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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2015 • Question 9

Question

Math 6010 • Linear Models Spring 2015 • Question 9

Let \(X_1,X_2,\ldots \) be independent and identically distributed random variables uniform on \([0,1]\). Show that for all \(\alpha >-1\) there are numerical sequences \(a_n\) and \(b_n\) such that \[ Y_n= \frac {\sum _{1\leq k\leq n} k^{\alpha }X_k-a_n}{b_n} \] converges in distribution to a standard normal random variable.

Raw LaTeX

Let \(X_1,X_2,\ldots\) be independent and identically distributed
random variables uniform on \([0,1]\). Show that for all \(\alpha>-1\) there are
numerical sequences \(a_n\) and \(b_n\) such that
\[
  Y_n=
  \frac{\sum_{1\leq k\leq n} k^{\alpha}X_k-a_n}{b_n}
\]
converges in distribution to a standard normal random variable.

Topic Tags

Central Limit Theorems Covariance and Independence Multivariate Normal Distributions Weak Convergence and Distributional Limits