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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1999 • Question 2

Question

Math 6010 • Linear Models 1999 • Question 2

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with a continuous distribution function \(F\). The ordered observations are \(X_{1,n}\leq X_{2,n}\leq \cdots \leq X_{n,n}\). Let \[ T_n=\sum _{i=1}^{n-1}\left (F(X_{i+1,n})-F(X_{i,n})\right )^2. \] Show that there are numerical sequences \(a_n\) and \(b_n\) such that \((T_n-a_n)/b_n\) converges in distribution to a standard normal random variable. (20 points)

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed random variables with a continuous distribution function \(F\). The ordered observations are \(X_{1,n}\leq X_{2,n}\leq\cdots\leq X_{n,n}\). Let
\[
T_n=\sum_{i=1}^{n-1}\left(F(X_{i+1,n})-F(X_{i,n})\right)^2.
\]
Show that there are numerical sequences \(a_n\) and \(b_n\) such that \((T_n-a_n)/b_n\) converges in distribution to a standard normal random variable. (20 points)

Topic Tags

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