polya theorem convergence in distribution implies uniform convergence

3 min read 24-01-2025

polya theorem convergence in distribution implies uniform convergence

Polya's theorem establishes a crucial link between convergence in distribution and uniform convergence for a sequence of distribution functions. This theorem provides sufficient conditions under which convergence in distribution implies a stronger form of convergence, namely, uniform convergence. Understanding this relationship is vital in probability theory and statistics, especially when dealing with approximations and limit theorems.

Understanding the Concepts

Before diving into Polya's theorem itself, let's clarify the key concepts involved:

1. Convergence in Distribution

A sequence of random variables {X_n} converges in distribution to a random variable X (denoted as X_n →^d X) if their cumulative distribution functions (CDFs) F_n(x) converge pointwise to the CDF F(x) of X at all points x where F(x) is continuous. This means:

lim_n→∞ F_n(x) = F(x) for all x where F is continuous.

This is a relatively weak form of convergence; it only requires convergence of the probabilities at individual points.

2. Uniform Convergence

A sequence of functions {F_n(x)} converges uniformly to a function F(x) if for any ε > 0, there exists an N such that for all n > N and for all x, |F_n(x) - F(x)| < ε. This is a stronger form of convergence, guaranteeing that the convergence is uniform across the entire domain.

Polya's Theorem Statement

Polya's theorem states:

If {F_n(x)} is a sequence of distribution functions converging pointwise to a continuous distribution function F(x), then the convergence is uniform. That is, if X_n →^d X and F(x) is continuous, then F_n(x) converges uniformly to F(x).

Proof of Polya's Theorem

The proof relies on the monotonicity and boundedness of distribution functions. We can outline the key steps:

Continuity of F(x): The assumption that F(x) is continuous is crucial. This ensures that there are no jumps in the limit distribution, which would prevent uniform convergence.
Approximation: For any ε > 0, we can find a finite set of points {x₁, x₂, ..., x_k} such that the intervals defined by these points cover the entire real line, and the variation of F(x) within each interval is less than ε/2.
Pointwise Convergence: Because of the pointwise convergence, for each x_i, there exists an N_i such that for all n > N_i, |F_n(x_i) - F(x_i)| < ε/2.
Monotonicity: Using the monotonicity property of distribution functions and the fact that the intervals cover the real line, we can show that for all x and n > max{N_i}, |F_n(x) - F(x)| < ε.
Uniform Convergence: This demonstrates that the convergence is uniform, completing the proof.

Implications and Applications

Polya's theorem has significant implications in several areas:

Approximation of Distributions: It justifies the use of approximations for distributions in various statistical applications. If we have a sequence of distributions converging in distribution to a continuous distribution, we can rely on the uniform convergence for error bounds.
Limit Theorems: Many limit theorems in probability theory rely on convergence in distribution. Polya's theorem strengthens these results by implying uniform convergence under the condition of continuity.
Statistical Inference: The uniform convergence guaranteed by Polya's theorem can be valuable in establishing consistency and asymptotic properties of estimators and test statistics.

Conclusion

Polya's theorem provides a powerful connection between convergence in distribution and uniform convergence. The condition of continuity of the limit distribution function is essential. This theorem offers a valuable tool for analyzing and understanding the behavior of sequences of random variables and their associated distribution functions. Its applications span various aspects of probability and statistics, solidifying its importance in the field.