Lévy's continuity theorem

Lévy's Continuity Theorem is a fundamental result in probability theory and stochastic processes, particularly in the study of characteristic functions. Named after the French mathematician Paul Lévy, this theorem provides a necessary and sufficient condition for a sequence of probability measures on the real line to converge weakly to a probability measure. It is a cornerstone in the theory of convergence of probability measures, with significant implications for the fields of statistics, quantum mechanics, and financial mathematics.

Statement of the Theorem

Lévy's Continuity Theorem states that a sequence of probability measures \(\{\mu_n\}\) on the real line converges weakly to a probability measure \(\mu\) if and only if the sequence of their characteristic functions \(\{\phi_n(t)\}\) converges pointwise to a function \(\phi(t)\) that is continuous at \(t = 0\), where \(\phi(t)\) is the characteristic function of \(\mu\).

Formally, if \(\mu_n\) are probability measures with characteristic functions \(\phi_n(t)\), and \(\mu\) is a probability measure with characteristic function \(\phi(t)\), then \[ \mu_n \xrightarrow{w} \mu \quad \text{if and only if} \quad \forall t \in \mathbb{R}, \quad \phi_n(t) \to \phi(t) \quad \text{and} \quad \phi(t) \text{ is continuous at } t=0. \]

Applications

Lévy's Continuity Theorem has wide-ranging applications across various domains:

- In statistics, it is used in the proof of the Central Limit Theorem and in the study of the convergence properties of distributions. - In financial mathematics, it aids in the valuation of derivatives and in the modeling of asset prices through the use of characteristic functions. - In quantum mechanics, the theorem finds applications in the study of quantum states and the evolution of systems.

Proof

The proof of Lévy's Continuity Theorem involves several key concepts from analysis and probability theory, including tightness of probability measures, the Portmanteau lemma, and properties of characteristic functions. The proof is structured to first show that convergence of characteristic functions implies weak convergence of measures, followed by demonstrating that weak convergence of measures implies the convergence of characteristic functions under the given conditions.

Related Theorems and Concepts

- Portmanteau lemma - Weak convergence - Characteristic function - Central Limit Theorem

See Also

- Probability theory - Stochastic process - Paul Lévy