Assume a finite set of complex random variables form a determinantal point process; we obtain a theorem on the limit of the empirical distribution of these random variables. The result is applied to two types of n-by-n random matrices as n goes to infinity. The first one is the product of m i.i.d. (complex) Ginibre ensembles, and the second one is the product of truncations of m independent Haar unitary matrices with sizes n j × n j for 1 ≤ j ≤ m. Assuming m depends on n, by using the special structures of the eigenvalues we developed, explicit limits of spectral distributions are obtained regardless of the speed of m compared to n. For the product of m Ginibre ensembles, as m is fixed, the limiting distribution is known by various authors, e.g., Götze and Tikhomirov (On the asymptotic spectrum of products of independent random matrices, 2010. http://arxiv.org/pdf/1012.2710v3.pdf), Bordenave (Electron
Commun Probab 16:104–113, 2011), O’Rourke and Soshnikov (Electron J Probab 16(81):2219–2245, 2011) and O’Rourke et al. (J Stat Phys 160(1):89–119, 2015). Our results hold for any m ≥ 1 which may depend on n. For the product of truncations of Haar-invariant unitary matrices, we show a rich feature of the limiting distribution.

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