Let \(\Omega \) be a bounded pseudoconvex domain in \({\mathbb {C}}^n\) , and let \(\phi \) be a strictly plurisubharmonic function on \(\Omega \) . For each \(k\in {\mathbb {N}}\) , we consider determinantal point process \(\Lambda _k\) with kernel \(K_{k\phi }\) , where \(K_{k\phi }\) is the reproducing kernel of infinite dimensional weighted Bergman space \(H(k\phi )\) with weight \(e^{-k\phi }\) . We show that the scaled cumulant generating function for \(\Lambda _k\) converges as \(k\rightarrow \infty \) to a certain limit, which can be explicitly expressed in terms of \(\phi \) and a test function u. Note that we need to restrict the class of test function u to those that are \(\phi \) -admissible, which amounts to imposing a positivity condition on their complex Hessians.
