Lévy's Inversion Formula/Integrable Characteristic Function

Theorem

Let $X$ be a real-valued random variable.

Let $P_X$ be the probability distribution of $X$.

Let $\phi : \R \to \C$ be the characteristic function of $X$.

Suppose that $\phi$ is Lebesgue integrable, i.e.:

$\ds \int _\R \cmod {\map \phi t} \rd t < + \infty$

Then $P_X$ is absolutely continuous with respect to the Lebesgue measure.

Moreover, the Radon-Nikodym derivative is given by:

$\ds \map g x := \dfrac 1 {2 \pi} \int_\R \map \phi t e^{- i t x} \rd t$

Proof

Let $a < b$ be such that:

$\map {P_X} {\set {a,b} } = 0$

Then:

\(\ds \int_a^b \map g x \rd x\)

\(=\)

\(\ds \dfrac 1 {2 \pi} \int_a^b \paren {\int_\R \map \phi t e^{- i t x} \rd t} \rd x\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 {2 \pi} \int_\R \map \phi t \paren {\int_a^b e^{- i t x} \rd x} \rd t\)

Fubini's Theorem

\(\ds \)

\(=\)

\(\ds \dfrac 1 {2 \pi} \int_\R \map \phi t \frac {e^{- i t a} - e^{- i t b} }{i t} \rd t\)

Now:

$\ds \forall t \in \R : \quad \cmod {\map \phi t \frac {e^{- i t a} - e^{- i t b} }{i t} } \le \cmod {\map \phi t} \paren {b - a}$

Thus:

\(\ds \dfrac 1 {2 \pi} \int_\R \map \phi t \frac {e^{- i t a} - e^{- i t b} }{i t} \rd t\)

\(=\)

\(\ds \lim_{T \mathop \to \infty} \dfrac 1 {2 \pi} \int_{-T}^T \map \phi t \frac {e^{- i t a} - e^{- i t b} }{i t} \rd t\)

Lebesgue's Dominated Convergence Theorem

\(\ds \)

\(=\)

\(\ds \map {P_X} {\hointl a b}\)

Lévy's Inversion Formula

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Source of Name

This entry was named for Paul Pierre Lévy.