We prove that, under fairly general conditions, properly rescaled determinantal random point field converges to a generalized Gaussian random process.

Let Yi, i ≥ 1, be i.i.d. random variables having values in an m-dimensional manifold M ⊂ Rd and consider sums ni=1 ξ(nYi, {nYi}i=1), where ξ is a real valued function defined on pairs (y,Y), with y ∈ Rd and Y ⊂ Rd locally finite. Subject to ξ satisfying a weak spatial dependence and continuity condition, we show that such sums satisfy weak laws of large numbers, variance asymptotics, and centra...