Probability theory and stochastic processes Modes of convergence Theorem

Lévy's continuity theorem

Builds on Characteristic function Convergence in distribution The characteristic function determines the distribution

Statement

Let $X, X_1, X_2, \dots$ be real random variables with characteristic functions $\varphi, \varphi_1, \varphi_2, \dots$ .

If $X_n \to X$ in distribution, then $\varphi_n(t) \to \varphi(t)$ for each $t \in \mathbb{R}$ .
Conversely, suppose that $\varphi_n(t)$ converges to a limit $\psi(t)$ for each $t \in \mathbb{R}$ , and that $\psi$ is continuous at $0$ . Then $\psi$ is the characteristic function of a random variable $Y$ , and $X_n \to Y$ in distribution.

Remarks

Part 1 follows from the definition of convergence in distribution with the bounded continuous functions $\cos(tx)$ and $\sin(tx)$ . The continuity at $0$ in part 2 is necessary. For $X_n \sim \mathcal{N}(0, n)$ , $\varphi_n(t) = e^{-nt^2/2}$ tends to $1_{\{t = 0\}}$ , which is not continuous at $0$ , and $X_n$ does not converge in distribution. The theorem holds in the same form for random vectors in $\mathbb{R}^k$ .

History

The theorem is named after Paul Lévy.

Reading path

To understand this page from the ground up, read these 55 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
Convergence in distribution 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
Lévy's continuity theorem statement

Where this leads

10 pages build on this one.

Tag	`0FPWF7TTMB`
URI	`m2://0FPWF7TTMB`
Aliases	`levy-continuity-theorem`
Kind	statement, theorem
Writing	W1 stated
MSC	60E10, 60F05
Status	This copy of the site has no build data. Run `m2 status 0FPWF7TTMB`.
Uses	Characteristic function, Convergence in distribution, The characteristic function determines the distribution
Same as	Wikidata Q1966978 Wikipedia Lévy's continuity theorem

Markdown · JSON · Edit on GitHub

What this page needs for W2

edge about out: the concepts that the statement is about (0 of 1)
edge proves in: at least one proof node (0 of 1)
edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
explanation L2 (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.