We will discuss the role of infinitely divisible laws in relation to the limit theorems of c-free and Boolean convolutions. As in the classical case, we show that the infinitely divisible laws arise as the limit distributions for infinitesimal triangular arrays. We will also discuss the necessary and sufficient conditions for the weak convergence of c-free and Boolean convolutions, and these conditions allow us to compare the asymptotic behavior between these convolutions and the classical convolution.

These results are obtained by using the complex analytic methods developed for the calculation of free convolution. The results on multiplicative c-free convolution are joint with Mihai Popa.