Probability theory and stochastic processes Expectation, moments and transforms Theorem

The characteristic function determines the distribution

Builds on Characteristic function Law of a random variable Cumulative distribution function

Statement

Let $X$ and $Y$ be real random variables with characteristic functions $\varphi_X$ and $\varphi_Y$ . If $\varphi_X(t) = \varphi_Y(t)$ for all $t \in \mathbb{R}$ , then $X$ and $Y$ have the same law.

Remarks

The proof uses the Lévy inversion formula. For real $a < b$ that are continuity points of the cumulative distribution function $F_X$ , $F_X(b) - F_X(a) = \lim_{T \to \infty} \frac{1}{2\pi} \int_{-T}^{T} \frac{e^{-ita} - e^{-itb}}{it} \, \varphi_X(t) \, dt.$ If $\int |\varphi_X(t)| \, dt < \infty$ , then $X$ has the continuous density $f_X(x) = \frac{1}{2\pi} \int e^{-itx} \varphi_X(t) \, dt$ . The same result holds for random vectors in $\mathbb{R}^n$ , with $\varphi_X(t) = E[e^{i \langle t, X \rangle}]$ .

Reading path

To understand this page from the ground up, read these 53 pages in order. The foundations come first.

Complex numbers object
Construction of the reals from Cauchy sequences concept
Element definition
Integers object
Ordered field definition
Proposition concept
Random experiment concept
Rational numbers object
Real numbers object
Euler's formula statement
Set definition
Ring of integers statement
Disjunction definition
Implication definition
Predicate definition
Ordered pair definition
Union definition
Sample space definition
The rationals form a field statement
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
The rationals form an ordered field statement
Uniqueness quantifier definition
Binary relation definition
Power set definition
Universal set definition
Order on the reals definition
Function definition
Complement definition
Interval definition
Injective function definition
Preimage of a set definition
Sigma-algebra definition
Countable set definition
Event definition
Borel sigma-algebra definition
Probability axioms concept
Probability measure definition
Probability distribution definition
Probability space definition
Random variable definition
Cumulative distribution function definition
Discrete random variable definition
Law of a random variable definition
Random variable with a density definition
Probability mass function definition
Expected value definition
Probability density function definition
Law of the unconscious statistician statement
Characteristic function definition
The characteristic function determines the distribution statement

Where this leads

12 pages build on this one.

Tag	`0T8BYAFEXX`
URI	`m2://0T8BYAFEXX`
Aliases	`uniqueness-theorem-for-characteristic-functions`, `levy-inversion-formula`
Kind	statement, theorem
Writing	W0 stub
MSC	60E10
Status	This copy of the site has no build data. Run `m2 status 0T8BYAFEXX`.
Uses	Characteristic function, Law of a random variable, Cumulative distribution function

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