Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6040 • Probability

Spring 2022 • Question 5

Question

Math 6040 • Probability Spring 2022 • Question 5

Fix an integer \(n \ge 1\). Let \(\{Y_{k}\}_{k \in \mathbb {N}}\) be i.i.d. uniform on \(\{1, 2, \ldots , n\}\). Let

\[ \sigma _{n} = \inf \{k \ge 2 : Y_{k} = Y_{m} \text { for some } 1 \le m < k\} \]

be the first time we get a repeated sample. Find an exponent \(\gamma \) and a nondegenerate distribution \(\mu \) such that, as \(n \to \infty \), \(n^{-\gamma }\sigma _{n}\) converges weakly to the distribution \(\mu \). (Hint: look at tail probabilities \(P(n^{-\gamma }\sigma _{n} \ge x)\).)

Raw LaTeX

Fix an integer \(n \ge 1\). Let \(\{Y_{k}\}_{k \in \mathbb{N}}\) be i.i.d. uniform on \(\{1, 2, \ldots, n\}\). Let

\[
\sigma_{n} = \inf\{k \ge 2 : Y_{k} = Y_{m} \text{ for some } 1 \le m < k\}
\]


be the first time we get a repeated sample. Find an exponent \(\gamma\) and a nondegenerate
distribution \(\mu\) such that, as \(n \to \infty\), \(n^{-\gamma}\sigma_{n}\) converges weakly to the distribution \(\mu\).
(Hint: look at tail probabilities \(P(n^{-\gamma}\sigma_{n} \ge x)\).)

Topic Tags

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