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In the Lévy-Itô decomposition it's necessary to compensate small jumps. That's clear. The small jumps are perhaps non-summable. But why are the jumps quared summalbe?

In the "ordinary" proofs of the Lévy Khintchine Formula or the Lévy-Itô Decomposition i can't get the point where we exactly use THIS and where the proof fails if we would not use this. I think there is a connection in both proofs, because the integrand in the L-K formula is near the origin something to the power of 2.

Are there basic results where it's proven that the x^2 integrated with respect to a Lévy measure is finite without using the Lévy-Khintchine Formula or the L-I decomposition?

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    That's the point i mentioned earlier... When all moments exists it should not be only square integrable, or not?! So it turns out that the compensated small jumps are summable (no surprise). Why i can deduce, if i have a compensated square integrable martingale, that the uncompensated process posseses square summable jumps and why this argument wouldn't work with the $L^1$-case? I think i've just a small fallacy, but i don't know where.2012-06-29

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