Convergence in distribution is the basic language of limit laws. It is weak enough to describe asymptotic shapes of random variables without requiring a pointwise coupling, yet strong enough to support stable consequences such as convergence of probabilities of continuity sets and convergence of expectations of bounded continuous test functions. The most direct tool for proving convergence in distribution is the characteristic function, the Fourier transform of a probability measure. Characteristic functions exist for every random variable, multiply under sums of independent variables, and determine the law uniquely.
This article explains weak convergence, why characteristic functions determine distributions, and how they yield a clean proof of the central limit theorem (CLT). The emphasis is on the logic: what needs to be shown, what is automatic, and where moment assumptions enter.
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Convergence in distribution and the portmanteau viewpoint
Write $X_n \Rightarrow X$ for convergence in distribution. By definition, this means
for all continuity points $t$ of the distribution function of $X$.
An equivalent and often more usable characterization is: for every bounded continuous function $f$,
This is one form of the portmanteau theorem. It clarifies that weak convergence is about convergence of integrals against a large class of test functions. This is exactly why Fourier transforms enter: they are integrals against the bounded continuous functions $x\mapsto e^{itx}$.
Weak convergence is weaker than convergence in probability. It cannot distinguish sequences that differ on sets of small probability, and it does not control moments unless additional uniform integrability-type bounds are supplied. Its strength is that it identifies limiting laws.
Tightness: preventing mass from escaping
A sequence of laws $\mu_n$ on $\mathbb{R}$ can fail to have convergent subsequences if mass escapes to infinity. Tightness rules this out: $\{\mu_n\}$ is tight if for every $\varepsilon>0$ there exists $M$ such that $\mu_n([-M,M])\ge 1-\varepsilon$ for all $n$. On $\mathbb{R}$, tightness plus pointwise convergence of distribution functions at continuity points is a robust route to weak convergence.
In many limit theorems, tightness is obtained from variance bounds via Chebyshev’s inequality or from uniform moment bounds.
Characteristic functions: definition and core properties
For a real-valued random variable $X$, the characteristic function is
It exists without moment assumptions because $|e^{itX}|=1$.
Key properties:
- $\varphi_X(0)=1$, and $|\varphi_X(t)|\le 1$.
- $\varphi_X$ is uniformly continuous.
- Scaling: $\varphi_{aX}(t)=\varphi_X(at)$.
- Independence: if $X$ and $Y$ are independent, $\varphi_{X+Y}(t)=\varphi_X(t)\varphi_Y(t)$.
The multiplicative property is the main algebraic advantage: sums become products.
Uniqueness: characteristic functions determine the law
If $\varphi_X(t)=\varphi_Y(t)$ for all $t$, then $X$ and $Y$ have the same distribution. There are several proofs; one route uses Fourier inversion for probability measures and approximations by smooth test functions. The important practical takeaway is that identifying the pointwise limit of characteristic functions identifies the limiting distribution.
Lévy’s continuity theorem: the convergence bridge
Lévy’s continuity theorem states:
- If $\varphi_{X_n}(t)\to \varphi(t)$ pointwise for all $t$, and $\varphi$ is continuous at 0, then $\varphi$ is the characteristic function of some random variable $X$, and $X_n\Rightarrow X$.
Conversely, if $X_n\Rightarrow X$, then $\varphi_{X_n}(t)\to \varphi_X(t)$ for all $t$.
This theorem is what makes characteristic functions a method for proving weak convergence: prove an analytic limit and recognize it.
The central limit theorem in this language
Let $X_1,X_2,\dots$ be independent identically distributed with $\mathbb{E}[X_i]=0$ and $\mathrm{Var}(X_i)=\sigma^2\in(0,\infty)$. Define
The CLT claims $S_n\Rightarrow Z$ where $Z$ is standard normal.
The characteristic function of $S_n$ is
Thus everything reduces to understanding $\varphi_{X_1}(u)$ near $u=0$.
The small-$u$ expansion
Using $e^{iuX}=1+iuX-\frac{u^2X^2}{2}+R(uX)$, where the remainder satisfies $|R(y)|\le C|y|^3$ for small $y$ and $|R(y)|\le 2$ globally, one can show under $\mathbb{E}[X^2]<\infty$ that
The mean-zero condition eliminates the linear term, and the variance supplies the quadratic term. The $o(u^2)$ term is where integrability and truncation arguments enter: one splits $X$ into a bounded part and a tail part and uses dominated convergence on the bounded part while controlling the tail using the finite second moment.
Passing to the $n$-th power
Set $u=t/(\sigma\sqrt{n})$. Then
Therefore,
The limit $e^{-t^2/2}$ is the characteristic function of the standard normal distribution. By Lévy’s theorem, $S_n\Rightarrow Z$.
This proof isolates the universality mechanism: after normalization by $\sqrt{n}$, only the second moment contributes at leading order to the characteristic function near 0.
Quantitative refinements and what weak convergence does not give
The characteristic-function proof identifies the limit but does not provide an error bound on $P(S_n\le t)-\Phi(t)$. Such quantitative control requires additional input (for instance, bounds involving third moments and smoothing inequalities). The gap is structural: weak convergence is a qualitative statement. To make it quantitative, one needs uniform control of the approximation error in Fourier space and a way to translate that control back to distribution functions.
It is still valuable to know what weak convergence does guarantee. If $f$ is bounded and continuous, $\mathbb{E}[f(S_n)]\to \mathbb{E}[f(Z)]$. Many asymptotic statistics are built by applying continuous mappings \to $S_n$; the continuous mapping theorem then transfers weak convergence through continuous transformations.
Why characteristic functions remain a practical tool
Even outside the CLT, characteristic functions offer a repeatable method:
- express the quantity of interest as a sum of independent components or as a scaled transformation,
- compute or estimate the characteristic function,
- identify its limit,
- invoke Lévy’s theorem.
This method excels for sums, for stable limits, and for problems where distribution functions are hard to handle directly but Fourier transforms are tractable.
Characteristic functions do not replace other methods, but they provide one of the clearest pipelines from probabilistic structure to limiting law: algebraic manipulation in the exponent, analytic limits, then an inversion theorem to return to probability.
Inversion and why continuity at zero matters
The statement “characteristic functions determine distributions” is not just a slogan; it comes from an inversion principle. For sufficiently nice densities $f$, Fourier inversion says
For general probability measures, one uses smoothed approximations and shows that integrals of test functions can be recovered from $\varphi_X$. The continuity-at-zero condition in Lévy’s theorem is a minimal regularity requirement that rules out pathological pointwise limits that fail to correspond to probability measures. Intuitively, continuity at zero ensures the limiting function behaves like an average of unit-modulus exponentials and therefore can be a characteristic function.
In many applications, continuity at zero is automatic because each $\varphi_{X_n}$ is continuous and the convergence is controlled well enough to preserve continuity at zero.
Slutsky and continuous mapping: how weak limits propagate
Two structural facts make weak convergence useful in applied probability.
- Slutsky’s theorem: If $X_n\Rightarrow X$ and $Y_n\to c$ in probability, then $X_n+Y_n\Rightarrow X+c$ and $X_nY_n\Rightarrow cX$.
- Continuous mapping theorem: If $X_n\Rightarrow X$ and $g$ is continuous, then $g(X_n)\Rightarrow g(X)$.
These results are why one can prove a CLT for sums and then immediately obtain a CLT for standardized statistics, studentized quantities under suitable conditions, and many functionals built from the sum.
Characteristic functions are compatible with these theorems. For example, convergence in probability \to a constant corresponds to characteristic functions converging pointwise \to $e^{itc}$, and products and compositions behave as expected.
Triangular arrays and the Lindeberg idea
The i.i.d. CLT is a baseline. Many applications involve sums of independent but not identically distributed terms. Consider a triangular array $X_{n,1},\dots,X_{n,n}$ independent with mean zero and variances $\sigma_{n,k}^2$, and let $s_n^2=\sum_{k=1}^n \sigma_{n,k}^2$. One studies
A central condition ensuring a normal limit is a Lindeberg-type tail control: for every $\varepsilon>0$,
This condition prevents rare large summands from dominating the normalized sum. In the characteristic-function proof, this is exactly what is needed to justify a uniform small-$u$ expansion: the quadratic term from variance is stable, and the remainder terms from large deviations vanish in aggregate.
The conceptual message is stable across formulations: normal limits arise when the sum has many small contributors and no single contributor carries a macroscopic fraction of the variance.
Tightness from variance bounds
In the CLT setting, tightness is straightforward: $\mathrm{Var}(S_n)=1$, so Chebyshev implies
Thus the laws of $S_n$ are tight. Tightness ensures that pointwise convergence of characteristic functions is not hiding mass escape. On $\mathbb{R}$, Lévy’s theorem already packages the needed compactness, but it is useful to remember that second-moment normalization carries a built-in tightness guarantee.
Why characteristic functions are more than a proof device
Characteristic functions provide a practical calculus of limits:
- They turn convolution (sums of independent variables) into multiplication.
- They turn scaling into a simple argument rescaling.
- They provide access to stable limits when densities or distribution functions are not tractable.
They are also a diagnostic tool. If you can compute or approximate $\log \varphi_{X_n}(t)$, then weak limits often become limits of exponentials. In many problems, $\log \varphi$ has an additive decomposition, which mirrors independence at the level of cumulants.
A common pitfall: weak convergence does not imply moment convergence
Even when $X_n\Rightarrow X$, it need not be true that $\mathbb{E}[X_n]\to \mathbb{E}[X]$ or that variances converge. Moment convergence requires uniform integrability or explicit moment bounds. In CLT contexts, normalization often forces variance to be controlled, but higher moments can still fail to converge.
A safe rule is:
- weak convergence plus uniform integrability of $|X_n|$ implies convergence of expectations,
- weak convergence plus uniform integrability of $X_n^2$ implies convergence of second moments.
This is one reason quantitative CLT bounds often assume finite third absolute moment: it controls tails strongly enough to translate Fourier error estimates into distribution-function error bounds.
The central picture
Weak convergence is convergence of laws. Characteristic functions encode laws via Fourier transforms and are stable under sums of independent components. Lévy’s continuity theorem makes pointwise characteristic-function limits equivalent to weak limits. The CLT follows from a second-order expansion near zero and a product-\to-exponential limit. Extensions to non-identical summands require a tail condition that keeps the quadratic variance term dominant. With these pieces, many limiting-distribution problems become analytic limit problems with a clear algebraic structure.
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