The discussion revolves around the characteristic function of the Cauchy distribution and the question of whether it has moments. Participants explore the relationship between the differentiability of the characteristic function and the existence of a Taylor series expansion around the origin.
Participants generally agree on the non-differentiability of the characteristic function at the origin, but there is no consensus on the implications of this observation regarding the existence of moments.
Participants reference graphical analysis and derivative calculations, but the discussion does not resolve the mathematical steps required to definitively prove the absence of moments.
Office_Shredder said:I can't say much about the first part of your post, but for there to be a Taylor series expansion the function has to be differentiable at the origin. It's not, just draw a graph and take a look at it.
To formally prove it, just calculate what the derivate is to the left of 0, and the right of 0, and observe they are not equal.
I have looked at the plot of the function. What would that do to help? As there is a sharp turn at ##t=0##, surely that further enhances my conclusion?Office_Shredder said:Have you tried drawing a plot of the function yet? There's a sharp corner at t=0. On the left of 0 it has one slope, on the right of 0 it has another slope. Similar to how the graph of |t| has a slope of -1 at 0 on the left, and 1 at 0 on the right.