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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1993 • Question 1

Question

Math 6010 • Linear Models 1993 • Question 1

Let \(X_0=1\) and define \(X_n\) inductively by declaring that \(X_{n+1}\) is uniformly distributed over \((0,X_n)\). Prove that \(n^{-1}\log X_n\to c\) a.s. and compute \(c\).

Raw LaTeX

Let \(X_0=1\) and define \(X_n\) inductively by declaring that \(X_{n+1}\) is uniformly distributed over \((0,X_n)\). Prove that \(n^{-1}\log X_n\to c\) a.s. and compute \(c\).

Topic Tags

Convergence in Probability and Almost Sure Convergence Uniform Integrability and Truncation Methods