In a recent paper of Salazar et al. (2014), The Diversification Delta: A Different Perspective, forthcoming in the Journal of Portfolio Management , the authors propose to use the exponential Shannon entropy as an uncertainty measure in portfolio selection and make the following claim:

If we define $$H(X)=−\int_xf(x)\text{ln}f(x)dx$$ to be the differential Shannon entropy, then we have that for random variables $X$ and $Y$ (pp.10): $$\text{exp}(H(X+Y))\leqslant \text{exp}(H(X))+\text{exp}(H(X))$$ After thorough research I haven't been able to find any proof of this sub-additive property which makes me wonder: is this property really correct? And would you have any proof of it? Moreover is seems that this sub-additive property is at odds with the entropy power inequality that says that for independent $X$ and $Y$: $$\text{exp}(2H(X+Y))⩾\text{exp}(2H(X))+\text{exp}(2H(X))$$

Also, I am interested in knowing how this property would generalize to Rényi entropy defined as $$H_\alpha(X)=\frac{1}{1-\alpha}\text{ln}∫_x(f(x))^αdx$$ Is $\text{exp}(H_\alpha(X))$ also sub-additive for some values of $\alpha$?

I would really appreciate any insights you might have on this problem.


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