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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2000 • Question 4

Question

Math 6010 • Linear Models 2000 • Question 4

Let \(X_1,X_2,X_3,\ldots \) be a sequence of random variables, each of which has all moments finite, and suppose that \(\lim _{n\to \infty }\mathbb {E}[(X_n)^k]=k!\) for each \(k\ge 1\). Show that \(X_n\) converges in distribution.

Raw LaTeX

Let \(X_1,X_2,X_3,\ldots\) be a sequence of random variables, each of
which has all moments finite, and suppose that
\(\lim_{n\to\infty}\E[(X_n)^k]=k!\) for each \(k\ge 1\). Show that \(X_n\) converges in
distribution.

Topic Tags

Estimator Distributions Multivariate Normal Distributions Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits