Covariance Function for Processes 

The CovarianceFunction at times s and t is given by Expectation[(x[s]-[s])(x[t]-[s]), x], where x[t] is the process state and [t] is the mean function.

A discrete-time and discrete-state random process.

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X CovarianceFunction[BinomialProcess[1/3], s, t]

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X CovarianceFunction[BinomialProcess[1/3], s, t]; DiscretePlot3D[%, {s, 0, 10}, {t, 0, 10}, ExtentSize -> 1/2, ColorFunction -> "Rainbow", ImageSize -> Medium, AxesLabel -> Automatic]

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A discrete-time and continuous-state random process.

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X CovarianceFunction[ARProcess[{2/3}, 1], s, t]

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X CovarianceFunction[ARProcess[{2/3}, 1], s, t]; DiscretePlot3D[%, {s, 0, 10}, {t, 0, 10}, ExtentSize -> 1/2, ColorFunction -> "Rainbow", ImageSize -> Medium, AxesLabel -> Automatic]

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A continuous-time and discrete-state random process.

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X CovarianceFunction[PoissonProcess[1], s, t]

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In[6]:=

X CovarianceFunction[PoissonProcess[1], s, t]; Plot3D[%, {s, 0, 10}, {t, 0, 10}, Mesh -> None, ColorFunction -> "Rainbow", ImageSize -> Medium, AxesLabel -> Automatic]

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A continuous-time and continuous-state random process.

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X CovarianceFunction[OrnsteinUhlenbeckProcess[0, 1, 1/5], s, t]

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In[8]:=

X CovarianceFunction[OrnsteinUhlenbeckProcess[0, 1, 1/5], s, t]; Plot3D[%, {s, 0, 10}, {t, 0, 10}, Mesh -> None, ColorFunction -> "Rainbow", ImageSize -> Medium, AxesLabel -> Automatic]

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