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Let ⊞, ⊠, and ⊎ be the free additive, free multiplicative, and boolean additive convolutions, respectively. For a probability measure μ on [0,∞) with finite second moment, we find a scaling limit of ${\left({\mu }^{⊠N}\right)}^{⊞N}$ as N goes to infinity. The -transform of its limit distribution can be represented by Lambert’s W-function. From this, we deduce that the limiting distribution is freely infinitely divisible, like the lognormal distribution in the classical case. We also show a similar limit theorem by replacing free...

We study relations between the Boolean convolution and the symmetrization and the pushforward of order 2. In particular we prove that if μ₁,μ₂ are probability measures on [0,∞) then ${\left(\mu ₁⨄\mu ₂\right)}^s=\mu {₁}^s⨄\mu {₂}^s$ and if ν₁,ν₂ are symmetric then ${\left(\nu ₁⨄\nu ₂\right)}^{\left(2\right)}=\nu {₁}^{\left(2\right)}⨄\nu {₂}^{\left(2\right)}$. Finally we investigate necessary and sufficient conditions under which the latter equality holds.