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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.


  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:

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


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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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