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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2018 • Question 5

Question

Math 6010 • Linear Models Fall 2018 • Question 5

Let \(\{X_i:i\geq 1\}\) be i.i.d. random variables distributed uniformly on \([0,1]\). Prove that the distribution of \[ \frac {4\sum _{i=1}^{n} iX_i-n^2}{n^{3/2}} \] converges weakly and identify the limiting distribution.

Raw LaTeX

Let \(\{X_i:i\geq1\}\) be i.i.d. random variables distributed
uniformly on \([0,1]\). Prove that the distribution of
\[
  \frac{4\sum_{i=1}^{n} iX_i-n^2}{n^{3/2}}
\]
converges weakly and identify the limiting distribution.

Topic Tags

Estimator Distributions Multivariate Normal Distributions Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits