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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1994 • Question 4

Question

Math 6010 • Linear Models 1994 • Question 4
Let \(X_1,X_2,\ldots\) be independent, identically distributed random variables. Show that the following statements are equivalent: **(a)** \(E|X_1|^\nu<\infty\); **(b)** \(X_n/n^{1/\nu}\to0\), almost surely; **(c)** \(\max_{1\leq i\leq n}|X_i|/n^{1/\nu}\to0\), almost surely.

Raw LaTeX

Let \(X_1,X_2,\ldots\) be independent, identically distributed random variables. Show that the following statements are equivalent:
\begin{parts}
\item \(E|X_1|^\nu<\infty\);
\item \(X_n/n^{1/\nu}\to0\), almost surely;
\item \(\max_{1\leq i\leq n}|X_i|/n^{1/\nu}\to0\), almost surely.
\end{parts}

Topic Tags

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