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| In [[probability theory]], for a [[probability measure]] '''P''' on a [[Hilbert space]] ''H'' with [[inner product]] <math>\langle \cdot,\cdot\rangle </math>, the '''covariance''' of '''P''' is the [[bilinear form]] Cov: ''H'' × ''H'' → '''R''' given by
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| :<math>\mathrm{Cov}(x, y) = \int_{H} \langle x, z \rangle \langle y, z \rangle \, \mathrm{d} \mathbf{P} (z)</math> | |
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| for all ''x'' and ''y'' in ''H''. The '''covariance operator''' ''C'' is then defined by
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| :<math>\mathrm{Cov}(x, y) = \langle Cx, y \rangle</math>
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| (from the [[Riesz representation theorem]], such operator exists if Cov is [[Bilinear form#On normed vector spaces|bounded]]). Since Cov is symmetric in its arguments, the covariance operator is
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| [[self-adjoint operator|self-adjoint]] (the infinite-dimensional analogy of the transposition symmetry in the finite-dimensional case). When '''P''' is a centred [[Gaussian measure]], ''C'' is also a [[nuclear operator]]. In particular, it is a [[compact operator]] of [[trace class]], that is, it has finite [[trace (mathematics)|trace]].
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| Even more generally, for a [[probability measure]] '''P''' on a [[Banach space]] ''B'', the covariance of '''P''' is the [[bilinear form]] on the [[algebraic dual]] ''B''<sup>#</sup>, defined by
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| :<math>\mathrm{Cov}(x, y) = \int_{B} \langle x, z \rangle \langle y, z \rangle \, \mathrm{d} \mathbf{P} (z)</math>
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| where <math> \langle x, z \rangle </math> is now the value of the linear functional ''x'' on the element ''z''.
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| Quite similarly, the [[covariance function]] of a function-valued [[random element]] (in special cases called [[random process]] or [[random field]]) ''z'' is
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| :<math>\mathrm{Cov}(x, y) = \int z(x) z(y) \, \mathrm{d} \mathbf{P} (z) = E(z(x) z(y))</math> | |
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| where ''z''(''x'') is now the value of the function ''z'' at the point ''x'', i.e., the value of the [[linear functional]] <math> u \mapsto u(x) </math> evaluated at ''z''.
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| {{probability-stub}}
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| [[Category:Probability theory]]
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| [[Category:Covariance and correlation]]
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| [[Category:Bilinear forms]]
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