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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1993 • Question 2

Question

Math 6010 • Linear Models 1993 • Question 2

Let \(X_1,X_2,\ldots \) be independent Poisson random variables with \(\mathbb {E} X_n=\lambda _n\), and let \(S_n=X_1+\cdots +X_n\). Show that if \(\sum \lambda _n=\infty \), then \(S_n/\mathbb {E} S_n\to 1\) a.s.

Hint: Show that (*) \(Y_n/c_n\to 1\) a.s., provided that \(Y_n\ge 0\) is nondecreasing in \(n\), and (*) holds for a subsequence \(n(k)\) that has \(c_{n(k+1)}/c_{n(k)}\to 1\).

Raw LaTeX

Let \(X_1,X_2,\ldots\) be independent Poisson random variables with
\(\E X_n=\lambda_n\), and let \(S_n=X_1+\cdots+X_n\). Show that if
\(\sum\lambda_n=\infty\), then \(S_n/\E S_n\to 1\) a.s.

Hint: Show that (*) \(Y_n/c_n\to 1\) a.s., provided that \(Y_n\ge 0\) is
nondecreasing in \(n\), and (*) holds for a subsequence \(n(k)\) that has
\(c_{n(k+1)}/c_{n(k)}\to 1\).

Topic Tags

Convergence in Probability and Almost Sure Convergence Covariance and Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods