Search Results for « Shot noise »

We investigate a shot noise process with subexponential shot marks occurring at renewal epochs. Our main result is a precise asymptotic formula for its tail probability. In doing so, some recent results regarding sums of randomly weighted subexponential random variables play a crucial role.

Citation: 'shot noise' in the IUPAC Compendium of Chemical Terminology, 5th ed.; International Union of Pure and Applied Chemistry; 2025. Online version 5.0.0, 2025. 10.1351/goldbook.09572 • License: The IUPAC Gold Book is licensed under Creative Commons Attribution-ShareAlike CC BY-SA 4.0 International for individual terms. Requests for commercial usage of the compendium should be directed to IUPAC.

Shot noise processes describe a stream of random impulses that arrive at Poisson times and decay through a deterministic response kernel. They provide a flexible and interpretable class of models for volatility, credit intensities, insurance reserves, neuronal activity, and photon counts. We give a self contained treatment of the probabilistic structure of these processes and develop a Bayesian framework for inference from discretely and noisily observed data. On the theory side we record the Laplace functional, cumulants, and Levy triplet, we identify the exponential kernel case with a non Gaussian Ornstein-Uhlenbeck process driven by a compound Poisson subordinator, and we show that the stationary law belongs to the Thorin class of generalized gamma convolutions whenever the mark tail is completely monotone. In the exponential mark case the stationary law is exactly gamma. Reading the same path as a hazard rate yields a closed form survival function and an increasing hazard with a finite plateau, linking the construction to reliability in dynamic environments. On the computation side the intractable likelihood is handled by treating the latent marked point process as auxiliary data. We give a data augmentation Gibbs sampler with birth, death, and move updates, together with a state space formulation that admits particle filtering and particle Markov chain Monte Carlo. We show that the same augmentation renders the sampler free of Metropolis steps, connect the shrinkage priors on the marks to the Lévy structure of the process itself, and, because the model is trivial to simulate but has an intractable likelihood, develop a likelihood-free generative Bayesian computation (GBC) scheme that trains a quantile neural network to produce amortized posterior draws. A simulation study recovers all structural parameters with well calibrated credible intervals and characterizes the finite sample near degeneracy between the arrival rate and the mark scale, and a further study validates the GBC estimator against particle Markov chain Monte Carlo and against the theory. We close with extensions to nonexponential kernels, generalized gamma convolution priors on the marks, and shot noise Cox processes.