Sorry if it is a very basic question. I was wondering under what circumstances the sum of all moments of a random variable converge, i.e. $$S_m=\sum_{n=0}^\infty m_n=\sum_{n=0}^\infty \mathbb{E}[X^n] < \infty$$ What about the sum of all cumulants, i.e. $$S_k=\sum_{n=0}^\infty k_n=\sum_{n=0}^\infty \frac{d^{(n)}}{dt^{(n)}}log\mathbb{E}[e^{tx}]\bigg|_{t=0} < \infty$$ What about the difference between the two, i.e. $S_k-S_m$?
2026-02-23 03:10:00.1771816200
Under what circumstances the sum of all moments (cumulants) converge?
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