Synopsis
A bridge between non-Gaussian models and Gaussian behavior
This work presents an infinite family of random processes that share the same correlation function, yet are non-Gaussian for most values of a key parameter. The construction starts from a shot-noise (Poisson) process and builds a stationary, random step function whose features can be calculated in principle. As the parameter grows, the processes gradually resemble a Gaussian process with the same power spectrum, highlighting the limits of using only second-order statistics to characterize randomness.
The approach lets you see how higher-order statistics matter in practice and how a single spectrum can correspond to many different stochastic behaviors. By specifying shot timings and rectangular shot shapes, the author shows how to derive univariate and multivariate distributions, and how sample functions look under different shot-density regimes. The results are grounded in a familiar method, yet give a complete, theory-backed view of a clearly non-Gaussian family that transitions toward Gaussianity.- Learn how a stationary shot-noise construction yields an isospectral family of processes
- See the relationship between correlation functions, power spectra, and higher-order statistics
- Discover how non-Gaussian models converge to Gaussian behavior as a parameter changes
- Explore visual examples illustrating low, medium, and high shot-density cases
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