# Levy's theorem of continuity

The **continuity theorem of Lévy** , some only briefly **continuity theorem** called, is a mathematical theorem from probability theory . It establishes a connection between the weak convergence of probability measures and the point-by-point convergence of the corresponding characteristic functions . The theorem is used, for example, as an aid in proving the central limit theorem . It is named after Paul Lévy .

## Preliminary remark

The law of continuity exists in several variants:

- It is sometimes only formulated for probability measures in , and partly for probability measures in .
- In some cases, the weak limit value of the sequence of probability measures and the corresponding characteristic functions are assumed to exist. These formulations are referred to as special formulation in this article. The general formulations then show the existence of a limit value and the characteristic function.

## One-dimensional case

Given probability measures on and the corresponding characteristic functions.

### Special case

It is equivalent:

- The sequence weakly converges against
- The sequence converges pointwise to .

### General case

It is equivalent:

- The sequence converges weakly
- The sequence converges point by point to a function continuous in 0

Then the function is the characteristic function of the weak limit of . That is, it applies

and .

## Higher dimensional case

Given probability measures on and the corresponding characteristic functions.

### Special case

Analogous to the one-dimensional case, it is equivalent:

- The sequence weakly converges against
- The sequence converges pointwise to .

### General case

One function

is called **partially continuous** in if for all the functions

are steadily in .

The law of continuity is now:

- If they converge pointwise to a function that is partially continuous in 0 , then this function is the characteristic function of a probability measure and it applies

- .

- Conversely , if converges weakly to a probability measure , then the characteristic functions on compact sets converge uniformly to .

## literature

- Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
- David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , doi : 10.1007 / b137972 .
- Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .
- Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , doi : 10.1007 / 978-3-642-21026-6 .