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.

 

[blue_raver22]

I would say that it is not necessarily true that the characteristic function is constant everywhere if it is constant at 0. It is possible for the characteristic function to have different values at other points in the domain, even if it is constant at 0. It all depends on the specific function and its properties.

In general, the characteristic function is a mathematical tool used to describe the distribution of a random variable. It is defined as the expectation value of the indicator function of a random variable, and it takes on values of either 0 or 1. Therefore, it is possible for the characteristic function to be constant at 0, but have different values at other points in the domain.

To determine if the characteristic function is constant everywhere, you would need to analyze the function and its properties. It is not a general rule that if the characteristic function is constant at one point, it is constant everywhere. So, it is important to carefully examine the proof and understand the assumptions and conditions under which the conclusion is made.

In terms of references, there are many books and online resources available on characteristic functions and their applications. I would recommend consulting textbooks on probability and statistics, as well as searching for specific topics related to the characteristic function that you are interested in. It is always best to consult multiple sources to gain a comprehensive understanding of a topic.