Abstract
We consider centered conditionally Gaussian d-dimensional vectors X with random covariance matrix Ξ having an arbitrary probability distribution law on the set of nonnegative definite symmetric d × d matrices M d +. The paper deals with the evaluation problem of mean values \( E\left[ {\prod\nolimits_{i = 1}^{2n} {\left( {{c_i},X} \right)} } \right] \) for c i ∈ ℝd, i = 1, …, 2n, extending the Wick theorem for a wide class of non-Gaussian distributions. We discuss in more detail the cases where the probability law ℒ(Ξ) is infinitely divisible, the Wishart distribution, or the inverse Wishart distribution. An example with Ξ \( = \sum\nolimits_{j = 1}^m {{Z_j}{\sum_j}} \), where random variables Z j , j = 1, …, m, are nonnegative, and Σ j ∈ M d +, j = 1, …, m, are fixed, includes recent results from Vignat and Bhatnagar, 2008.
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Dedicated to the memory of Vytautas Statulevičius
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Grigelionis, B. On the Wick theorem for mixtures of centered Gaussian distributions. Lith Math J 49, 372–380 (2009). https://doi.org/10.1007/s10986-009-9064-6
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DOI: https://doi.org/10.1007/s10986-009-9064-6