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Gaussian limit for determinantal random point fields. (English) Zbl 1033.60063
A random point field on a locally compact Hausdorff space $$E$$ with a measure $$\mu$$ is considered. The appearance of $$n$$ points of the field at $$n$$ disjoint subsets of $$E$$ is proposed to be determined by the $$n$$-dimensional density $$p_n$$ with respect to $$\mu^n$$. The author assumes that there exists an integral operator with a kernel $$K$$ such that for any $$n\geq 2$$ the density $$p_n$$ is representable in the form of determinant of the kernel $$K$$. Three theorems are proved. The first one in the case of general $$E$$ gives sufficient conditions for the sequence of centered and normed linear functionals of the point field to converge in distribution to a Gaussian random value. The second one for $$E= R^d$$ $$(d\geq 2)$$ and “almost” homogeneous kernel $$K$$ gives conditions for the weak convergence of the sequence of the rescaled point fields to the Gaussian white noise. And the third one for $$E=R^1$$ and homogeneous kernel $$K$$ asserts the weak convergence of a sequence of rescaled point processes to a self-similar Gaussian process with the spectral density $$| k|^\alpha$$ $$(0< \alpha< 1)$$.

MSC:
 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G18 Self-similar stochastic processes
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References:
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