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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2018 • Question 8

Question

Math 6010 • Linear Models Spring 2018 • Question 8

Let \((X_n)\) be i.i.d. uniform random variables on \([0,1]\). For each \(n\geq 1\), define \[ Y_n=n\min \{X_k:1\leq k\leq n\}. \] Prove that as \(n\to \infty \), \(Y_n\) converges weakly to an exponential random variable with mean one.

Raw LaTeX

Let \((X_n)\) be i.i.d. uniform random variables on \([0,1]\). For each
\(n\geq1\), define
\[
  Y_n=n\min\{X_k:1\leq k\leq n\}.
\]
Prove that as \(n\to\infty\), \(Y_n\) converges weakly to an exponential random
variable with mean one.

Topic Tags

Exponential Martingales Measure-Theoretic Probability Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits