Couplings for determinantal point processes and their reduced Palm distributions with a view to quantifying repulsiveness

For a determinantal point process (DPP) X with a kernel K whose spectrum is strictly less than one, André Goldman has established a coupling to its reduced Palm process at a point u with 0]> so that, almost surely, is obtained by removing a finite number of points from X. We sharpen this result, assuming weaker conditions and establishing that can be obtained by removing at most one point from X, where we specify the distribution of the difference. This is used to discuss the degree of repulsiveness in DPPs in terms of, including Ginibre point processes and other specific parametric models for DPPs.