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Keywords:
determinantal point process; permanental point process; trivial tail-$\sigma $-field; exponential moment; shot-noise process; Berry-Esseen bound; multiparameter $K$-function; kernel-type product density estimator; goodness-of-fit test
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.
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