Is the characteristic function constant everywhere if it is constant at 0?

[wayneckm]

Hi there,


Recently I have come across a proof with application of characteristic function.

After some steps in the proof, it concluded that there is a neighborhood of 0 such that the characteristic function is constant at 1, then it said the characteristic function is constant at 1 everywhere over the domain.

I suspect that "If there exists a neighborhood of 0 such that the characteristic function is constant, it is constant everywhere." Is this correct?

I have tried to search from the web regarding this but found nothing. Would anyone suggest me some good reference on characteristic function as well.

Thanks.


Wayne

 

[SW VandeCarr]

After some steps in the proof, it concluded that there is a neighborhood of 0 such that the characteristic function is constant at 1, then it said the characteristic function is constant at 1 everywhere over the domain.

I suspect that "If there exists a neighborhood of 0 such that the characteristic function is constant, it is constant everywhere." Is this correct?

I have tried to search from the web regarding this but found nothing. Would anyone suggest me some good reference on characteristic function as well.

Thanks.Wayne

It's true that

$$\varphi (0)=\int_{-\infty}^\infty f_{X}(x)dx=1$$.

However, I'm not aware that the ChF is constant anywhere for any PDF. Can you provide the source? Perhaps you are you talking about the (Dirac)delta distribution?

EDIT: The characteristic function is apparently defined differently in a non-probabilistic context. See the first paragraph of the following:

http://mathworld.wolfram.com/CharacteristicFunction.html

 

[Landau]

The question doesn't make any sense, since you don't specify what topological space you are working in.

 

[bpet]

... it concluded that there is a neighborhood of 0 such that the characteristic function is constant at 1, then it said the characteristic function is constant at 1 everywhere over the domain.

An interesting generalization of the result is proved in Feller (vol 2 p 475), namely that any c.f. with constant absolute value is of the form $$\psi_X(t) = e^{ibt}$$, i.e. $$|\psi|=1$$ and the distribution of X is concentrated at b; moreover any c.f. that achieves absolute value 1 away from t=0 is periodic and represents a distribution concentrated on a lattice.