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Characteristic function

  1. Aug 30, 2010 #1
    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
     
  2. jcsd
  3. Aug 31, 2010 #2
    It's true that

    [tex]\varphi (0)=\int_{-\infty}^\infty f_{X}(x)dx=1[/tex].

    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
     
    Last edited: Sep 1, 2010
  4. Sep 4, 2010 #3

    Landau

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    The question doesn't make any sense, since you don't specify what topological space you are working in.
     
  5. Sep 5, 2010 #4
    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 [tex]\psi_X(t) = e^{ibt}[/tex], i.e. [tex]|\psi|=1[/tex] 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.
     
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