1 Statement of Lévy’s continuity theorem

Lévy’s continuity theorem gives a criterion for when a sequence of probability distributions converges weakly by examining their characteristic functions. In broad terms, it states that the weak limit can be read off from pointwise limits of characteristic functions, provided the limiting function behaves appropriately near the origin.

1.1 Characteristic functions and pointwise convergence

Let \((X_n)_{n\ge 1}\) be a sequence of real-valued random variables with distributions \(\mu_n\). The characteristic function of \(X_n\) is \[ \varphi_n(t)=\mathbb{E}\big[e^{itX_n}\big], \quad t\in\mathbb{R}. \] Lévy’s theorem assumes that there exists a function \(\varphi:\mathbb{R}\to\mathbb{C}\) such that, for every \(t\), \[ \varphi_n(t)\to \varphi(t) \quad \text{as } n\to\infty, \] i.e., \(\varphi_n\) converges pointwise to \(\varphi\).

1.2 Continuity at the origin and limiting characteristic functions

The key additional requirement is that \(\varphi\) is continuous at \(0\), typically with \(\varphi(0)=1\). In the standard formulation, one assumes \(\varphi\) is a characteristic function candidate arising as the pointwise limit of characteristic functions, and that it is continuous at the origin. Under these conditions, there exists a probability measure \(\mu\) whose characteristic function is exactly \(\varphi\).

1.3 Weak convergence interpretation

With \(\mu_n\) the laws of \(X_n\) and \(\mu\) the law corresponding to \(\varphi\), Lévy’s continuity theorem asserts that \[ \mu_n \Rightarrow \mu, \] meaning \(X_n\) converges in distribution to a random variable \(X\) with characteristic function \(\varphi\). Thus, convergence of characteristic functions (in the pointwise sense) yields convergence of distributions (in the weak sense).

2 Mathematical prerequisites and definitions

Lévy’s theorem sits at the intersection of measure-theoretic probability and Fourier analysis. The underlying framework is weak convergence of probability measures and properties of characteristic functions.

2.1 Probability measures on metric spaces

Many statements can be made on \(\mathbb{R}^d\) (or more general metric spaces), where weak convergence is defined by testing measures against bounded continuous functions. For Lévy’s continuity theorem, the characteristic function perspective is most commonly used on \(\mathbb{R}\) or \(\mathbb{R}^d\).

2.1.1 Convergence in distribution (weak convergence)

A sequence of probability measures \(\mu_n\) on a metric space converges weakly to \(\mu\), written \(\mu_n \Rightarrow \mu\), if \[ \int f\,d\mu_n \to \int f\,d\mu \] for every bounded continuous function \(f\). When \(\mu_n\) are laws of random variables \(X_n\), this is equivalent to convergence in distribution of \(X_n\).

2.2 Characteristic functions: definition and properties

Characteristic functions provide a Fourier transform of probability measures, encoding all distributional information in a complex-valued function.

2.2.1 Uniform boundedness and basic identities

For any random variable \(X\), the characteristic function satisfies \[

\varphi(t)= \big\mathbb{E}[e^{itX}]\big\le \mathbb{E}[e^{itX}]=1,

\] so characteristic functions are uniformly bounded by \(1\) in magnitude. Also, \(\varphi(0)=1\) for every characteristic function.

2.2.2 Continuity and normalization at zero

Characteristic functions are continuous functions of \(t\). In particular, continuity at \(0\) is automatic for genuine characteristic functions and takes the form \(\varphi(t)\to \varphi(0)=1\) as \(t\to 0\). Lévy’s theorem uses continuity at the origin as a practical check that the limiting function indeed corresponds to some probability measure.

2.3 Tightness and how it relates to convergence

Weak convergence is often studied via tightness: a family of probability measures is tight if for every \(\varepsilon>0\) there exists a compact set \(K\) such that each measure assigns mass at least \(1-\varepsilon\) to \(K\). In many limit arguments, tightness prevents “mass from escaping to infinity,” allowing subsequence limits to be identified and combined. Lévy’s continuity theorem, phrased in terms of characteristic functions, is powerful partly because characteristic function convergence constrains such escape behavior.

3 Main implications and uses

Lévy’s theorem is widely used as a convergence criterion because characteristic functions are easier to compute or manipulate than distribution functions.

3.1 From characteristic functions to distributional limits

When the characteristic function of each \(X_n\) can be written explicitly and a candidate limit exists, Lévy’s theorem supplies a direct route to the limiting distribution without requiring pointwise convergence of distribution functions.

3.1.1 Identifying the limit distribution via the limit characteristic function

Given pointwise convergence \(\varphi_n(t)\to \varphi(t)\) with \(\varphi\) continuous at \(0\), the theorem guarantees that \(\varphi\) is the characteristic function of the weak limit. The limiting distribution can then be studied either abstractly (as the unique measure with characteristic function \(\varphi\)) or via inversion methods when feasible.

3.2 Convergence tests for sequences of random variables

Lévy’s continuity theorem serves as a test: to prove \(X_n\Rightarrow X\), it suffices to show that \(\varphi_n(t)\to \varphi_X(t)\) for all \(t\), where \(\varphi_X\) is the characteristic function of the proposed limit \(X\). Because characteristic functions encode convergence in distribution, this reduces distributional proofs to analytic ones.

3.3 Stability under transformations

Many common constructions preserve convergence in distribution. Since characteristic functions behave predictably under linear transformations and summation of independent variables, Lévy’s theorem often underpins results such as:

  • If \(X_n\Rightarrow X\), then \(aX_n+b\Rightarrow aX+b\) for constants \(a,b\).
  • For independent sums, characteristic functions multiply, turning limit computations into products.

4 Proof strategies (high-level)

Proofs vary in presentation, but they generally combine Fourier-analytic ideas with uniqueness and approximation of test functions.

4.1 Inversion of characteristic functions and distribution recovery

A central theme is that characteristic functions determine distributions uniquely and can be used to recover distributional information.

4.1.1 Uniqueness of characteristic functions

Two probability measures on \(\mathbb{R}\) (or \(\mathbb{R}^d\)) that share the same characteristic function must be equal. This injectivity means that once the limiting characteristic function is identified, the corresponding limiting distribution is determined.

4.2 Continuity theorem via test functions

Another common proof route uses test functions and approximation. Weak convergence can be established by showing convergence of integrals against bounded continuous functions. Characteristic functions enter via Fourier inversion or related transforms, allowing one to express certain classes of test functions in terms of \(\varphi_n\) and then pass to the limit.

4.3 Role of continuity at the origin

Continuity at \(0\) is the technical bridge ensuring that the pointwise limit corresponds to a true characteristic function. Without it, a pointwise limit of characteristic functions might fail to be the Fourier transform of any probability measure, leaving the conclusion about weak convergence unsupported.

5 Examples and applications

The theorem is most effective when characteristic functions are tractable and converge to a recognizable limit.

5.1 Convergence to a normal distribution

A standard application is to show that a normalized sum converges to a Gaussian distribution. If \(X_n\) are appropriately standardized and their characteristic functions converge pointwise to the characteristic function of a normal law, Lévy’s continuity theorem yields \(X_n\Rightarrow N(\mu,\sigma^2)\). This approach complements or re-derives the central limit behavior using analytic convergence.

5.2 Convergence of sums of independent random variables

For independent random variables, the characteristic function of a sum is the product of the individual characteristic functions. If \[ S_n=\sum_{k} Y_{n,k} \] with suitable independence structure, one can compute \[ \varphi_{S_n}(t)=\prod_{k} \varphi_{Y_{n,k}}(t) \] and then analyze the limit of this product as \(n\to\infty\). Once pointwise convergence of \(\varphi_{S_n}\) is established and the limiting function is continuous at \(0\), Lévy’s theorem implies convergence in distribution of \(S_n\).

5.3 Limit theorems using characteristic functions

Many classical and modern limit results rely on identifying candidate limiting characteristic functions.

5.3.1 Stable laws as limiting cases

Stable distributions arise as limits of normalized sums, particularly when classical moment conditions fail. Characteristic functions of stable laws have a characteristic form that often appears directly in the limit of characteristic functions of partial sums. Lévy’s continuity theorem then translates that analytic convergence into weak convergence to the stable law.

Lévy’s theorem is part of a broader framework of equivalent or complementary characterizations of weak convergence.

6.1 Scheffé-type ideas and alternative convergence characterizations

While Scheffé’s lemma is most often discussed for almost-sure or \(L^1\) convergence of densities, the general philosophy is similar: convergence of certain transforms or densities can imply convergence of distributions under additional conditions. Lévy’s theorem fits this pattern by using characteristic functions as the transform.

6.2 Portmanteau theorem connections

The Portmanteau theorem provides several equivalent ways to express weak convergence, such as convergence of integrals against bounded continuous functions, or inequalities involving limsups of probabilities of closed sets. Lévy’s continuity theorem supplies a different criterion—based on characteristic functions—that can be used to verify one of the Portmanteau conditions indirectly.

6.3 Connections to the central limit theorem

The central limit theorem is frequently proved by studying characteristic functions and taking limits. Lévy’s theorem formalizes the step from “the characteristic functions converge” to “the distributions converge,” making it a natural companion result to Fourier-based central limit arguments.

7 Common pitfalls and technical conditions

The theorem’s conditions are not merely technicalities; omitting them can lead to incorrect conclusions.

7.1 When pointwise convergence is not sufficient

Pointwise convergence \(\varphi_n(t)\to \varphi(t)\) alone does not always guarantee that \(\varphi\) is a characteristic function. Without additional validation of the limiting function, weak convergence may fail or the limit distribution may not exist in the desired form.

7.2 Ensuring the limit function is a valid characteristic function

In applications, verifying that \(\varphi(0)=1\) and that \(\varphi\) is continuous at \(0\) is crucial. These checks ensure that \(\varphi\) can serve as the Fourier transform of some probability measure. Depending on the setting, further structure (such as positive definiteness) can also be relevant, but continuity at the origin is the typical focal condition in Lévy’s theorem’s standard statement.

7.3 Checking continuity at zero in practice

Continuity at the origin is often easier than it seems: if \(\varphi_n(t)\to \varphi(t)\) and each \(\varphi_n\) satisfies \(\varphi_n(0)=1\), then establishing that \(\varphi(t)\to 1\) as \(t\to 0\) confirms the continuity requirement. In many analytic limit computations (e.g., using logarithms of characteristic functions or expansions near \(0\)), this behavior is visible from the form of the candidate limit.