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Thread: Taylor expansion?

  1. #1
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    Taylor expansion?

    How did the authors expand $\displaystyle \phi(\omega)$ below? (Related to the characteristic function derived from the moment generating function)

    I'm self studying this, and not sure what area of math covers these topics? I know how to take a Taylor expansion of a typical function, but what about this improper integral? Or the sum? I'm not sure what they did.



    EDIT: $\displaystyle \phi_X(\omega)=M_X(i\omega)=E(e^{i\omega X})$

    Ah, nevermind, I figured it out -- they used several theorems relating to expectation and started from the E(e^{i\omega X}) instead of the integral or sum.

    But how do they know the integral converges? What test did they use?
    Last edited by scorpion007; Jun 27th 2010 at 12:28 AM.
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  2. #2
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    The integral (and series) converge only for some functions f(x). Here, I assume that f(x) is some probability density function and so $\displaystyle \sum f(x)= 1$ or $\displaystyle \int f(x)= 1dx$. In that case, you can use the "comparison test" for both sum and integral, comparing $\displaystyle e^{e\omega x}f(x)$ to $\displaystyle f(x)$.
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  3. #3
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    Thanks a lot!

    So because exp(iwx)f(x) <= f(x) on (-infty .. infty) and int {f(x)} converges, so does the former function.
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