The Living Thing / Notebooks :

Branching processes and their statistics

A diverse class of stochastic models that I am mildly obsessed with.

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.

I will assume linear intensity models except where otherwise indicated.


To learn

Discrete index, discrete state: The Galton-Watson process and friends

There are many standard expositions. Two good ones:

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. Consider also change of time.

See my masters thesis.

Continuous index, continuous state: The CSBP type of Lévy process

Aldous does a no-nonsense expo on these. Super trendy at the moment: It turns out that growing trees is connected in a deep but purportedly simple way to “glueing together” excursions of random processes, oh, and a bunch of trippy fractals and random trees and stuff.

Lee and Hopcraft (LeHJ08) also found an analogous result for discrete state branching processes.

Inference theory

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 even. For CSBPs I am not aware of any specific examples except in very specific constrained cases, notably, the discrete state case, which rather defeats the purpose. Can you do more with general nonparametric Lévy measure inference in this continuous case? How much more? Over98 seems to be one of the few refs. Surely the finance folks are onto this?

Discrete index, continuous state

Umm. Is this well-defined? I suppose so. Can’t find any literature references though. It surely has a fancy name. “Marked Galton-Watson Process”? Some kind of compound Poisson, I imagine.

Special issues for multivariate branching processes

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


Measure-valued state or something? Are these even branching processes, or did they just seem to be so because I ran into them at the end of a branching process seminar? Can’t recall, must investigate later.

Dynk04, Dynk91 and Ethe00 were recommended to me for this.

Classic data sets


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

Spatstat is for spatial point processes.


Aldous, D. (1991) The Continuum Random Tree I. The Annals of Probability, 19(1), 1–28. DOI.
Aldous, D. (1993) The Continuum Random Tree III. The Annals of Probability, 21(1), 248–289. DOI.
Al-Osh, M. A., & Aly, E.-E. A. A.(1992) First order autoregressive time series with negative binomial and geometric marginals. Communications in Statistics - Theory and Methods, 21(9), 2483–2492. DOI.
Al-Osh, M. A., & Alzaid, A. A.(1987) First-Order Integer-Valued Autoregressive (INAR(1)) Process. Journal of Time Series Analysis, 8(3), 261–275. DOI.
Aly, E.-E. A. A., & Bouzar, N. (2005) Stationary solutions for integer-valued autoregressive processes. International Journal of Mathematics and Mathematical Sciences, 2005(1), 1–18. DOI.
Alzaid, A., & Al-Osh, M. (1988) First-Order Integer-Valued Autoregressive (INAR (1)) Process: Distributional and Regression Properties. Statistica Neerlandica, 42(1), 53–61. DOI.
Applebaum, D. (2004) Lévy processes-from probability to finance and quantum groups. Notices of the AMS, 51(11), 1336–1347.
Aragón, T. J.(2012) Applied epidemiology using R. . MedEpi Publishing. http://www. medepi. net/epir/index. html. Calendar Time. Accessed
Athreya, K. B., & Keiding, N. (1977) Estimation Theory for Continuous-Time Branching Processes. Sankhyā: The Indian Journal of Statistics, Series A (1961-2002), 39(2), 101–123.
Athreya, K. B., & Vidyashankar, A. N.(1997) Large Deviation Rates for Supercritical and Critical Branching Processes. In K. B. Athreya & P. Jagers (Eds.), Classical and Modern Branching Processes (pp. 1–18). Springer New York
Bacry, E., Dayri, K., & Muzy, J. F.(2012) Non-parametric kernel estimation for symmetric Hawkes processes Application to high frequency financial data. The European Physical Journal B, 85(5), 1–12. DOI.
Bacry, E., Delattre, S., Hoffmann, M., & Muzy, J. F.(2013a) Modelling microstructure noise with mutually exciting point processes. Quantitative Finance, 13(1), 65–77. DOI.
Bacry, E., Delattre, S., Hoffmann, M., & Muzy, J. F.(2013b) Some limit theorems for Hawkes processes and application to financial statistics. Stochastic Processes and Their Applications, 123(7), 2475–2499. DOI.
Bacry, E., Jaisson, T., & Muzy, J.-F. (2014) Estimation of slowly decreasing Hawkes kernels: Application to high frequency order book modelling. arXiv:1412.7096 [Q-Fin, Stat].
Bacry, E., & Muzy, J.-F. (2014a) Hawkes model for price and trades high-frequency dynamics. Quantitative Finance, 14(7), 1147–1166. DOI.
Bacry, E., & Muzy, J.-F. (2014b) Second order statistics characterization of Hawkes processes and non-parametric estimation. arXiv:1401.0903 [Physics, Q-Fin, Stat].
Baddeley, A. (2007) Spatial Point Processes and their Applications. In W. Weil (Ed.), Stochastic Geometry (pp. 1–75). Springer Berlin Heidelberg
Barndorff-Nielsen, O. E., & Sørensen, M. (1994) A Review of Some Aspects of Asymptotic Likelihood Theory for Stochastic Processes. International Statistical Review / Revue Internationale de Statistique, 62(1), 133–165. DOI.
Bhat, B. R., & Adke, S. R.(1981) Maximum Likelihood Estimation for Branching Processes with Immigration. Advances in Applied Probability, 13(3), 498–509. DOI.
Bhattacharjee, M. C.(1987) The Time to Extinction of Branching Processes and Log-Convexity: I. Probability in the Engineering and Informational Sciences, 1(3), 265–278. DOI.
Bibby, B. M., & Sørensen, M. (1995) Martingale Estimation Functions for Discretely Observed Diffusion Processes. Bernoulli, 1(1/2), 17–39. DOI.
Böckenholt, U. (1998) Mixed INAR(1) Poisson regression models: Analyzing heterogeneity and serial dependencies in longitudinal count data. Journal of Econometrics, 89(1–2), 317–338. DOI.
Böttcher, B. (2013) Feller evolution systems: Generators and approximation. Stochastics and Dynamics, 14(3), 1350025. DOI.
Bowman, K. O., & Shenton, L. R.(1989) The distribution of a moment estimator for a parameter of the generalized poision distribution. Communications in Partial Differential Equations, 14(4), 867–893. DOI.
Brown, B. M., & Hewitt, J. I.(1975) Inference for the Diffusion Branching Process. Journal of Applied Probability, 12(3), 588–594. DOI.
Burridge, J. (2013) Cascade sizes in a branching process with Gamma distributed generations. arXiv:1304.3741 [Math].
Caballero, M. E., & Chaumont, L. (2006) Conditioned Stable Lévy Processes and the Lamperti Representation. Journal of Applied Probability, 43(4), 967–983.
Caballero, M. E., Garmendia, J. L. P., & Bravo, G. U.(2013) A Lamperti-type representation of continuous-state branching processes with immigration. The Annals of Probability, 41(3A), 1585–1627. DOI.
Caballero, M.-E., Lambert, A., & Bravo, G. U.(2009) Proof(s) of the Lamperti representation of Continuous-State Branching Processes. Probability Surveys, 6, 62–89. DOI.
Chen, F., & Hall, P. (2016) Nonparametric Estimation for Self-Exciting Point Processes—A Parsimonious Approach. Journal of Computational and Graphical Statistics, 25(1), 209–224. DOI.
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), 640–648. DOI.
Çinlar, E. (1975) Exceptional Paper—Markov Renewal Theory: A Survey. Management Science, 21(7), 727–752. DOI.
Cohn, H. (1997) Stochastic Monotonicity and Branching Processes. In K. B. Athreya & P. Jagers (Eds.), Classical and Modern Branching Processes (pp. 51–56). 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., & Famoye, F. (1992) Generalized poisson regression model. Communications in Statistics - Theory and Methods, 21(1), 89–109. DOI.
Consul, P. C., & Famoye, F. (2006) Lagrangian probability distributions. . Boston: Birkhäuser
Consul, P. C., & Felix, F. (1989) Minimum variance unbiased estimation for the lagrange power series distributions. Statistics, 20(3), 407–415. DOI.
Consul, P. C., & Shenton, L. R.(1973) Some interesting properties of Lagrangian distributions. Communications in Statistics, 2(3), 263–272. DOI.
Consul, P. C., & Shoukri, M. M.(1984) Maximum likelihood estimation for the generalized poisson distribution. Communications in Statistics - Theory and Methods, 13(12), 1533–1547. DOI.
Consul, P. C., & Shoukri, M. M.(1988) Some Chance Mechanisms Related to a Generalized Poisson Probability Model. American Journal of Mathematical and Management Sciences, 8(1–2), 181–202. DOI.
Consul, P., & Shenton, L. (1972) Use of Lagrange Expansion for Generating Discrete Generalized Probability Distributions. SIAM Journal on Applied Mathematics, 23(2), 239–248. DOI.
Crane, R., Schweitzer, F., & Sornette, D. (2010) Power law signature of media exposure in human response waiting time distributions. Physical Review E, 81(5), 56101. DOI.
Crisan, D., Del Moral, P., & Lyons, T. (1999) Discrete filtering using branching and interacting particle systems. Markov Processes and Related Fields, 5(3), 293–318.
Cui, Y., & Lund, R. (2009) A new look at time series of counts. Biometrika, 96(4), 781–792. DOI.
Curien, N., & Le Gall, J.-F. (2013) The Brownian Plane. Journal of Theoretical Probability, 27(4), 1249–1291. DOI.
Daley, D. J., & Vere-Jones, D. (2003) An introduction to the theory of point processes. (2nd ed., Vol. 1. Elementary theory and methods). New York: Springer
Daley, D. J., & Vere-Jones, D. (2008) An introduction to the theory of point processes. (2nd ed., Vol. 2. General theory and structure). New York: Springer
Dassios, A., & Zhao, H. (2011) A dynamic contagion process. Advances in Applied Probability, 43(3), 814–846. DOI.
Dekking, F. M., & Speer, E. R.(1997) On the Shape of the Wavefront of Branching Random Walk. In K. B. Athreya & P. Jagers (Eds.), Classical and Modern Branching Processes (pp. 73–88). Springer New York
Del Moral, P., & Miclo, L. (2000) Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering. In Séminaire de Probabilités XXXIV (pp. 1–145). Springer
Deschâtres, F., & Sornette, D. (2005) Dynamics of book sales: Endogenous versus exogenous shocks in complex networks. Physical Review E, 72(1), 16112. DOI.
Doney, R. A., & Kyprianou, A. E.(2006) Overshoots and undershoots of Lévy processes. The Annals of Applied Probability, 16(1), 91–106. DOI.
Drost, F. C., Akker, R. van den, & Werker, B. J. M.(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), 467–485. DOI.
Du, N., Farajtabar, M., Ahmed, A., Smola, A. J., & Song, L. (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 (pp. 219–228). New York, NY, USA: ACM DOI.
Duembgen, M., & Podolskij, M. (2015) High-frequency asymptotics for path-dependent functionals of Itô semimartingales. Stochastic Processes and Their Applications, 125(4), 1195–1217. DOI.
Dwass, M. (1969) The Total Progeny in a Branching Process and a Related Random Walk. Journal of Applied Probability, 6(3), 682–686. DOI.
Dynkin, E. B.(1991) Branching Particle Systems and Superprocesses. The Annals of Probability, 19(3), 1157–1194. DOI.
Dynkin, E. B.(2004) Superdiffusions and positive solutions of nonlinear partial differential equations. . Providence, R.I: American Mathematical Society
Eden, U., Frank, L., Barbieri, R., Solo, V., & Brown, E. (2004) Dynamic Analysis of Neural Encoding by Point Process Adaptive Filtering. Neural Computation, 16(5), 971–998. DOI.
Embrechts, P., Liniger, T., & Lin, L. (2011) Multivariate Hawkes processes: an application to financial data. Journal of Applied Probability, 48A, 367–378. DOI.
Etheridge, A. (2000) An introduction to superprocesses. . Providence, RI: American Mathematical Society
Evans, S. N.(2008) Probability and real trees. (Vol. 1920). Berlin: Springer
Falkner, N., & Teschl, G. (2012) On the substitution rule for Lebesgue–Stieltjes integrals. Expositiones Mathematicae, 30(4), 412–418. DOI.
Feigin, P. D.(1976) Maximum Likelihood Estimation for Continuous-Time Stochastic Processes. Advances in Applied Probability, 8(4), 712–736. DOI.
Filimonov, V., Bicchetti, D., Maystre, N., & Sornette, D. (2014) Quantification of the high level of endogeneity and of structural regime shifts in commodity markets. Journal of International Money and Finance, 42, 174–192. DOI.
Filimonov, V., Wheatley, S., & Sornette, D. (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. DOI.
Fleet, L. (2014) Networks: Improve your virality. Nature Physics, 10(6), 415–415. DOI.
Fokianos, K. (2011) Some recent progress in count time series. Statistics, 45(1), 49–58. DOI.
Freeland, R. K., & McCabe, B. P. M.(2004) Analysis of low count time series data by Poisson autoregression. Journal of Time Series Analysis, 25(5), 701–722. DOI.
Fukasawa, T., & Basawa, I. V.(2002) Estimation for a class of generalized state-space time series models. Statistics & Probability Letters, 60(4), 459–473. DOI.
Gehler, P. V., Holub, A. D., & Welling, M. (2006) The Rate Adapting Poisson Model for Information Retrieval and Object Recognition. In Proceedings of the 23rd International Conference on Machine Learning (pp. 337–344). New York, NY, USA: ACM DOI.
Geiger, J., & Kauffmann, L. (2004) The Shape of Large Galton-Watson Trees with Possibly Infinite Variance. Random Struct. Algorithms, 25(3), 311–335. DOI.
Godoy, B. I., Solo, V., Min, J., & Pasha, S. A.(2016) Local likelihood estimation of time-variant Hawkes models. (pp. 4199–4203). Presented at the 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE DOI.
Guttorp, P. (1991) Statistical inference for branching processes. . New York: Wiley
Haccou, P., Jagers, P., & Vatutin, V. A.(2005) Branching Processes: Variation, Growth, and Extinction of Populations. . Cambridge: Cambridge University Press
Hall, A., Scotto, M., & Cruz, J. (2009) Extremes of integer-valued moving average sequences. TEST, 19(2), 359–374. DOI.
Halpin, P. F., & Boeck, P. D.(2013) Modelling dyadic Interaction with Hawkes Processes. Psychometrika, 78(4), 793–814. DOI.
Hansen, N. R., Reynaud-Bouret, P., & Rivoirard, V. (2015) Lasso and probabilistic inequalities for multivariate point processes. Bernoulli, 21(1), 83–143. DOI.
Hardiman, S. J., Bercot, N., & Bouchaud, J.-P. (2013) Critical reflexivity in financial markets: a Hawkes process analysis. The European Physical Journal B, 86(10), 1–9. DOI.
Hardiman, S. J., & Bouchaud, J.-P. (2014) Branching-ratio approximation for the self-exciting Hawkes process. Physical Review E, 90(6), 62807. DOI.
Hawkes, A. G.(1971) Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1), 83–90. DOI.
Hawkes, A. G., & Oakes, D. (1974) A cluster process representation of a self-exciting process. Journal of Applied Probability, 11(3), 493. DOI.
Heyde, C. C., & Seneta, E. (2010) Estimation Theory for Growth and Immigration Rates in a Multiplicative Process. In R. Maller, I. Basawa, P. Hall, & E. Seneta (Eds.), Selected Works of C.C. Heyde (pp. 214–235). Springer New York
Imoto, T. (2016) Properties of Lagrangian distributions. Communications in Statistics - Theory and Methods, 45(3), 712–721. DOI.
Iribarren, J. L., & Moro, E. (2011) Branching dynamics of viral information spreading. Physical Review E, 84(4), 46116. DOI.
Jacod, J. (1997) On continuous conditional Gaussian martingales and stable convergence in law. In J. Azéma, M. Yor, & M. Emery (Eds.), Séminaire de Probabilités XXXI (pp. 232–246). Springer Berlin Heidelberg
Jacod, J., Podolskij, M., & Vetter, M. (2010) Limit theorems for moving averages of discretized processes plus noise. The Annals of Statistics, 38(3), 1478–1545. DOI.
Jagers, P. (1969) Renewal theory and the almost sure convergence of branching processes. Arkiv För Matematik, 7(6), 495–504. DOI.
Jagers, P. (1997) Towards Dependence in General Branching Processes. In K. B. Athreya & P. Jagers (Eds.), Classical and Modern Branching Processes (pp. 127–139). Springer New York
János Engländer. (2007) Branching diffusions, superdiffusions and random media. Probability Surveys, 4, 303–364. DOI.
Jánossy, L., & Messel, H. (1950) Fluctuations of the Electron-Photon Cascade - Moments of the Distribution. Proceedings of the Physical Society. Section A, 63(10), 1101. DOI.
Jánossy, L., & Messel, H. (1951) Investigation into the Higher Moments of a Nucleon Cascade. Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, 54, 245–262.
Kedem, B., & Fokianos, K. (2002) Regression models for time series analysis. . Chichester; Hoboken, NJ: John Wiley & Sons
Keener, R. W.(2009) Curved Exponential Families. In Theoretical Statistics (pp. 85–99). Springer New York
Kesten, H. (1973) Random difference equations and Renewal theory for products of random matrices. Acta Mathematica, 131(1), 207–248. DOI.
Kraus, A., & Panaretos, V. M.(2014) Frequentist estimation of an epidemic’s spreading potential when observations are scarce. Biometrika, 101(1), 141–154. DOI.
Kvitkovičová, A., & Panaretos, V. M.(2011) Asymptotic inference for partially observed branching processes. Advances in Applied Probability, 43(4), 1166–1190. DOI.
Lakshmanan, K. C., Sadtler, P. T., Tyler-Kabara, E. C., Batista, A. P., & Yu, B. M.(2015) Extracting Low-Dimensional Latent Structure from Time Series in the Presence of Delays. Neural Computation, 27(9), 1825–1856. DOI.
Lamperti, J. (1967a) Continuous-state branching processes. Bull. Amer. Math. Soc, 73(3), 382–386.
Lamperti, J. (1967b) The Limit of a Sequence of Branching Processes. Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete, 7(4), 271–288. DOI.
Laredo, C., David, O., & Garnier, A. (2009) Inference for Partially Observed Multitype Branching Processes and Ecological Applications. arXiv:0902.4520 [Stat].
Latour, A. (1998) Existence and Stochastic Structure of a Non-negative Integer-valued Autoregressive Process. Journal of Time Series Analysis, 19(4), 439–455. DOI.
Laub, P. J., Taimre, T., & Pollett, P. K.(2015) Hawkes Processes. arXiv:1507.02822 [Math, Q-Fin, Stat].
Le Gall, J.-F. (2005) Random trees and applications. Probability Surveys, 2, 245–311. DOI.
Le Gall, J.-F. (2013) Uniqueness and universality of the Brownian map. The Annals of Probability, 41(4), 2880–2960. DOI.
Le Gall, J.-F., & Miermont, G. (2012) Scaling limits of random trees and planar maps. Probability and Statistical Physics in Two and More Dimensions, 15, 155–211.
Lee, W. H., Hopcraft, K. I., & Jakeman, E. (2008) Continuous and discrete stable processes. Physical Review E, 77(1), 11109. DOI.
Levina, A., & Herrmann, J. M.(2013) The Abelian distribution. Stochastics and Dynamics, 14(3), 1450001. DOI.
Lewis, E., & Mohler, G. (2011) A nonparametric EM algorithm for multiscale Hawkes processes. Preprint.
Li, S., Famoye, F., & Lee, C. (2010) On the generalized Lagrangian probability distributions. Journal of Probability and Statistical Science, 8(1), 113–123.
Li, Y., & Mykland, P. A.(2007) Are volatility estimators robust with respect to modeling assumptions?. Bernoulli, 13(3), 601–622. DOI.
Li, Z. (2012) Continuous-state branching processes. arXiv:1202.3223 [Math].
Li, Z. (2014) Path-valued branching processes and nonlocal branching superprocesses. The Annals of Probability, 42(1), 41–79. DOI.
Li, Z.-H. (2000) Asymptotic Behaviour of Continuous Time and State Branching Processes. Journal of the Australian Mathematical Society (Series A), 68(1), 68–84. DOI.
Liniger, T. J.(2009) Multivariate Hawkes processes. . Diss., Eidgenössische Technische Hochschule ETH Zürich, Nr. 18403, 2009
Lyons, R. (1990) Random Walks and Percolation on Trees. The Annals of Probability, 18(3), 931–958. DOI.
Lyons, R. (2011) Probability on trees and networks.
Marsan, D., & Lengliné, O. (2008) Extending earthquakes’ reach through cascading. Science, 319(5866), 1076–1079. DOI.
McKenzie, E. (1986) Autoregressive Moving-Average Processes with Negative-Binomial and Geometric Marginal Distributions. Advances in Applied Probability, 18(3), 679–705. DOI.
McKenzie, E. (1988) Some ARMA Models for Dependent Sequences of Poisson Counts. Advances in Applied Probability, 20(4), 822–835. DOI.
McKenzie, E. (2003) Discrete variate time series. In B.-H. of Statistics (Ed.), Handbook of Statistics, C. Raoand D. Shanbhag, Eds., ElsevierScience, Amsterdam, 573–606. MR1973555 (Vol. 21, pp. 573–606). Elsevier
Meiners, M. (2009) Weighted branching and a pathwise renewal equation. Stochastic Processes and Their Applications, 119(8), 2579–2597. DOI.
Messel, H. (1951) On the Fluctuation of a Nucleon Cascade in Homogeneous Nuclear Matter and Calculation of Average Numbers: Part I. Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, 54, 125–135.
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. DOI.
Messel, H., & Potts, R. B.(1952a) Note on the Fluctuation Problem in Cascade Theory. Proceedings of the Physical Society. Section A, 65(10), 854. DOI.
Messel, H., & Potts, R. B.(1952b) The Solution of the Fluctuation Problem in Electron-Photon Shower Theory. Physical Review, 86(6), 847–851. DOI.
Mohler, G. O., Short, M. B., Brantingham, P. J., Schoenberg, F. P., & Tita, G. E.(2011) Self-exciting point process modeling of crime. Journal of the American Statistical Association, 106(493), 100–108. DOI.
Monteiro, M., Scotto, M. G., & Pereira, I. (2012) Integer-Valued Self-Exciting Threshold Autoregressive Processes. Communications in Statistics - Theory and Methods, 41(15), 2717–2737. DOI.
Motoike, I. N., & Imamura, H. T.(2010) Branching pattern formation that reflects the history of signal propagation. Physical Review E, 82(4), 46205. DOI.
Mutafchiev, L. (1995) Local limit approximations for Lagrangian distributions. Aequationes Mathematicae, 49(1), 57–85. DOI.
Nanthi, K., & Wasan, M. T.(1984) Branching processes. Stochastic Processes and Their Applications, 18(2), 189. DOI.
Nastić, A. S., Ristić, M. M., & Bakouch, H. S.(2012) A combined geometric INAR(p) model based on negative binomial thinning. Mathematical and Computer Modelling, 55(5–6), 1665–1672. DOI.
Neuts, M. F.(1978) Renewal processes of phase type. Naval Research Logistics Quarterly, 25(3), 445–454. DOI.
Neyman, J. (1965) Certain Chance Mechanisms Involving Discrete Distributions. Sankhyā: The Indian Journal of Statistics, Series A (1961-2002), 27(2/4), 249–258.
Nolan, J. P.(2001) Maximum Likelihood Estimation and Diagnostics for Stable Distributions. In O. E. Barndorff-Nielsen, S. I. Resnick, & T. Mikosch (Eds.), Lévy Processes (pp. 379–400). Birkhäuser Boston
Oakes, D. (1975) The Markovian self-exciting process. Journal of Applied Probability, 12(1), 69. DOI.
Ogata, Y. (1978) The asymptotic behaviour of maximum likelihood estimators for stationary point processes. Annals of the Institute of Statistical Mathematics, 30(1), 243–261. DOI.
Ogata, Y. (1988) Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association, 83(401), 9–27. DOI.
Ogata, Y. (1999) Seismicity analysis through point-process modeling: a review. Pure and Applied Geophysics, 155(2–4), 471–507. DOI.
Ogata, Y., & Akaike, H. (1982) On linear intensity models for mixed doubly stochastic Poisson and self-exciting point processes. Journal of the Royal Statistical Society, Series B, 44, 269–274. DOI.
Olofsson, P. (2005) Probability, statistics, and stochastic processes. . Hoboken, N.J: Hoboken, N.J.: Wiley-Interscience
Otter, R. (1948) The Number of Trees. Annals of Mathematics, 49(3), 583–599. DOI.
Otter, R. (1949) The Multiplicative Process. The Annals of Mathematical Statistics, 20(2), 206–224. DOI.
Overbeck, L. (1998) Estimation for Continuous Branching Processes. Scandinavian Journal of Statistics, 25(1), 111–126. DOI.
Ozaki, T. (1979) Maximum likelihood estimation of Hawkes’ self-exciting point processes. Annals of the Institute of Statistical Mathematics, 31(1), 145–155. DOI.
Panaretos, V. M., & Kraus, A. (n.d.) Estimating the Spreading Potential of an Epidemic When Observations Are Scarce.
Pazsit, I. (1987) Note on the calculation of the variance in linear collision cascades. Journal of Physics D: Applied Physics, 20(2), 151. DOI.
Pinto, J. C. L., & Chahed, T. (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 (pp. 339–346). Washington, DC, USA: IEEE Computer Society DOI.
Podolskij, M., & Vetter, M. (2010) Understanding limit theorems for semimartingales: a short survey: Limit theorems for semimartingales. Statistica Neerlandica, 64(3), 329–351. DOI.
Priesemann, V., Munk, M. H., & Wibral, M. (2009) Subsampling effects in neuronal avalanche distributions recorded in vivo. BMC Neuroscience, 10(1), 40. DOI.
Ramakrishnan, A., & Srinivasan, S. K.(1956) A new approach to the cascade theory. In Proceedings of the Indian Academy of Sciences-Section A (Vol. 44, pp. 263–273). Springer
Rasmussen, C. E., & Williams, C. K. I.(2006) Gaussian processes for machine learning. . Cambridge, Mass: MIT Press
Reynaud-Bouret, P., & Schbath, S. (2010) Adaptive estimation for Hawkes processes; application to genome analysis. The Annals of Statistics, 38(5), 2781–2822. DOI.
Ristić, M. M., Nastić, A. S., & Bakouch, H. S.(2012) Estimation in an Integer-Valued Autoregressive Process with Negative Binomial Marginals (NBINAR(1)). Communications in Statistics - Theory and Methods, 41(4), 606–618. DOI.
Ruan, Z., Iniguez, G., Karsai, M., & Kertesz, J. (2015) Kinetics of Social Contagion. arXiv:1506.00251 [Physics].
Saichev, A., Helmstetter, A., & Sornette, D. (2005) Power-law Distributions of Offspring and Generation Numbers in Branching Models of Earthquake Triggering. Pure and Applied Geophysics, 162(6–7), 1113–1134. DOI.
Saichev, A. I., & Sornette, D. (2010) Generation-by-generation dissection of the response function in long memory epidemic processes. The European Physical Journal B, 75(3), 343–355. DOI.
Saichev, A., Malevergne, Y., & Sornette, D. (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].
Saichev, A., & Sornette, D. (2011a) Generating functions and stability study of multivariate self-excited epidemic processes. arXiv:1101.5564 [Cond-Mat, Physics:physics].
Saichev, A., & Sornette, D. (2011b) Hierarchy of temporal responses of multivariate self-excited epidemic processes. arXiv:1101.1611 [Cond-Mat, Physics:physics].
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. DOI.
Shoukri, M. M., & Consul, P. C.(1987) Some Chance Mechanisms Generating the Generalized Poisson Probability Models. In I. B. MacNeill, G. J. Umphrey, A. Donner, & V. K. Jandhyala (Eds.), Biostatistics (pp. 259–268). Dordrecht: Springer Netherlands
Sibuya, M., Miyawaki, N., & Sumita, U. (1994) Aspects of Lagrangian Probability Distributions. Journal of Applied Probability, 31, 185–197. DOI.
Silva, I., & Silva, M. E.(2006) Asymptotic distribution of the Yule–Walker estimator for INAR processes. Statistics & Probability Letters, 76(15), 1655–1663. DOI.
Soltani, A. R., Shirvani, A., & Alqallaf, F. (2009) A class of discrete distributions induced by stable laws. Statistics & Probability Letters, 79(14), 1608–1614. DOI.
Sood, V., Mathieu, M., Shreim, A., Grassberger, P., & Paczuski, M. (2010) Interacting branching process as a simple model of innovation. Physical Review Letters, 105(17), 178701. DOI.
Sornette, D. (2006) Endogenous versus exogenous origins of crises. In Extreme events in nature and society (pp. 95–119). Springer
Sornette, D., Deschâtres, F., Gilbert, T., & Ageon, Y. (2004) Endogenous versus exogenous shocks in complex networks: An empirical test using book sale rankings. Physical Review Letters, 93(22), 228701. DOI.
Sornette, D., & Helmstetter, A. (2003) Endogenous versus exogenous shocks in systems with memory. Physica A: Statistical Mechanics and Its Applications, 318(3–4), 577–591. DOI.
Sornette, D., Malevergne, Y., & Muzy, J. F.(2002) Volatility fingerprints of large shocks: Endogeneous versus exogeneous. arXiv:cond-Mat/0204626.
Sornette, D., Malevergne, Y., & Muzy, J.-F. (2004) Volatility fingerprints of large shocks: endogenous versus exogenous. In H. Takayasu (Ed.), The Application of Econophysics (pp. 91–102). Springer Japan
Sornette, D., & Utkin, S. (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), 61110. DOI.
Steutel, F. W., & van Harn, K. (1979) Discrete Analogues of Self-Decomposability and Stability. The Annals of Probability, 7(5), 893–899. DOI.
The Theano Development Team, Al-Rfou, R., Alain, G., Almahairi, A., Angermueller, C., Bahdanau, D., … Zhang, Y. (2016) Theano: A Python framework for fast computation of mathematical expressions. arXiv:1605.02688 [Cs].
Turkman, K. F., Scotto, M. G., & Bermudez, P. de Z. (2014) Models for Integer-Valued Time Series. In Non-Linear Time Series (pp. 199–244). Springer International Publishing
Unser, M. A., & Tafti, P. (2014) An introduction to sparse stochastic processes. . New York: Cambridge University Press
van Harn, K., & Steutel, F. W.(1993) Stability equations for processes with stationary independent increments using branching processes and Poisson mixtures. Stochastic Processes and Their Applications, 45(2), 209–230. DOI.
van Harn, K., Steutel, F. W., & Vervaat, W. (1982) Self-decomposable discrete distributions and branching processes. Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete, 61(1), 97–118. DOI.
Veen, A., & Schoenberg, F. P.(2008) Estimation of Space–Time Branching Process Models in Seismology Using an EM–Type Algorithm. Journal of the American Statistical Association, 103(482), 614–624. DOI.
Watanabe, S. (1968) A limit theorem of branching processes and continuous state branching processes. Journal of Mathematics of Kyoto University, 8(1), 141–167.
Wei, C. Z., & Winnicki, J. (1990) Estimation of the Means in the Branching Process with Immigration. The Annals of Statistics, 18(4), 1757–1773. DOI.
Weiner, H. J.(1965) An Integral Equation in Age Dependent Branching Processes. The Annals of Mathematical Statistics, 36(5), 1569–1573. DOI.
Weiß, C. H.(2008) Thinning operations for modeling time series of counts—a survey. AStA Advances in Statistical Analysis, 92(3), 319–341. DOI.
Weiß, C. H.(2009) A New Class of Autoregressive Models for Time Series of Binomial Counts. Communications in Statistics - Theory and Methods, 38(4), 447–460. DOI.
Wheatley, S. (2013, July) 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. DOI.
Yaari, G., Nowak, A., Rakocy, K., & Solomon, S. (2008) Microscopic study reveals the singular origins of growth. The European Physical Journal B, 62(4), 505–513. DOI.
Yang, S.-H., & Zha, H. (2013) Mixture of Mutually Exciting Processes for Viral Diffusion. In Proceedings of The 30th International Conference on Machine Learning (Vol. 28, pp. 1–9).
Zeger, S. L.(1988) A regression model for time series of counts. Biometrika, 75(4), 621–629. DOI.
Zeger, S. L., & Qaqish, B. (1988) Markov Regression Models for Time Series: A Quasi-Likelihood Approach. Biometrics, 44(4), 1019–1031. DOI.
Zhao, Z., & Singer, A. (2013) Fourier–Bessel rotational invariant eigenimages. Journal of the Optical Society of America A, 30(5), 871. DOI.
Zheng, H., & Basawa, I. V.(2008) First-order observation-driven integer-valued autoregressive processes. Statistics & Probability Letters, 78(1), 1–9. DOI.
Zheng, H., Basawa, I. V., & Datta, S. (2006) Inference for pth-order random coefficient integer-valued autoregressive processes. Journal of Time Series Analysis, 27(3), 411–440. DOI.
Zheng, H., Basawa, I. V., & Datta, S. (2007) First-order random coefficient integer-valued autoregressive processes. Journal of Statistical Planning and Inference, 137(1), 212–229. DOI.