Continuous kernel point processes with spectral transform

Discrete kernel point processes (DKPPs) assign probabilities to random subsets through spectral functions of kernel submatrices, unifying determinantal point processes (DPP) and parts of Boltzmann machines under a single family. We study what changes when the ground set is a continuous space. When the number of selected points is fixed, the construction is always well defined under mild boundedness assumptions and recovers the continuous determinantal point process at the logarithmic transform. Allowing the number of points to vary is more restrictive, and finiteness of the normalizing constant is no longer guaranteed. For the Box–Cox family, we show that the range 0 ≤ λ ≤ 1 gives finite normalizing constants under bounded kernels, while for every λ > 1 there exists a bounded positive semidefinite kernel for which the normalizing constant is infinite. A cardinality penalty of order k λ is sufficient and, in a worst-case kernel, necessary to recover finiteness. These results identify a finiteness boundary in the continuous setting that has no analogue in the finite theory.