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From Transform to Distribution

We have seen that transforms are useful for computing with sums of independent random variables. But this is only useful if we can recover the distribution from the transform. The inversion theorem tells us how, and the uniqueness theorem guarantees that no information is lost in the transform — knowing the CF is the same as knowing the CDF.

Theorem: Fourier Inversion Formula

If $X$ has a continuous PDF $f(x)$ and $\phi_X \in L^1(\mathbb{R})$ (i.e., $\int_{-\infty}^{\infty} |\phi_X(u)|\,du < \infty$ ), then

$f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \phi_X(u)\,e^{-jux}\,du.$

The PDF and CF form a Fourier transform pair (up to sign convention).

The characteristic function decomposes the distribution into "frequency components." The inversion formula reassembles the PDF from these components, exactly as Fourier synthesis reconstructs a signal from its spectrum.

Proof

Apply Fourier inversion

Since $\phi_X(u) = \int e^{jux}f(x)\,dx$ and $\phi_X \in L^1$ , the standard Fourier inversion theorem gives:

$f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \phi_X(u)\,e^{-jux}\,du.$

Theorem: General Inversion Formula (Levy)

Let $\overline{F}_X(x) = \frac{1}{2}[F(x) + \lim_{h \downarrow 0}F(x - h)]$ be the symmetrized CDF. Then for any $a < b$ :

$\overline{F}_X(b) - \overline{F}_X(a) = \lim_{T \to \infty}\int_{-T}^{T} \frac{e^{-jau} - e^{-jbu}}{2\pi ju}\,\phi_X(u)\,du.$

At continuity points of $F$ , $\overline{F}_X = F$ , so the formula recovers the CDF itself.

This general formula works even when the CF is not in $L^1$ — for example, for the Cauchy distribution. The price is a limiting truncation of the integral, rather than convergence of the improper integral.

Proof

Compute the truncated integral

For fixed $T$ , the integral $I_T = \int_{-T}^{T}\frac{e^{-jau} - e^{-jbu}}{2\pi ju}\phi_X(u)\,du$ is well-defined since the integrand is bounded.

Substitute the definition of $\ntn{cf}$

Substituting $\phi_X(u) = \int e^{jux}\,dF(x)$ and exchanging the order of integration (justified by Fubini), the inner integral evaluates to a function that converges to the indicator of $(a, b]$ (with half-values at discontinuities) as $T \to \infty$ .

Take $T \to \infty$

By the Dirichlet integral and dominated convergence, $I_T \to \overline{F}_X(b) - \overline{F}_X(a)$ .

,

Theorem: Uniqueness of the Characteristic Function

Two random variables $X$ and $Y$ have the same CDF if and only if they have the same characteristic function:

$\phi_X(u) = \phi_Y(u) \;\text{for all } u \in \mathbb{R} \quad\Longleftrightarrow\quad F(x) = F_Y(x) \;\text{for all } x \in \mathbb{R}.$

The CF is an injective map from distributions to functions. This is what makes the product-of-CFs approach definitive: if we compute the CF of a sum and recognize it as the CF of a known distribution, we have identified the distribution of the sum with certainty.

Proof

Forward: same CDF implies same CF

If $F = F_Y$ , then $\phi_X(u) = \int e^{jux}\,dF(x) = \int e^{jux}\,dF_Y(x) = \phi_Y(u)$ .

Converse: same CF implies same CDF

If $\phi_X = \phi_Y$ , then by the general inversion formula (Theorem TGeneral Inversion Formula (Levy)), $\overline{F}_X(b) - \overline{F}_X(a) = \overline{F}_Y(b) - \overline{F}_Y(a)$ for all $a < b$ . Let $a \to -\infty$ : $\overline{F}_X(b) = \overline{F}_Y(b)$ . At continuity points (which are dense), $F(b) = F_Y(b)$ . By right-continuity of CDFs, equality holds everywhere.

Key Takeaway

The characteristic function uniquely determines the distribution. This is the foundation of the "identify by CF" strategy: compute the CF of a random variable, recognize it as belonging to a known family, and conclude that the random variable has that distribution.

Example: Deriving the Chi-Squared Distribution via the CF

Let $X_1, \ldots, X_{2m}$ be i.i.d. $\mathcal{N}(0, 1)$ . Find the distribution of $Y = \sum_{i=1}^{2m} X_i^2$ using characteristic functions.

Solution

CF of $Z = X^2$ for $X \sim \ntn{gauss}(0,1)$

$\phi_Z(u) = \mathbb{E}[e^{juX^2}] = \int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}} e^{-(1/2 - ju)x^2}\,dx$ .

The Gaussian integral gives $\phi_Z(u) = (1 - 2ju)^{-1/2}$ (for $\text{Im}(1 - 2ju) > 0$ , which holds for all real $u$ ).

CF of $Y = \sum X_i^2$ by independence

$\phi_Y(u) = [\phi_Z(u)]^{2m} = (1 - 2ju)^{-m}$ .

Identify the distribution via Laplace inversion

The Laplace transform $\mathcal{L}[f_{Y}](s) = (1 + 2s)^{-m}$ has a pole of order $m$ at $s = -1/2$ . Residue computation yields

$f_{Y}(y) = \frac{y^{m-1}e^{-y/2}}{2^m \Gamma(m)}, \qquad y \geq 0.$

This is the $\text{Gamma}(m, 1/2)$ density, i.e., $Y \sim \chi^2(2m)$ .

Visualizing the Fourier Inversion Formula

This animation shows how the inverse Fourier integral reconstructs the PDF from the characteristic function. As the truncation

T

increases, the partial reconstruction converges to the true density, exhibiting Gibbs-like oscillations at discontinuities.

Reconstruction of

f(x)

from

\phi_X(u)

via the inverse Fourier integral with increasing truncation

T

.

Historical Note: Paul Levy and the Triumph of Characteristic Functions

20th century

Paul Levy (1886--1971) established characteristic functions as the central tool of probability theory in the early 20th century. His Calcul des Probabilites (1925) proved the continuity theorem — that convergence of CFs implies convergence in distribution — which gave the first clean proof of the central limit theorem. Levy also pioneered the study of stable distributions, which are precisely those distributions whose CFs have the form $\exp(j\mu u - c|u|^\alpha(1 - j\beta\text{sgn}(u)\tan(\pi\alpha/2)))$ . The Gaussian ( $\alpha = 2$ ) and Cauchy ( $\alpha = 1$ ) are special cases.

Common Mistake: Not Every CF Has a Simple Inverse

Mistake:

Assuming that the inversion integral always yields a closed-form PDF. In practice, many CFs (e.g., the CF of a sum of unlike independent random variables) do not invert to standard distributions.

Correction:

The inversion theorem guarantees the existence of the inverse, not that it has a closed form. When closed-form inversion fails, use numerical Fourier inversion (e.g., the FFT) or Gil-Pelaez inversion for computing tail probabilities directly from the CF.

Fourier Transform Pair

A PDF $f(x)$ and its characteristic function $\phi_X(u)$ are Fourier transform pairs: $\phi_X(u) = \mathcal{F}[f](-u)$ and $f(x) = \frac{1}{2\pi}\int \phi_X(u) e^{-jux}\,du$ .

Related: Characteristic Function

Inversion and Uniqueness Theorems

From Transform to Distribution

Theorem: Fourier Inversion Formula

Apply Fourier inversion

Theorem: General Inversion Formula (Levy)

Compute the truncated integral

Substitute the definition of $\ntn{cf}$

Take $T \to \infty$

Theorem: Uniqueness of the Characteristic Function

Forward: same CDF implies same CF

Converse: same CF implies same CDF

Key Takeaway

Example: Deriving the Chi-Squared Distribution via the CF

CF of $Z = X^2$ for $X \sim \ntn{gauss}(0,1)$

CF of $Y = \sum X_i^2$ by independence

Identify the distribution via Laplace inversion

Visualizing the Fourier Inversion Formula

Historical Note: Paul Levy and the Triumph of Characteristic Functions

Common Mistake: Not Every CF Has a Simple Inverse

Fourier Transform Pair