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Multidimensional Hermite polynomials of complex Gaussian variables. (English) Zbl 0830.60031

Multidimensional Hermite polynomials of complex Gaussian variables are defined and some properties are investigated. Let \(\{Z(t), t \in R^d\}\) be a complex-valued second-order Gaussian random field. Then \[ E \bigl( Z(t) \mid Z(s) \bigr) = Z (s \wedge t) \quad \text{and} \quad \biggl \langle H_p \bigl( Z(s) \bigr),\;H_p \bigl( Z(t) \bigr) \biggr \rangle = \biggl |H_p \bigl( Z(s \wedge t) \bigr) \biggr |^2, \] where \(H_p\) is a one-dimensional Hermite polynomial.
Reviewer: N.Leonenko (Kiev)

MSC:

60G15 Gaussian processes
62M40 Random fields; image analysis
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