Non-gaussian white noise examples

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What are some examples of a continuous-time non-Gaussian white noise process?

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A commonly used non-Gaussian white noise process is a Lévy flight. There, the increments are independent at different times, and follow a non-Gaussian distribution, namely a stable distribution, which has power-law tails. This type of nosie also comes up in research problems such as here. Disclaimer: I am one of the authors of this work.