Karhunen-Loève expansion of Random Measures
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We present an orthogonal expansion for real regular second-order finite random measures over ℝ^d. Such expansion, which may be seen as a Karhunen-Loève decomposition, consists in a series expansion of deterministic real finite measures weighted by uncorrelated real random variables with summable variances. The convergence of the series is in a mean-square-ℳ_B(ℝ^d)^*-weak^* sense, with ℳ_B(ℝ^d) being the space of bounded measurable functions over ℝ^d. This is proven profiting the extra requirement for a regular random measure that its covariance structure is identified with a covariance measure over ℝ^d×ℝ^d. We also obtain a series decomposition of the covariance measure which converges in a separately ℳ_B(ℝ^d)^*-weak^*-total-variation sense. We then obtain an analogous result for function-regulated regular random measures.
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