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Covariation-characteristic functions of random measures and their applications to stochastic geometry. (English) Zbl 0962.60034

Let \(\mathcal G\) be the space, endowed with weak topology, of all probability measures on the \(\sigma\)-algebra of Borel sets \({\mathcal B}^{d}\) in \({\mathbb R}^{d}\). Then a random measure \(\psi\) on \({\mathcal B}^{d}\) is a \(\mathcal G\)-valued random element. The covariation-characteristic function of the random measure \(\psi\) is defined as \[ E\int\dots\int\exp\{i(z_1 x_1+\dots+z_{n}x_{n})\}\psi(dx_1)\dots \psi(dx_{n}), \quad n\in{\mathbb N},\;z_{j}\in {\mathbb R}^{d}. \] One-to-one correspondence between the distribution of random measures \(\psi\) and their characteristic functions is proved. Applications to random measures generated by random fields are proposed.

MSC:

60G57 Random measures
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