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Math Help - Characteristic Function of a Multivariate Cauchy

  1. #1
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    Acolman, Mexico
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    Characteristic Function of a Multivariate Cauchy

    Hello, I am very lost with the following problem, is there a neat way of doing it, or just solving integral after integral is the way to go?
    Any hint would be greatly appreciated.

    Let X \sim N_p(0,\Sigma) be independent of Z \sim N(0,1). Show that Y=X/Z has characteristic function

    \varphi_Y(t) = \exp\{-(t^T \Sigma t)^{1/2}\}

    Thanks in advance,
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  2. #2
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    Re: Characteristic Function of a Multivariate Cauchy

    Hey akolman.

    Recall that the Moment Generating Function of a variable Y is E[e^(tY)] and in this case Y=X/Z and the expectation of E[X/Z] = Integral (over whole of R^2) (x/z)*f(x,z)dxdz.

    The characteristic function of a random variable is simply MGF_Y(it) where MGF_Y(t) = E[e^(tY)] so the characteristic function is simply E[e^(itY)].

    To start, setup your integral and see how you go.
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