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You can't have a precise argument without a precise definition. In general, the appropriate notion of integral here is the Lebesgue-Stieltjes integral. In a fairly general setup, let $F: \mathbb R \to \mathbb R$ be a right-continuous function that is of locally bounded variation, that is V_F([a,b]) := \sup\lbrace \sum_{i=1}^n \vert F(x_{i+1}) - F(x_i ) \...
Even if you assume null cokurtosis terms, your equality is still off: \begin{align} \operatorname{Kurt}[X+Y] = {1 \over \sigma_{X+Y}^4} \big( & \sigma_X^4\operatorname{Kurt}[X] + \sigma_Y^4\operatorname{Kurt}[Y] \big). \end{align} Note that you need $\sigma_{X+Y}^2$. You already have $\sigma_X^2$ and $\sigma_Y^2$ (computed in the paper). Full formula is:...