j) \] exists a.s. and in \(L^2(\mathbb {P})\).

Raw LaTeX

Let \(X_1,X_2,\ldots\) be independent random variables with 2 finite
moments, and
\[
  \sum_{j=1}^{\infty}\Var(X_j)<\infty.
\]
Prove that
\[
  \lim_{n\to\infty}\sum_{j=1}^{n}(X_j-\E X_j)
\]
exists a.s. and in \(L^2(\Prob)\).

Topic Tags

Convergence in Probability and Almost Sure Convergence Covariance and Independence Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods