  We consider smooth linear statistics of determinantal point processes on the
complex plane, and their large scale asymptotics. We prove asymptotic normality
in the finite variance case, where Soshnikov's theorem is not applicable. The
setting is similar to that of Rider and Vir\'ag [Electron. J. Probab., 12, no.
45, 1238--1257, (2007)] for the complex plane, but replaces analyticity
conditions by the assumption that the correlation kernel is reproducing. Our
proof is a streamlined version of that of Ameur, Hedenmalm and Makarov [Duke
Math J., 159, 31--81, (2011)] for eigenvalues of normal random matrices. In our
case, the reproducing property is brought to bear to compensate for the lack of
analyticity and radial symmetries.
