New limit theorems related to free multiplicative convolution

Let $⊞$, $⊠$ and $\uplus$ be the free additive, free multiplicative, and boolean additive convolutions, respectively. For a probability measure $\mu$ on $[0,\mathrm{\infty })$ with finite second moment, we find the scaling limit of $({\mu }^{⊠N}{)}^{⊞N}$ as $N$ goes to infinity. The $\mathcal{R}$–transform of the limit distribution can be represented by the Lambert’s $W$ function. We also find similar limit theorem by replacing the free additive convolution with the boolean convolution.