A Probabilistic Characterization of Negative Definite Functions

Area of Science:

• Mathematical Analysis
• Probability Theory
• Functional Analysis

Background:

• Negative definite functions are crucial in various fields, including probability and statistics.
• Characterizing these functions is essential for theoretical advancements.
• Previous work by Lifshits et al. established part of the condition.

Purpose of the Study:

• To provide a complete and simplified characterization of continuous negative definite functions on ℝ^d.
• To establish the equivalence between polynomial boundedness, a specific probabilistic inequality, and the property of being negative definite.
• To offer a new proof for the 'if' part of the theorem.

Main Methods:

• Utilizing Fourier transforms of tempered distributions.
• Applying concepts from stochastic processes and random vector analysis.
• Leveraging advanced mathematical analysis techniques.

Main Results:

• A continuous function f on ℝ^d is negative definite if and only if it is polynomially bounded.
• The function must also satisfy the inequality E[(f(X) - f(Y))²] ≤ C E[|X - Y|²] for all i.i.d. random vectors X and Y.
• The proof establishes a novel connection between analytical properties and probabilistic inequalities.

Conclusions:

• The study provides a comprehensive and elegant characterization of negative definite functions.
• The findings simplify the identification and application of negative definite functions in theoretical research.
• The proof methodology offers new insights into the interplay between function theory and probability.