Question

Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables having common distribution function \( F(x) \). Let \( a<b \) be two real numbers such that \( F(x) = 0 \) for all \( x \leq a \), \( 0<F(x)<1 \) for all \( a<x<b \), and \( F(x) = 1 \) for all \( x \geq b \). Let \( S_n(x) \) be the empirical distribution function at \( x \) based on \( X_1, X_2, \dots, X_n \), \( n \geq 1 \). Then which one of the following statements is NOT true?

• A. \( P \left( \lim_{n \to \infty} \sup_{-\infty<x<\infty} |S_n(x) - F(x)| = 0 \right) = 1 \)
• B. For fixed \( x \in (a, b) \) and \( t \in (-\infty, \infty) \), \( P \left( \lim_{n \to \infty} \sqrt{n} |S_n(x) - F(x)| \leq t \right) = P(Z \leq t) \), where \( Z \) is the standard normal random variable.
• C. The covariance between \( S_n(x) \) and \( S_n(y) \) equals \( \frac{1}{n} F(x)(1 - F(y)) \) for all \( n \geq 2 \) and for fixed \( -\infty<x, y<\infty \).
• D. If \( Y_n = \sup_{-\infty<x<\infty} (S_n(x) - F(x))^2 \), then \( \{n Y_n\}_{n \geq 1} \) converges in distribution to a central chi-square random variable with 2 degrees of freedom.

Hint:

In empirical distribution functions, the Glivenko-Cantelli theorem guarantees uniform convergence, while the central limit theorem describes the asymptotic distribution of the differences between \( S_n(x) \) and \( F(x) \).

Answer 1

The Correct Option is C

We are given a sequence of independent and identically distributed random variables with common distribution function \( F(x) \). We need to analyze the given options. 
Step 1: Analyzing option A. 
This statement is correct. By the Glivenko-Cantelli theorem, the empirical distribution function \( S_n(x) \) converges uniformly to the true distribution function \( F(x) \) almost surely as \( n \to \infty \). 
Step 2: Analyzing option B. 
This statement is correct. It represents the central limit theorem for the empirical distribution function, where \( \sqrt{n} |S_n(x) - F(x)| \) converges in distribution to a normal distribution. 
Step 3: Analyzing option C. 
This statement is correct. The covariance between \( S_n(x) \) and \( S_n(y) \) is given by the expression \( \frac{1}{n} F(x)(1 - F(y)) \), which is a standard result for empirical distribution functions. 
Step 4: Analyzing option D. 
This statement is incorrect. The sequence \( \{n Y_n\}_{n \geq 1} \) does not converge to a chi-square distribution with 2 degrees of freedom. Instead, the limiting distribution is a different form, so this option is not true. 
Step 5: Conclusion. 
The correct answer is (D) as it is the statement that is not true.