Convolutional subordinator processes


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.

References

Barndorff-Nielsen, O. E., and J. Schmiegel. 2004. “Lévy-Based Spatial-Temporal Modelling, with Applications to Turbulence.” Russian Mathematical Surveys 59 (1): 65. https://doi.org/10.1070/RM2004v059n01ABEH000701.
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Higdon, Dave. 2002. “Space and Space-Time Modeling Using Process Convolutions.” In Quantitative Methods for Current Environmental Issues, edited by Clive W. Anderson, Vic Barnett, Philip C. Chatwin, and Abdel H. El-Shaarawi, 37–56. London: Springer. https://doi.org/10.1007/978-1-4471-0657-9_2.
James, Lancelot F. 2005. “Bayesian Poisson Process Partition Calculus with an Application to Bayesian Lévy Moving Averages.” Annals of Statistics 33 (4): 1771–99. https://doi.org/10.1214/009053605000000336.
Nieto-Barajas, Luis E., Igor Prünster, and Stephen G. Walker. 2004. “Normalized Random Measures Driven by Increasing Additive Processes.” Annals of Statistics 32 (6): 2343–60. https://doi.org/10.1214/009053604000000625.
Ranganath, Rajesh, and David M. Blei. 2018. “Correlated Random Measures.” Journal of the American Statistical Association 113 (521): 417–30. https://doi.org/10.1080/01621459.2016.1260468.
Roychowdhury, Anirban, and Brian Kulis. 2015. “Gamma Processes, Stick-Breaking, and Variational Inference.” In Artificial Intelligence and Statistics, 800–808. PMLR. http://proceedings.mlr.press/v38/roychowdhury15.html.
Walker, Stephen G., Paul Damien, PuruShottam W. Laud, and Adrian F. M. Smith. 1999. “Bayesian Nonparametric Inference for Random Distributions and Related Functions.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 61 (3): 485–527. https://doi.org/10.1111/1467-9868.00190.
Wolpert, R. 1998. “Poisson/Gamma Random Field Models for Spatial Statistics.” Biometrika 85 (2): 251–67. https://doi.org/10.1093/biomet/85.2.251.
Wolpert, Robert L. 2006. “Stationary Gamma Processes,” 13.
Wolpert, Robert L., and Katja Ickstadt. 1998. “Simulation of Lévy Random Fields.” In Practical Nonparametric and Semiparametric Bayesian Statistics, edited by Dipak Dey, Peter Müller, and Debajyoti Sinha, 227–42. Lecture Notes in Statistics. New York, NY: Springer. https://doi.org/10.1007/978-1-4612-1732-9_12.

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