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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2015 • Question 5

Question

Math 6010 • Linear Models Spring 2015 • Question 5

Let \(X_1,X_2,\ldots \) be independent and identically distributed random variables with distribution function \(F(x)=\mathbb {P}\{X_1\leq x\}\). Show that \[ \sup _{-\infty <x<\infty } |F_n(x)-F(x)| \to 0 \qquad \text {a.s.} \] as \(n\to \infty \), where \[ F_n(x)=\frac {1}{n}\sum _{i=1}^{n} I\{X_i\leq x\}. \]

Raw LaTeX

Let \(X_1,X_2,\ldots\) be independent and identically distributed
random variables with distribution function \(F(x)=\Prob\{X_1\leq x\}\). Show
that
\[
  \sup_{-\infty<x<\infty} |F_n(x)-F(x)| \to 0
  \qquad\text{a.s.}
\]
as \(n\to\infty\), where
\[
  F_n(x)=\frac{1}{n}\sum_{i=1}^{n} I\{X_i\leq x\}.
\]

Topic Tags

Convergence in Probability and Almost Sure Convergence Covariance and Independence Laws of Large Numbers Uniform Integrability and Truncation Methods