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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6040 • Probability

Spring 2023 • Question 5

Question

Math 6040 • Probability Spring 2023 • Question 5

Prove that if \(\{X_{i}\}_{i=1}^{\infty }\) are i.i.d. Uniform-\([0,1]\) random variables, then \[ \frac {4\sum _{i=1}^{n} iX_{i} - n^{2}}{n^{3/2}} \] converges weakly, and identify the limiting distribution. (Recall \(\sum _{i=1}^{n} i = \frac {n(n+1)}{2}\) and \(\sum _{i=1}^{n} i^{2} = \frac {n(n+1)(2n+1)}{6}\).)

Raw LaTeX

Prove that if \(\{X_{i}\}_{i=1}^{\infty}\) are i.i.d. Uniform-\([0,1]\) random variables, then
\[
\frac{4\sum_{i=1}^{n} iX_{i} - n^{2}}{n^{3/2}}
\]
converges weakly, and identify the limiting distribution. (Recall \(\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\)
and
\(\sum_{i=1}^{n} i^{2} = \frac{n(n+1)(2n+1)}{6}\).)

Topic Tags

Laws of Large Numbers Measure-Theoretic Probability Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits