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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 \(X_1,X_2,\cdots \) be independent, identically distributed random variables. Show that the following statements are equivalent:

(a) \(\mathbb {E}|X_1|^\nu <\infty \);

(b) \(X_n/n^{1/\nu }\to 0\), almost surely;

(c) \(\max _{1\le i\le n}|X_i|/n^{1/\nu }\to 0\), almost surely.

Raw LaTeX

Let \(X_1,X_2,\cdots\) be independent, identically distributed random
variables. Show that the following statements are equivalent:

(a) \(\E|X_1|^\nu<\infty\);

(b) \(X_n/n^{1/\nu}\to 0\), almost surely;

(c) \(\max_{1\le i\le n}|X_i|/n^{1/\nu}\to 0\), almost surely.

Topic Tags

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