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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2015 • Question 7

Question

Math 6010 • Linear Models Spring 2015 • Question 7

Let \(X_1,X_2,\ldots \) be independent identically distributed positive random variables with \(\mathbb {E} X_1=\mu \). Define \[ N(t)=\min \left \{k:\sum _{i=1}^{k} X_i>t\right \}. \] Show that \[ \lim _{t\to \infty }\frac {1}{t}N(t)=\frac {1}{\mu } \qquad \text {a.s.} \]

Raw LaTeX

Let \(X_1,X_2,\ldots\) be independent identically distributed
positive random variables with \(\E X_1=\mu\). Define
\[
  N(t)=\min\left\{k:\sum_{i=1}^{k} X_i>t\right\}.
\]
Show that
\[
  \lim_{t\to\infty}\frac{1}{t}N(t)=\frac{1}{\mu}
  \qquad\text{a.s.}
\]

Topic Tags

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