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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2000 • Question 3

Question

Math 6010 • Linear Models 2000 • Question 3

Let \(X_1,X_2,X_3,\ldots \) be i.i.d. nonnegative random variables with mean 1 and variance 1, and put \(S_n=X_1+X_2+\cdots +X_n\). Show that \(2(\sqrt {S_n}-\sqrt {n})\) converges in distribution to a standard normal as \(n\to \infty \).

Raw LaTeX

Let \(X_1,X_2,X_3,\ldots\) be i.i.d. nonnegative random variables with
mean 1 and variance 1, and put \(S_n=X_1+X_2+\cdots+X_n\). Show that
\(2(\sqrt{S_n}-\sqrt{n})\) converges in distribution to a standard normal as
\(n\to\infty\).

Topic Tags

Central Limit Theorems Multivariate Normal Distributions Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits