Branching processes



A diverse class of stochastic models that I am mildly obsessed with, where over some index set (usually time, space or both) there are distributed births of some kind, and we count the total population.

In particular, I am interested in “pure birth” branching processes, where each event leads to certain numbers of offspring with a certain probability. These correspond to certain types of “cluster” and “self excitation” processes.

These come in Markov and non-Markov flavours, depending on, loosely, whether the notional particles in the system have a memoryless life cycle or not.

To learn

  • Basic handling of process defined on a multidimensional index set, i.e. space-time processes and branching random fields. (“cluster processes”) Maybe I’ve done that over at spatial point processes by now?
  • The various connections to trees, and hence the connection to networks.
  • Connection to stable processes and Lévy processes.

We do not care about time

Cascade models.

Discrete index, discrete state, Markov: The Galton-Watson process

This section got long enough to break out separately. See my notes on long-memory Galton-Watson process.

Continuous index, discrete state: the Hawkes Process

If you have a integer-valued state space, but a continuous time index, and linear intensities, then this is a Hawkes point process, the cluster point process. See my masters thesis, or my Hawkes process notes

Continuous index, continuous state

Aldous gives an expo on the Continuous State Branching process. I do not know much about these. Perhaps I could know more if I read Z. Li (2011), which introduces CSBPs as a special case of Measure-valued branching processes, and also connects them with supecprocesses (Etheridge 2000; Dynkin 2004, 1991) were recommended to me for the latter.

Parameter estimation

I’m curious about this, and Lévy process inference in general. It’s interesting because such processes are always incompletely sampled; What’s the best you can do with finitely many samples from a continuous branching process? For the simple case of the Weiner process (as a Lévy process) there is a well-understood estimation theory, with twiddly flourishes on top. For CSBPs I am not aware of any general methods. (Overbeck 1998) seems to be one of the few refs and is rather constrained. Surely the finance folks are onto this?

Discrete index, continuous state

Popular in physics as a contagion model. See Burridge (2013a);Burridge (2013b) for some handy relations between these models, Gamma processes, martingales and limits of negative binomial distributions via renewal theory and Kendall’s identity.

Special issues for multivariate branching processes

If you are looking at cross-excitation between variables then I have some additional matter at contagion processes.

Classic data sets

Data sets which might be explored for their branching process nature tend towards the epidemiological.

Implementations

IHSEP is Feng Chen’s software to continuous index, discrete state branching processes.

Spatstat is for spatial point processes.

References

Aldous, David. 1991. The Continuum Random Tree. I.” The Annals of Probability 19 (1): 1–28.
———. 1993. The Continuum Random Tree III.” The Annals of Probability 21 (1): 248–89.
Aldous, David, and Jim Pitman. 1998. Tree-Valued Markov Chains Derived from Galton-Watson Processes.” Annales de l’Institut Henri Poincare (B) Probability and Statistics 34 (5): 637–86.
Al-Osh, M. A., and A. A. Alzaid. 1987. First-Order Integer-Valued Autoregressive (INAR(1)) Process.” Journal of Time Series Analysis 8 (3): 261–75.
Al-Osh, Mohamed A., and Emad-Eldin A. A. Aly. 1992. First Order Autoregressive Time Series with Negative Binomial and Geometric Marginals.” Communications in Statistics - Theory and Methods 21 (9): 2483–92.
Aly, Emad-Eldin A. A., and Nadjib Bouzar. 2005. Stationary Solutions for Integer-Valued Autoregressive Processes.” International Journal of Mathematics and Mathematical Sciences 2005 (1): 1–18.
Alzaid, A., and M. Al-Osh. 1988. First-Order Integer-Valued Autoregressive (INAR (1)) Process: Distributional and Regression Properties.” Statistica Neerlandica 42 (1): 53–61.
Applebaum, David. 2004. Lévy Processes — from Probability to Finance and Quantum Groups.” Notices of the AMS 51 (11): 1336–47.
———. 2009. Lévy Processes and Stochastic Calculus. 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge ; New York: Cambridge University Press.
Aragón, Tomás J. 2012. Applied Epidemiology Using R. MedEpi Publishing. http://www. medepi. net/epir/index. html. Calendar Time. Accessed.
Athreya, K. B., and Niels Keiding. 1977. Estimation Theory for Continuous-Time Branching Processes.” Sankhyā: The Indian Journal of Statistics, Series A (1961-2002) 39 (2): 101–23.
Athreya, K. B., and A. N. Vidyashankar. 1997. Large Deviation Rates for Supercritical and Critical Branching Processes.” In Classical and Modern Branching Processes, edited by Krishna B. Athreya and Peter Jagers, 1–18. The IMA Volumes in Mathematics and Its Applications 84. Springer New York.
Bacry, E., K. Dayri, and J. F. Muzy. 2012. Non-Parametric Kernel Estimation for Symmetric Hawkes Processes. Application to High Frequency Financial Data.” The European Physical Journal B 85 (5): 1–12.
Bacry, E., S. Delattre, M. Hoffmann, and J. F. Muzy. 2013a. Modelling Microstructure Noise with Mutually Exciting Point Processes.” Quantitative Finance 13 (1): 65–77.
———. 2013b. Some Limit Theorems for Hawkes Processes and Application to Financial Statistics.” Stochastic Processes and Their Applications, A Special Issue on the Occasion of the 2013 International Year of Statistics, 123 (7): 2475–99.
Bacry, Emmanuel, Martin Bompaire, Stéphane Gaïffas, and Jean-Francois Muzy. 2020. Sparse and Low-Rank Multivariate Hawkes Processes.” Journal of Machine Learning Research 21 (50): 1–32.
Bacry, Emmanuel, Thibault Jaisson, and Jean-Francois Muzy. 2014. Estimation of Slowly Decreasing Hawkes Kernels: Application to High Frequency Order Book Modelling.” arXiv:1412.7096 [q-Fin, Stat], December.
Bacry, Emmanuel, and Jean-François Muzy. 2014. Hawkes Model for Price and Trades High-Frequency Dynamics.” Quantitative Finance 14 (7): 1147–66.
———. 2016. First- and Second-Order Statistics Characterization of Hawkes Processes and Non-Parametric Estimation.” IEEE Transactions on Information Theory 62 (4): 2184–2202.
Baddeley, Adrian. 2007. Spatial Point Processes and Their Applications.” In Stochastic Geometry, edited by Wolfgang Weil, 1–75. Lecture Notes in Mathematics 1892. Springer Berlin Heidelberg.
Barndorff-Nielsen, O. E., and M. Sørensen. 1994. A Review of Some Aspects of Asymptotic Likelihood Theory for Stochastic Processes.” International Statistical Review / Revue Internationale de Statistique 62 (1): 133–65.
Bauwens, Luc, and Nikolaus Hautsch. 2006. Stochastic Conditional Intensity Processes.” Journal of Financial Econometrics 4 (3): 450–93.
Bertoin, Jean, Marc Yor, et al. 2001. On Subordinators, Self-Similar Markov Processes and Some Factorizations of the Exponential Variable.” Electron. Comm. Probab 6 (95): 106.
Bhat, B. R., and S. R. Adke. 1981. Maximum Likelihood Estimation for Branching Processes with Immigration.” Advances in Applied Probability 13 (3): 498–509.
Bhattacharjee, M. C. 1987. The Time to Extinction of Branching Processes and Log-Convexity: I.” Probability in the Engineering and Informational Sciences 1 (03): 265–78.
Bibby, Bo Martin, and Michael Sørensen. 1995. Martingale Estimation Functions for Discretely Observed Diffusion Processes.” Bernoulli 1 (1/2): 17–39.
Böckenholt, Ulf. 1998. Mixed INAR(1) Poisson Regression Models: Analyzing Heterogeneity and Serial Dependencies in Longitudinal Count Data.” Journal of Econometrics 89 (1–2): 317–38.
Bowman, K.O., and L.R. Shenton. 1989. The Distribution of a Moment Estimator for a Parameter of the Generalized Poision Distribution.” Communications in Partial Differential Equations 14 (4): 867–93.
Bowsher, Clive G. 2007. Modelling Security Market Events in Continuous Time: Intensity Based, Multivariate Point Process Models.” Journal of Econometrics 141 (2): 876–912.
Brown, B. M., and J. I. Hewitt. 1975. Inference for the Diffusion Branching Process.” Journal of Applied Probability 12 (3): 588–94.
Burridge, James. 2013a. Cascade Sizes in a Branching Process with Gamma Distributed Generations.” arXiv:1304.3741 [Math], April.
———. 2013b. Crossover Behavior in Driven Cascades.” Physical Review E 88 (3): 032124.
Caballero, M. E., and L. Chaumont. 2006. Conditioned Stable Lévy Processes and the Lamperti Representation.” Journal of Applied Probability 43 (4): 967–83.
Caballero, M. Emilia, José Luis Pérez Garmendia, and Gerónimo Uribe Bravo. 2013. A Lamperti-Type Representation of Continuous-State Branching Processes with Immigration.” The Annals of Probability 41 (3A): 1585–1627.
Caballero, Maria-Emilia, Amaury Lambert, and Geronimo Uribe Bravo. 2009. Proof(s) of the Lamperti Representation of Continuous-State Branching Processes.” Probability Surveys 6: 62–89.
Chen, Feng, and Peter Hall. 2016. Nonparametric Estimation for Self-Exciting Point Processes—A Parsimonious Approach.” Journal of Computational and Graphical Statistics 25 (1): 209–24.
Chistyakov, V. 1964. A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes.” Theory of Probability & Its Applications 9 (4): 640–48.
Çinlar, Erhan. 1975. Exceptional Paper—Markov Renewal Theory: A Survey.” Management Science 21 (7): 727–52.
Cohn, Harry. 1997. Stochastic Monotonicity and Branching Processes.” In Classical and Modern Branching Processes, edited by Krishna B. Athreya and Peter Jagers, 51–56. The IMA Volumes in Mathematics and Its Applications 84. Springer New York.
Consul, P. C. 2014. Lagrange and Related Probability Distributions.” In Wiley StatsRef: Statistics Reference Online. John Wiley & Sons, Ltd.
Consul, P. C., and Felix Famoye. 1992. Generalized Poisson Regression Model.” Communications in Statistics - Theory and Methods 21 (1): 89–109.
———. 2006. Lagrangian Probability Distributions. Boston: Birkhäuser.
Consul, P. C., and Famoye Felix. 1989. Minimum Variance Unbiased Estimation for the Lagrange Power Series Distributions.” Statistics 20 (3): 407–15.
Consul, P. C., and L. R. Shenton. 1973. Some Interesting Properties of Lagrangian Distributions.” Communications in Statistics 2 (3): 263–72.
Consul, P.C., and M. M. Shoukri. 1984. Maximum Likelihood Estimation for the Generalized Poisson Distribution.” Communications in Statistics - Theory and Methods 13 (12): 1533–47.
Consul, P.C., and M.M. Shoukri. 1988. Some Chance Mechanisms Related to a Generalized Poisson Probability Model.” American Journal of Mathematical and Management Sciences 8 (1-2): 181–202.
Consul, P., and L. Shenton. 1972. Use of Lagrange Expansion for Generating Discrete Generalized Probability Distributions.” SIAM Journal on Applied Mathematics 23 (2): 239–48.
Crane, Riley, Frank Schweitzer, and Didier Sornette. 2010. Power Law Signature of Media Exposure in Human Response Waiting Time Distributions.” Physical Review E 81 (5): 056101.
Crisan, D, P Del Moral, and T Lyons. 1999. “Discrete Filtering Using Branching and Interacting Particle Systems.” Markov Processes and Related Fields 5 (3): 293–318.
Cui, Yunwei, and Robert Lund. 2009. A New Look at Time Series of Counts.” Biometrika 96 (4): 781–92.
Curien, Nicolas, and Jean-François Le Gall. 2013. The Brownian Plane.” Journal of Theoretical Probability 27 (4): 1249–91.
Daley, Daryl J., and David Vere-Jones. 2003. An introduction to the theory of point processes. 2nd ed. Vol. 1. Elementary theory and methods. New York: Springer.
———. 2008. An Introduction to the Theory of Point Processes. 2nd ed. Vol. 2. General theory and structure. Probability and Its Applications. New York: Springer.
Dekking, F. M., and E. R. Speer. 1997. On the Shape of the Wavefront of Branching Random Walk.” In Classical and Modern Branching Processes, edited by Krishna B. Athreya and Peter Jagers, 73–88. The IMA Volumes in Mathematics and Its Applications 84. Springer New York.
Del Moral, Pierre, and Laurent Miclo. 2000. Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering.” In Séminaire de Probabilités XXXIV, 1–145. Lecture Notes in Mathematics 1729. Springer.
Deschâtres, Fabrice, and Didier Sornette. 2005. Dynamics of Book Sales: Endogenous Versus Exogenous Shocks in Complex Networks.” Physical Review E 72 (1): 016112.
Doney, R. A., and A. E. Kyprianou. 2006. Overshoots and Undershoots of Lévy Processes.” The Annals of Applied Probability 16 (1): 91–106.
Drost, Feike C., Ramon van den Akker, and Bas J. M. Werker. 2009. Efficient Estimation of Auto-Regression Parameters and Innovation Distributions for Semiparametric Integer-Valued AR(p) Models.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 71 (2): 467–85.
Du, Nan, Mehrdad Farajtabar, Amr Ahmed, Alexander J. Smola, and Le Song. 2015. Dirichlet-Hawkes Processes with Applications to Clustering Continuous-Time Document Streams.” In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 219–28. KDD ’15. New York, NY, USA: ACM.
Duembgen, Moritz, and Mark Podolskij. 2015. High-Frequency Asymptotics for Path-Dependent Functionals of Itô Semimartingales.” Stochastic Processes and Their Applications 125 (4): 1195–1217.
Dwass, Meyer. 1969. The Total Progeny in a Branching Process and a Related Random Walk.” Journal of Applied Probability 6 (3): 682–86.
Dynkin, E. B. 1991. Branching Particle Systems and Superprocesses.” The Annals of Probability 19 (3): 1157–94.
———. 2004. Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations. University Lecture Series, v. 34. Providence, R.I: American Mathematical Society.
Eden, U, L Frank, R Barbieri, V Solo, and E Brown. 2004. Dynamic Analysis of Neural Encoding by Point Process Adaptive Filtering.” Neural Computation 16 (5): 971–98.
Eichler, Michael, Rainer Dahlhaus, and Johannes Dueck. 2016. Graphical Modeling for Multivariate Hawkes Processes with Nonparametric Link Functions.” Journal of Time Series Analysis, January, n/a–.
Embrechts, Paul, Thomas Liniger, and Lu Lin. 2011. Multivariate Hawkes Processes: An Application to Financial Data.” Journal of Applied Probability 48A (August): 367–78.
Etheridge, Alison. 2000. An Introduction to Superprocesses. University Lecture Series, v. 20. Providence, RI: American Mathematical Society.
Evans, Steven N. 2008. Probability and Real Trees. Vol. 1920. Lecture Notes in Mathematics 1920. Berlin: Springer.
Falkner, Neil, and Gerald Teschl. 2012. On the Substitution Rule for Lebesgue–Stieltjes Integrals.” Expositiones Mathematicae 30 (4): 412–18.
Feigin, Paul David. 1976. Maximum Likelihood Estimation for Continuous-Time Stochastic Processes.” Advances in Applied Probability 8 (4): 712–36.
Filimonov, Vladimir, David Bicchetti, Nicolas Maystre, and Didier Sornette. 2014. Quantification of the High Level of Endogeneity and of Structural Regime Shifts in Commodity Markets.” Journal of International Money and Finance, Understanding International Commodity Price Fluctuations, 42 (April): 174–92.
Filimonov, Vladimir, Spencer Wheatley, and Didier Sornette. 2015. Effective Measure of Endogeneity for the Autoregressive Conditional Duration Point Processes via Mapping to the Self-Excited Hawkes Process.” Communications in Nonlinear Science and Numerical Simulation 22 (1–3): 23–37.
Fokianos, Konstantinos. 2011. Some Recent Progress in Count Time Series.” Statistics 45 (1): 49–58.
Freeland, R. K., and B. P. M. McCabe. 2004. Analysis of Low Count Time Series Data by Poisson Autoregression.” Journal of Time Series Analysis 25 (5): 701–22.
Gehler, Peter V., Alex D. Holub, and Max Welling. 2006. The Rate Adapting Poisson Model for Information Retrieval and Object Recognition.” In Proceedings of the 23rd International Conference on Machine Learning, 337–44. ICML ’06. New York, NY, USA: ACM.
Geiger, Jochen, and Lars Kauffmann. 2004. The Shape of Large Galton-Watson Trees with Possibly Infinite Variance.” Random Struct. Algorithms 25 (3): 311–35.
Godoy, Boris I., Victor Solo, Jason Min, and Syed Ahmed Pasha. 2016. Local Likelihood Estimation of Time-Variant Hawkes Models.” In, 4199–4203. IEEE.
Guttorp, Peter. 1991. Statistical Inference for Branching Processes. Wiley Series in Probability and Mathematical Statistics. New York: Wiley.
Haccou, Patsy, Peter Jagers, and Vladimir A. Vatutin. 2005. Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge: Cambridge University Press.
Hall, Andreia, Manuel Scotto, and João Cruz. 2009. Extremes of Integer-Valued Moving Average Sequences.” TEST 19 (2): 359–74.
Halpin, Peter F., and Paul De Boeck. 2013. Modelling Dyadic Interaction with Hawkes Processes.” Psychometrika 78 (4): 793–814.
Hansen, Niels Richard, Patricia Reynaud-Bouret, and Vincent Rivoirard. 2015. Lasso and Probabilistic Inequalities for Multivariate Point Processes.” Bernoulli 21 (1): 83–143.
Hardiman, Stephen J., Nicolas Bercot, and Jean-Philippe Bouchaud. 2013. Critical Reflexivity in Financial Markets: A Hawkes Process Analysis.” The European Physical Journal B 86 (10): 1–9.
Hardiman, Stephen J., and Jean-Philippe Bouchaud. 2014. Branching-Ratio Approximation for the Self-Exciting Hawkes Process.” Physical Review E 90 (6): 062807.
Harn, K. van, and F. W. Steutel. 1993. Stability Equations for Processes with Stationary Independent Increments Using Branching Processes and Poisson Mixtures.” Stochastic Processes and Their Applications 45 (2): 209–30.
Harn, K. van, F. W. Steutel, and W. Vervaat. 1982. Self-Decomposable Discrete Distributions and Branching Processes.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete 61 (1): 97–118.
Hawkes, Alan G. 1971. Spectra of Some Self-Exciting and Mutually Exciting Point Processes.” Biometrika 58 (1): 83–90.
Hawkes, Alan G., and David Oakes. 1974. A Cluster Process Representation of a Self-Exciting Process.” Journal of Applied Probability 11 (3): 493.
Heyde, C. C., and E. Seneta. 2010. Estimation Theory for Growth and Immigration Rates in a Multiplicative Process.” In Selected Works of C.C. Heyde, edited by Ross Maller, Ishwar Basawa, Peter Hall, and Eugene Seneta, 214–35. Selected Works in Probability and Statistics. Springer New York.
Houdré, Christian. 2002. Remarks on Deviation Inequalities for Functions of Infinitely Divisible Random Vectors.” The Annals of Probability 30 (3): 1223–37.
Imoto, Tomoaki. 2016. Properties of Lagrangian Distributions.” Communications in Statistics - Theory and Methods 45 (3): 712–21.
Iribarren, José Luis, and Esteban Moro. 2011. Branching Dynamics of Viral Information Spreading.” Physical Review E 84 (4): 046116.
Jacod, Jean. 1997. On Continuous Conditional Gaussian Martingales and Stable Convergence in Law.” In Séminaire de Probabilités XXXI, edited by Jacques Azéma, Marc Yor, and Michel Emery, 232–46. Lecture Notes in Mathematics 1655. Springer Berlin Heidelberg.
Jacod, Jean, Mark Podolskij, and Mathias Vetter. 2010. Limit Theorems for Moving Averages of Discretized Processes Plus Noise.” The Annals of Statistics 38 (3): 1478–1545.
Jagers, Peter. 1969. Renewal Theory and the Almost Sure Convergence of Branching Processes.” Arkiv För Matematik 7 (6): 495–504.
———. 1997. Towards Dependence in General Branching Processes.” In Classical and Modern Branching Processes, edited by Krishna B. Athreya and Peter Jagers, 127–39. The IMA Volumes in Mathematics and Its Applications 84. Springer New York.
János Engländer. 2007. Branching Diffusions, Superdiffusions and Random Media.” Probability Surveys 4: 303–64.
Jánossy, L., and H. Messel. 1950. Fluctuations of the Electron-Photon Cascade - Moments of the Distribution.” Proceedings of the Physical Society. Section A 63 (10): 1101.
———. 1951. Investigation into the Higher Moments of a Nucleon Cascade.” Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences 54: 245–62.
Kedem, Benjamin, and Konstantinos Fokianos. 2002. Regression models for time series analysis. Chichester; Hoboken, NJ: John Wiley & Sons.
Keener, Robert W. 2009. Curved Exponential Families.” In Theoretical Statistics, 85–99. Springer Texts in Statistics. Springer New York.
Kesten, Harry. 1973. Random Difference Equations and Renewal Theory for Products of Random Matrices.” Acta Mathematica 131 (1): 207–48.
Kratz, Peter, and Etienne Pardoux. 2016. Large Deviations for Infectious Diseases Models.” arXiv:1602.02803 [Math], February.
Kraus, Andrea, and Victor M. Panaretos. 2014. Frequentist Estimation of an Epidemic’s Spreading Potential When Observations Are Scarce.” Biometrika 101 (1): 141–54.
Kvitkovičová, Andrea, and Victor M. Panaretos. 2011. Asymptotic Inference for Partially Observed Branching Processes.” Advances in Applied Probability 43 (4): 1166–90.
Lakshmanan, Karthik C., Patrick T. Sadtler, Elizabeth C. Tyler-Kabara, Aaron P. Batista, and Byron M. Yu. 2015. Extracting Low-Dimensional Latent Structure from Time Series in the Presence of Delays.” Neural Computation 27 (9): 1825–56.
Lamperti, John. 1967a. Continuous-State Branching Processes.” Bull. Amer. Math. Soc 73 (3): 382–86.
———. 1967b. The Limit of a Sequence of Branching Processes.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete 7 (4): 271–88.
Laredo, Catherine, Olivier David, and Aurélie Garnier. 2009. Inference for Partially Observed Multitype Branching Processes and Ecological Applications.” arXiv:0902.4520 [Stat], February.
Latour, Alain. 1998. Existence and Stochastic Structure of a Non-Negative Integer-Valued Autoregressive Process.” Journal of Time Series Analysis 19 (4): 439–55.
Laub, Patrick J., Thomas Taimre, and Philip K. Pollett. 2015. Hawkes Processes.” arXiv:1507.02822 [Math, q-Fin, Stat], July.
Le Gall, Jean-François. 2005. Random Trees and Applications.” Probability Surveys 2: 245–311.
———. 2013. Uniqueness and Universality of the Brownian Map.” The Annals of Probability 41 (4): 2880–960.
Le Gall, Jean-François, and Grégory Miermont. 2012. Scaling Limits of Random Trees and Planar Maps.” Probability and Statistical Physics in Two and More Dimensions 15: 155–211.
Lee, W. H., K. I. Hopcraft, and E. Jakeman. 2008. Continuous and Discrete Stable Processes.” Physical Review E 77 (1): 011109.
Levina, Anna, and J. Michael Herrmann. 2013. The Abelian Distribution.” Stochastics and Dynamics 14 (03): 1450001.
Lewis, Erik, and George Mohler. 2011. A Nonparametric EM Algorithm for Multiscale Hawkes Processes.” Preprint.
Li, S, F Famoye, and C Lee. 2010. “On the Generalized Lagrangian Probability Distributions.” Journal of Probability and Statistical Science 8 (1): 113–23.
Li, Yingying, and Per A. Mykland. 2007. Are Volatility Estimators Robust with Respect to Modeling Assumptions? Bernoulli 13 (3): 601–22.
Li, Zeng-Hu. 2000. Asymptotic Behaviour of Continuous Time and State Branching Processes.” Journal of the Australian Mathematical Society (Series A) 68 (01): 68–84.
Li, Zenghu. 2011. Measure-Valued Branching Markov Processes. Probability and Its Applications. Heidelberg ; New York: Springer.
———. 2012. Continuous-State Branching Processes.” arXiv:1202.3223 [Math], February.
———. 2014. Path-Valued Branching Processes and Nonlocal Branching Superprocesses.” The Annals of Probability 42 (1): 41–79.
Liniger, Thomas Josef. 2009. Multivariate Hawkes Processes.” Diss., Eidgenössische Technische Hochschule ETH Zürich, Nr. 18403, 2009.
Lyons, Russell. 1990. Random Walks and Percolation on Trees.” The Annals of Probability 18 (3): 931–58.
Marsan, David, and Olivier Lengliné. 2008. Extending Earthquakes’ Reach Through Cascading.” Science 319 (5866): 1076–79.
McKenzie, Ed. 1986. Autoregressive Moving-Average Processes with Negative-Binomial and Geometric Marginal Distributions.” Advances in Applied Probability 18 (3): 679–705.
———. 1988. Some ARMA Models for Dependent Sequences of Poisson Counts.” Advances in Applied Probability 20 (4): 822–35.
McKenzie, Eddie. 2003. Discrete Variate Time Series.” In Handbook of Statistics, edited by c Raoand and d Shanbhag, 21:573–606. Stochastic Processes: Modelling and Simulation. Elsevier.
Meiners, Matthias. 2009. Weighted Branching and a Pathwise Renewal Equation.” Stochastic Processes and Their Applications 119 (8): 2579–97.
Messel, H. 1952. The Solution of the Fluctuation Problem in Nucleon Cascade Theory: Homogeneous Nuclear Matter.” Proceedings of the Physical Society. Section A 65 (7): 465.
Messel, H., and R. B. Potts. 1952. Note on the Fluctuation Problem in Cascade Theory.” Proceedings of the Physical Society. Section A 65 (10): 854.
Mishra, Swapnil, Marian-Andrei Rizoiu, and Lexing Xie. 2016. Feature Driven and Point Process Approaches for Popularity Prediction.” In Proceedings of the 25th ACM International Conference on Information and Knowledge Management, 1069–78. CIKM ’16. New York, NY, USA: ACM.
Mohler, G. O., M. B. Short, P. J. Brantingham, F. P. Schoenberg, and G. E. Tita. 2011. Self-Exciting Point Process Modeling of Crime.” Journal of the American Statistical Association 106 (493): 100–108.
Monteiro, Magda, Manuel G. Scotto, and Isabel Pereira. 2012. Integer-Valued Self-Exciting Threshold Autoregressive Processes.” Communications in Statistics - Theory and Methods 41 (15): 2717–37.
Mutafchiev, Ljuben. 1995. Local Limit Approximations for Lagrangian Distributions.” Aequationes Mathematicae 49 (1): 57–85.
Nanthi, K., and M.T. Wasan. 1984. Branching Processes.” Stochastic Processes and Their Applications 18 (2): 189.
Neuts, Marcel F. 1978. Renewal Processes of Phase Type.” Naval Research Logistics Quarterly 25 (3): 445–54.
Neyman, Jerzy. 1965. Certain Chance Mechanisms Involving Discrete Distributions.” Sankhyā: The Indian Journal of Statistics, Series A (1961-2002) 27 (2/4): 249–58.
Nolan, John P. 2001. Maximum Likelihood Estimation and Diagnostics for Stable Distributions.” In Lévy Processes, edited by Ole E. Barndorff-Nielsen, Sidney I. Resnick, and Thomas Mikosch, 379–400. Birkhäuser Boston.
Oakes, David. 1975. The Markovian Self-Exciting Process.” Journal of Applied Probability 12 (1): 69.
Ogata, Y. 1999. Seismicity Analysis Through Point-Process Modeling: A Review.” Pure and Applied Geophysics 155 (2-4): 471–507.
Ogata, Yoshiko. 1978. The Asymptotic Behaviour of Maximum Likelihood Estimators for Stationary Point Processes.” Annals of the Institute of Statistical Mathematics 30 (1): 243–61.
Ogata, Yosihiko. 1988. Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes.” Journal of the American Statistical Association 83 (401): 9–27.
Ogata, Yosihiko, and Hirotugu Akaike. 1982. On Linear Intensity Models for Mixed Doubly Stochastic Poisson and Self-Exciting Point Processes.” Journal of the Royal Statistical Society, Series B 44: 269–74.
Olofsson, Peter. 2005. Probability, Statistics, and Stochastic Processes. Hoboken, N.J: Hoboken, N.J. : Wiley-Interscience.
Otter, Richard. 1948. The Number of Trees.” Annals of Mathematics 49 (3): 583–99.
———. 1949. The Multiplicative Process.” The Annals of Mathematical Statistics 20 (2): 206–24.
Overbeck, Ludger. 1998. Estimation for Continuous Branching Processes.” Scandinavian Journal of Statistics 25 (1): 111–26.
Ozaki, T. 1979. Maximum Likelihood Estimation of Hawkes’ Self-Exciting Point Processes.” Annals of the Institute of Statistical Mathematics 31 (1): 145–55.
Pakes, A. G. 1971a. On the Critical Galton-Watson Process with Immigration.” Journal of the Australian Mathematical Society 12 (4): 476–82.
———. 1971b. On a Theorem of Quine and Seneta for the Galton-Watson Process With Immigration.” Australian Journal of Statistics 13 (3): 159–64.
Pardoux, Etienne, and Brice Samegni-Kepgnou. 2016. Large Deviation Principle for Poisson Driven SDEs in Epidemic Models.” arXiv:1606.01619 [Math], June.
———. 2017. Large Deviation Principle for Epidemic Models.” Journal of Applied Probability 54 (3): 905–20.
Pazsit, I. 1987. Note on the Calculation of the Variance in Linear Collision Cascades.” Journal of Physics D: Applied Physics 20 (2): 151.
Pinto, Julio Cesar Louzada, and Tijani Chahed. 2014. Modeling Multi-Topic Information Diffusion in Social Networks Using Latent Dirichlet Allocation and Hawkes Processes.” In Proceedings of the 2014 Tenth International Conference on Signal-Image Technology and Internet-Based Systems, 339–46. SITIS ’14. Washington, DC, USA: IEEE Computer Society.
Podolskij, Mark, and Mathias Vetter. 2010. Understanding Limit Theorems for Semimartingales: A Short Survey: Limit Theorems for Semimartingales.” Statistica Neerlandica 64 (3): 329–51.
Ramakrishnan, Alladi, and S. K. Srinivasan. 1956. A New Approach to the Cascade Theory.” In Proceedings of the Indian Academy of Sciences-Section A, 44:263–73. Springer.
Rasmussen, Carl Edward, and Christopher K. I. Williams. 2006. Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning. Cambridge, Mass: MIT Press.
Reynaud-Bouret, Patricia, Vincent Rivoirard, Franck Grammont, and Christine Tuleau-Malot. 2014. Goodness-of-Fit Tests and Nonparametric Adaptive Estimation for Spike Train Analysis.” The Journal of Mathematical Neuroscience 4 (1): 3.
Reynaud-Bouret, Patricia, and Emmanuel Roy. 2007. “Some Non Asymptotic Tail Estimates for Hawkes Processes.” Bulletin of the Belgian Mathematical Society - Simon Stevin 13 (5): 883–96.
Reynaud-Bouret, Patricia, and Sophie Schbath. 2010. Adaptive Estimation for Hawkes Processes; Application to Genome Analysis.” The Annals of Statistics 38 (5): 2781–2822.
Riabiz, Marina, Tohid Ardeshiri, and Simon Godsill. 2016. A Central Limit Theorem with Application to Inference in α-Stable Regression Models.” In, 70–82.
Rizoiu, Marian-Andrei, Lexing Xie, Scott Sanner, Manuel Cebrian, Honglin Yu, and Pascal Van Hentenryck. 2017. Expecting to Be HIP: Hawkes Intensity Processes for Social Media Popularity.” In World Wide Web 2017, International Conference on, 1–9. WWW ’17. Perth, Australia: International World Wide Web Conferences Steering Committee.
Saichev, A. I., and D. Sornette. 2010. Generation-by-Generation Dissection of the Response Function in Long Memory Epidemic Processes.” The European Physical Journal B 75 (3): 343–55.
Saichev, A., A. Helmstetter, and D. Sornette. 2005. Power-Law Distributions of Offspring and Generation Numbers in Branching Models of Earthquake Triggering.” Pure and Applied Geophysics 162 (6-7): 1113–34.
Saichev, A., Y. Malevergne, and D. Sornette. 2008. Theory of Zipf’s Law and of General Power Law Distributions with Gibrat’s Law of Proportional Growth.” arXiv:0808.1828 [Physics, q-Fin], August.
Saichev, A., and D. Sornette. 2011a. Hierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes.” arXiv:1101.1611 [Cond-Mat, Physics:physics], January.
———. 2011b. Generating Functions and Stability Study of Multivariate Self-Excited Epidemic Processes.” arXiv:1101.5564 [Cond-Mat, Physics:physics], January.
Sandkühler, J., and A. A. Eblen-Zajjur. 1994. Identification and Characterization of Rhythmic Nociceptive and Non-Nociceptive Spinal Dorsal Horn Neurons in the Rat.” Neuroscience 61 (4): 991–1006.
Sevast’yanov, B. A. 1968. Renewal Equations and Moments of Branching Processes.” Mathematical Notes of the Academy of Sciences of the USSR 3 (1): 3–10.
Shoukri, M. M., and P. C. Consul. 1987. Some Chance Mechanisms Generating the Generalized Poisson Probability Models.” In Biostatistics, edited by Ian B. MacNeill, Gary J. Umphrey, Allan Donner, and V. Krishna Jandhyala, 259–68. Dordrecht: Springer Netherlands.
Sibuya, Masaaki, Norihiko Miyawaki, and Ushio Sumita. 1994. Aspects of Lagrangian Probability Distributions.” Journal of Applied Probability 31: 185–97.
Soltani, A. R., A. Shirvani, and F. Alqallaf. 2009. A Class of Discrete Distributions Induced by Stable Laws.” Statistics & Probability Letters 79 (14): 1608–14.
Sood, Vishal, Myléne Mathieu, Amer Shreim, Peter Grassberger, and Maya Paczuski. 2010. Interacting Branching Process as a Simple Model of Innovation.” Physical Review Letters 105 (17): 178701.
Sornette, D, and A Helmstetter. 2003. Endogenous Versus Exogenous Shocks in Systems with Memory.” Physica A: Statistical Mechanics and Its Applications 318 (3–4): 577–91.
Sornette, Didier. 2006. Endogenous Versus Exogenous Origins of Crises.” In Extreme Events in Nature and Society, 95–119. The Frontiers Collection. Springer.
Sornette, Didier, Fabrice Deschâtres, Thomas Gilbert, and Yann Ageon. 2004. Endogenous Versus Exogenous Shocks in Complex Networks: An Empirical Test Using Book Sale Rankings.” Physical Review Letters 93 (22): 228701.
Sornette, D., Y. Malevergne, and J.-F. Muzy. 2004. Volatility Fingerprints of Large Shocks: Endogenous Versus Exogenous.” In The Application of Econophysics, edited by Hideki Takayasu, 91–102. Springer Japan.
Sornette, D., and S. Utkin. 2009. Limits of Declustering Methods for Disentangling Exogenous from Endogenous Events in Time Series with Foreshocks, Main Shocks, and Aftershocks.” Physical Review E 79 (6): 061110.
Steutel, F. W., and K. van Harn. 1979. Discrete Analogues of Self-Decomposability and Stability.” The Annals of Probability 7 (5): 893–99.
Turkman, Kamil Feridun, Manuel González Scotto, and Patrícia de Zea Bermudez. 2014. “Models for Integer-Valued Time Series.” In Non-Linear Time Series, 199–244. Springer International Publishing.
Unser, Michael A., and Pouya Tafti. 2014. An Introduction to Sparse Stochastic Processes. New York: Cambridge University Press.
Veen, Alejandro, and Frederic P Schoenberg. 2008. Estimation of Space–Time Branching Process Models in Seismology Using an EM–Type Algorithm.” Journal of the American Statistical Association 103 (482): 614–24.
Watanabe, Shinzo. 1968. A Limit Theorem of Branching Processes and Continuous State Branching Processes.” Journal of Mathematics of Kyoto University 8 (1): 141–67.
Wei, C. Z., and J. Winnicki. 1990. Estimation of the Means in the Branching Process with Immigration.” The Annals of Statistics 18 (4): 1757–73.
Weiner, H. J. 1965. An Integral Equation in Age Dependent Branching Processes.” The Annals of Mathematical Statistics 36 (5): 1569–73.
Weiß, Christian H. 2008. Thinning Operations for Modeling Time Series of Counts—a Survey.” Advances in Statistical Analysis 92 (3): 319–41.
———. 2009. A New Class of Autoregressive Models for Time Series of Binomial Counts.” Communications in Statistics - Theory and Methods 38 (4): 447–60.
Wheatley, Spencer. 2013. “Quantifying Endogeneity in Market Prices with Point Processes: Methods & Applications.” Masters Thesis. ETH Zürich.
Winnicki, J. 1991. Estimation of the Variances in the Branching Process with Immigration.” Probability Theory and Related Fields 88 (1): 77–106.
Yaari, G., A. Nowak, K. Rakocy, and S. Solomon. 2008. Microscopic Study Reveals the Singular Origins of Growth.” The European Physical Journal B 62 (4): 505–13.
Yang, Shuang-Hong, and Hongyuan Zha. 2013. Mixture of Mutually Exciting Processes for Viral Diffusion. In Proceedings of The 30th International Conference on Machine Learning, 28:1–9.
Zeger, Scott L. 1988. A Regression Model for Time Series of Counts.” Biometrika 75 (4): 621–29.
Zeger, Scott L., and Bahjat Qaqish. 1988. Markov Regression Models for Time Series: A Quasi-Likelihood Approach.” Biometrics 44 (4): 1019–31.
Zhao, Zhizhen, and Amit Singer. 2013. Fourier–Bessel Rotational Invariant Eigenimages.” Journal of the Optical Society of America A 30 (5): 871.
Zheng, Haitao, and Ishwar V. Basawa. 2008. First-Order Observation-Driven Integer-Valued Autoregressive Processes.” Statistics & Probability Letters 78 (1): 1–9.
Zheng, Haitao, Ishwar V. Basawa, and Somnath Datta. 2007. First-Order Random Coefficient Integer-Valued Autoregressive Processes.” Journal of Statistical Planning and Inference 137 (1): 212–29.
Zhou, Ke, Hongyuan Zha, and Le Song. 2013. Learning Triggering Kernels for Multi-Dimensional Hawkes Processes.” In Proceedings of the 30th International Conference on Machine Learning (ICML-13), 1301–9.

No comments yet. Why not leave one?

GitHub-flavored Markdown & a sane subset of HTML is supported.