Mean of a Distribution Question

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The discussion centers on the relationship between a probability distribution function F and its characteristic function φ. It establishes that if the mean of F exists, there exists a positive u such that the integral of the absolute difference between 1 and φ(t) divided by t is finite for all t within the bounds of |t| < u. The key insight is the approximation φ(t) ~ 1 + iFt as t approaches 0, which is crucial for understanding the behavior of the characteristic function near the origin.

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Epsilon36819
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Here goes:

If F is a probability distribution function and /phi is its integrable characteristic function. If the mean of F exists, why can we say that there exists u>0 st int[abs(1- /phi(t))/t] < infinity, where the integral is over the set of all t st abs(t)<u ?

(abs = absolute value)

I just. Can`t. See. It.

Thanks in advance for your help.
 
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I'm not completely sure of details, but you would use phi(t) ~ 1 +iFt for t near 0.
 

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