# 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 .${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R} ^ {d}}$
• 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. ${\ displaystyle P, P_ {1}, P_ {2}, P_ {3}, \ dots}$${\ displaystyle (\ mathbb {R}, {\ mathcal {B}} (\ mathbb {R}))}$${\ displaystyle \ Phi, \ Phi _ {1}, \ Phi _ {2}, \ Phi _ {3}, \ dots}$

### Special case

It is equivalent:

• The sequence weakly converges against${\ displaystyle (P_ {n}) _ {n \ in \ mathbb {N}}}$${\ displaystyle P}$
• The sequence converges pointwise to .${\ displaystyle (\ Phi _ {n}) _ {n \ in \ mathbb {N}}}$ ${\ displaystyle \ Phi}$

### General case

It is equivalent:

• The sequence converges weakly${\ displaystyle (P_ {n}) _ {n \ in \ mathbb {N}}}$
• The sequence converges point by point to a function continuous in 0${\ displaystyle (\ Phi _ {n}) _ {n \ in \ mathbb {N}}}$${\ displaystyle f}$

Then the function is the characteristic function of the weak limit of . That is, it applies ${\ displaystyle f}$${\ displaystyle (P_ {n}) _ {n \ in \ mathbb {N}}}$

${\ displaystyle w {\ text {-}} \! \ lim _ {n \ to \ infty} P_ {n} = Q}$

and . ${\ displaystyle \ Phi _ {Q} = f}$

## Higher dimensional case

Given probability measures on and the corresponding characteristic functions. ${\ displaystyle P, P_ {1}, P_ {2}, P_ {3}, \ dots}$${\ displaystyle (\ mathbb {R} ^ {d}, {\ mathcal {B}} (\ mathbb {R} ^ {d}))}$${\ displaystyle \ Phi, \ Phi _ {1}, \ Phi _ {2}, \ Phi _ {3}, \ dots}$

### Special case

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

• The sequence weakly converges against${\ displaystyle (P_ {n}) _ {n \ in \ mathbb {N}}}$${\ displaystyle P}$
• The sequence converges pointwise to .${\ displaystyle (\ Phi _ {n}) _ {n \ in \ mathbb {N}}}$${\ displaystyle \ Phi}$

### General case

One function

${\ displaystyle f: \ mathbb {R} ^ {d} \ to \ mathbb {R}}$

is called partially continuous in if for all the functions ${\ displaystyle x '}$${\ displaystyle j = 1, \ dots, d}$

${\ displaystyle y_ {j} \ mapsto f (x_ {1}, \ dots, x_ {j-1}, y_ {j}, x_ {j + 1}, \ dots, x_ {d})}$

are steadily in . ${\ displaystyle y_ {j} = x '_ {j}}$

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${\ displaystyle \ Phi _ {n}}$${\ displaystyle f}$${\ displaystyle f = \ Phi _ {Q}}$${\ displaystyle Q}$
${\ displaystyle w {\ text {-}} \! \ lim _ {n \ to \ infty} P_ {n} = Q}$.
• Conversely , if converges weakly to a probability measure , then the characteristic functions on compact sets converge uniformly to .${\ displaystyle (P_ {n}) _ {n \ in \ mathbb {N}}}$${\ displaystyle P}$${\ displaystyle \ Phi _ {P}}$

## Individual evidence

1. ^ Schmidt: Measure and Probability. 2011, p. 404.
2. ^ Schmidt: Measure and Probability. 2011, p. 404.
3. Kusolitsch: Measure and Probability Theory. 2014, p. 306.
4. Meintrup Schäffler: Stochastics. 2005, p. 184.
5. Klenke: Probability Theory 2013, p. 316.