Apply ARMA Filter to a Heavy-Tailed White Noise Process 

Sample the process of independent and identically distributed random variables from Student distribution.

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X wn\[ScriptCapitalP] = WhiteNoiseProcess[StudentTDistribution[5.3]]; whiteNoise = RandomFunction[wn\[ScriptCapitalP], {0, 10^5}]

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Apply ARMA filter to the white noise signal.

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X arma\[ScriptCapitalP] = ARMAProcess[{5/6, -1/6}, {2/3}, 1];

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X path = RecurrenceFilter[TransferFunctionModel[arma\[ScriptCapitalP]], whiteNoise]

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Extract the tail of the path, where the sequence has settled into stationary regime.

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X pathTail = TimeSeriesWindow[path, {10^5 - 10^4, 10^5}];

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X ListPlot[pathTail, Filling -> Axis, ImageSize -> Medium]

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Compare slice distributions of the unfiltered and the filtered signals.

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X whiteNoiseTail = TimeSeriesWindow[whiteNoise, {10^5 - 10^4, 10^5}];

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X SmoothHistogram[{pathTail["Values"], whiteNoiseTail["Values"]}, PlotRange -> All, Filling -> Axis, PlotLegends -> {"Student t white noise", "ARMA filtered noise"}]

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