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On the strong Brillinger-mixing property of \(\alpha\)-determinantal point processes and some applications. (English) Zbl 06644006
Summary: First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function \(C(x,y)\) defining an \(\alpha\)-determinantal point process (DPP). Assuming absolute integrability of the function \(C_0(x)=C(o,x)\), we show that a stationary \(\alpha\)-DPP with kernel function \(C_0(x)\) is “strongly” Brillinger-mixing, implying, among others, that its tail-\(\sigma\)-field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch some applications to statistical second-order analysis of \(\alpha\)-DPPs.

MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05 Central limit and other weak theorems
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