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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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##### References:
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