Convolutional subordinator processes

March 8, 2021 — December 1, 2021

functional analysis
kernel tricks
machine learning
PDEs
physics
point processes
regression
spatial
statistics
stochastic processes
time series
uncertainty

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Figure 1

Defining stochastic processes by convolution of noise with smoothing kernels, where the driving noise is a Lévy subordinator.

Why would we want this? One reason is that this gives us a way to create nonparametric distributions over measures.

A related but distinct technique is that we can create interesting Generalized Gamma convolution.

1 References

Barndorff-Nielsen, Ole E., Maejima, and Sato. 2006. “Some Classes of Multivariate Infinitely Divisible Distributions Admitting Stochastic Integral Representations.” Bernoulli.
Barndorff-Nielsen, Ole E., Pedersen, and Sato. 2001. “Multivariate Subordination, Self-Decomposability and Stability.” Advances in Applied Probability.
Barndorff-Nielsen, O. E., and Schmiegel. 2004. “Lévy-Based Spatial-Temporal Modelling, with Applications to Turbulence.” Russian Mathematical Surveys.
Bondesson. 2012. Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics 76.
Buchmann, Kaehler, Maller, et al. 2015. “Multivariate Subordination Using Generalised Gamma Convolutions with Applications to V.G. Processes and Option Pricing.” arXiv:1502.03901 [Math, q-Fin].
Çinlar. 1979. “On Increasing Continuous Processes.” Stochastic Processes and Their Applications.
Higdon. 2002. “Space and Space-Time Modeling Using Process Convolutions.” In Quantitative Methods for Current Environmental Issues.
James. 2005. “Bayesian Poisson Process Partition Calculus with an Application to Bayesian Lévy Moving Averages.” Annals of Statistics.
James, Roynette, and Yor. 2008. “Generalized Gamma Convolutions, Dirichlet Means, Thorin Measures, with Explicit Examples.” Probability Surveys.
Nieto-Barajas, Prünster, and Walker. 2004. “Normalized Random Measures Driven by Increasing Additive Processes.” Annals of Statistics.
Ranganath, and Blei. 2018. “Correlated Random Measures.” Journal of the American Statistical Association.
Roychowdhury, and Kulis. 2015. “Gamma Processes, Stick-Breaking, and Variational Inference.” In Artificial Intelligence and Statistics.
Steutel, and van Harn. 2003. Infinite Divisibility of Probability Distributions on the Real Line.
Thorin. 1977a. “On the Infinite Divisbility of the Pareto Distribution.” Scandinavian Actuarial Journal.
———. 1977b. “On the Infinite Divisibility of the Lognormal Distribution.” Scandinavian Actuarial Journal.
Walker, Damien, Laud, et al. 1999. “Bayesian Nonparametric Inference for Random Distributions and Related Functions.” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Wolpert, and Ickstadt. 1998. “Simulation of Lévy Random Fields.” In Practical Nonparametric and Semiparametric Bayesian Statistics. Lecture Notes in Statistics.