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
 Basic handling of process defined on a multidimensional index set, i.e. spacetime 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. Amazingly this works in continuous state spaces too.
 Theory of inference for the continuous state case; (Is there any? Perhaps in the Lévy process literature?)
 connection to the nonstochastic epidemiological SIR DE models (What to they do about noise?)
 Connection to stable processes and Lévy processes.
Discrete index, discrete state: The GaltonWatson process and friends
There are many standard expositions. Two good ones:
This section got long enough to break out separately. See my notes on longmemory GaltonWatson process.
Continuous index, discrete state: the Hawkes Process
If you have a integervalued 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.
As an example, here is a timebased one, the Hawkes process.
We start with the log likelihood of a generic point process, with occurrence times
I recall the the intensity for the Hawkes process in in particular:
where is the influence kernel with parameter , the branching ratio, and the star is shorthand for .
Timeinhomogeneous extension
My nonparametric of choice here will be of convolution kernel type. That is, I will introduce an additional convolution kernel , and functions of the form
for some set of kernel bandwidths , kernel weights , kernel locations .
There are many kernels available. We start with the top hat kernel, the piecewiseconstant function.
giving the following background intensity
I augment the parameter vector to include the kernel weights We could also try to infer kernel locations and bandwidths.
The hypothesized generating model now has conditional intensity process
Continuous index, continuous state: The CSBP type of Lévy process
Aldous gives an 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.
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 wellunderstood estimation theory, with twiddly flourishes on top even. For CSBPs I am not aware of any specific examples except 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 welldefined? I suppose so. Can't find any literature references though. It surely has a fancy name. “Marked GaltonWatson Process”? Some kind of compound Poisson, I imagine.
Special issues for multivariate branching processes
If you are looking at crossexcitation between variables then I have some additional matter at contagion processes.
Superprocesses
Measurevalued 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

Tomás Aragón's free online epidemiology textbook (Arag12) lists, among others,

In the R package
tscount
one may findcampy
,ecoli
,ehec
,influenza
andmeasles
. 
Gapminder hosts some disease datasets including tubercolosis, malaria, and diarrhoea .

If you include lifecycle questions in your model (death and birth rates) you might use population data sets. The UN department of Economic and Social Affairs has global population data sets.
Implementations
IHSEP is Feng Chen's software to continuous index, discrete state branching processes.
Spatstat is for spatial point processes.
Refs
 RiAG16: Marina Riabiz, Tohid Ardeshiri, Simon Godsill (2016) A central limit theorem with application to inference in αstable regression models. (pp. 70–82).
 SoSA09: A. R. Soltani, A. Shirvani, F. Alqallaf (2009) A class of discrete distributions induced by stable laws. Statistics & Probability Letters, 79(14), 1608–1614. DOI
 HaOa74: Alan G. Hawkes, David Oakes (1974) A cluster process representation of a selfexciting process. Journal of Applied Probability, 11(3), 493. DOI
 CaGB13: M. Emilia Caballero, José Luis Pérez Garmendia, Gerónimo Uribe Bravo (2013) A Lampertitype representation of continuousstate branching processes with immigration. The Annals of Probability, 41(3A), 1585–1627. DOI
 Wata68: Shinzo Watanabe (1968) A limit theorem of branching processes and continuous state branching processes. Journal of Mathematics of Kyoto University, 8(1), 141–167.
 RaSr56: Alladi Ramakrishnan, S. K. Srinivasan (1956) A new approach to the cascade theory. In Proceedings of the Indian Academy of SciencesSection A (Vol. 44, pp. 263–273). Springer
 Weiß09: Christian H. Weiß (2009) A New Class of Autoregressive Models for Time Series of Binomial Counts. Communications in Statistics  Theory and Methods, 38(4), 447–460. DOI
 CuLu09: Yunwei Cui, Robert Lund (2009) A new look at time series of counts. Biometrika, 96(4), 781–792. DOI
 LeMo11: Erik Lewis, George Mohler (2011) A nonparametric EM algorithm for multiscale Hawkes processes. Preprint.
 Zege88: Scott L. Zeger (1988) A regression model for time series of counts. Biometrika, 75(4), 621–629. DOI
 BaSø94: O. E. BarndorffNielsen, M. Sørensen (1994) A Review of Some Aspects of Asymptotic Likelihood Theory for Stochastic Processes. International Statistical Review / Revue Internationale de Statistique, 62(1), 133–165. DOI
 Chis64: V. Chistyakov (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
 ReSc10: Patricia ReynaudBouret, Sophie Schbath (2010) Adaptive estimation for Hawkes processes; application to genome analysis. The Annals of Statistics, 38(5), 2781–2822. DOI
 Wein65: H. J. Weiner (1965) An Integral Equation in Age Dependent Branching Processes. The Annals of Mathematical Statistics, 36(5), 1569–1573. DOI
 UnTa14: Michael A. Unser, Pouya Tafti (2014) An introduction to sparse stochastic processes. New York: Cambridge University Press
 Ethe00: Alison Etheridge (2000) An introduction to superprocesses. Providence, RI: American Mathematical Society
 DaVe03: Daryl J. Daley, David VereJones (2003) An introduction to the theory of point processes (Vol. 1. Elementary theory and methods). New York: Springer
 DaVe08: Daryl J. Daley, David VereJones (2008) An introduction to the theory of point processes (Vol. 2. General theory and structure). New York: Springer
 FrMc04: R. K. Freeland, B. P. M. McCabe (2004) Analysis of low count time series data by Poisson autoregression. Journal of Time Series Analysis, 25(5), 701–722. DOI
 Arag12: Tomás J. Aragón (2012) Applied epidemiology using R. MedEpi Publishing. http://www. medepi. net/epir/index. html. Calendar Time. Accessed
 LiMy07: Yingying Li, Per A. Mykland (2007) Are volatility estimators robust with respect to modeling assumptions? Bernoulli, 13(3), 601–622. DOI
 SiMS94: Masaaki Sibuya, Norihiko Miyawaki, Ushio Sumita (1994) Aspects of Lagrangian Probability Distributions. Journal of Applied Probability, 31, 185–197. DOI
 Li00: ZengHu Li (2000) Asymptotic Behaviour of Continuous Time and State Branching Processes. Journal of the Australian Mathematical Society (Series A), 68(01), 68–84. DOI
 KvPa11: Andrea Kvitkovičová, Victor M. Panaretos (2011) Asymptotic inference for partially observed branching processes. Advances in Applied Probability, 43(4), 1166–1190. DOI
 Mcke86: Ed McKenzie (1986) Autoregressive MovingAverage Processes with NegativeBinomial and Geometric Marginal Distributions. Advances in Applied Probability, 18(3), 679–705. DOI
 DeMi00: Pierre Del Moral, Laurent Miclo (2000) Branching and interacting particle systems approximations of FeynmanKac formulae with applications to nonlinear filtering. In Séminaire de Probabilités XXXIV (pp. 1–145). Springer
 Jáno07: János Engländer (2007) Branching diffusions, superdiffusions and random media. Probability Surveys, 4, 303–364. DOI
 IrMo11: José Luis Iribarren, Esteban Moro (2011) Branching dynamics of viral information spreading. Physical Review E, 84(4), 046116. DOI
 Dynk91: E. B. Dynkin (1991) Branching Particle Systems and Superprocesses. The Annals of Probability, 19(3), 1157–1194. DOI
 NaWa84: K. Nanthi, M.T. Wasan (1984) Branching processes. Stochastic Processes and Their Applications, 18(2), 189. DOI
 HaJV05: Patsy Haccou, Peter Jagers, Vladimir A. Vatutin (2005) Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge: Cambridge University Press
 HaBo14: Stephen J. Hardiman, JeanPhilippe Bouchaud (2014) Branchingratio approximation for the selfexciting Hawkes process. Physical Review E, 90(6), 062807. DOI
 Burr13: James Burridge (2013) Cascade sizes in a branching process with Gamma distributed generations. ArXiv:1304.3741 [Math].
 Neym65: Jerzy Neyman (1965) Certain Chance Mechanisms Involving Discrete Distributions. Sankhyā: The Indian Journal of Statistics, Series A (19612002), 27(2/4), 249–258.
 CaCh06: M. E. Caballero, L. Chaumont (2006) Conditioned Stable Lévy Processes and the Lamperti Representation. Journal of Applied Probability, 43(4), 967–983.
 LeHJ08: W. H. Lee, K. I. Hopcraft, E. Jakeman (2008) Continuous and discrete stable processes. Physical Review E, 77(1), 011109. DOI
 Lamp67a: John Lamperti (1967a) Continuousstate branching processes. Bull. Amer. Math. Soc, 73(3), 382–386.
 Li12: Zenghu Li (2012) Continuousstate branching processes. ArXiv:1202.3223 [Math].
 HaBB13: Stephen J. Hardiman, Nicolas Bercot, JeanPhilippe Bouchaud (2013) Critical reflexivity in financial markets: a Hawkes process analysis. The European Physical Journal B, 86(10), 1–9. DOI
 Keen09: Robert W. Keener (2009) Curved Exponential Families. In Theoretical Statistics (pp. 85–99). Springer New York DOI
 DFAS15: Nan Du, Mehrdad Farajtabar, Amr Ahmed, Alexander J. Smola, Le Song (2015) DirichletHawkes Processes with Applications to Clustering ContinuousTime 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
 StHa79: F. W. Steutel, K. van Harn (1979) Discrete Analogues of SelfDecomposability and Stability. The Annals of Probability, 7(5), 893–899. DOI
 CrDL99: D Crisan, P Del Moral, T Lyons (1999) Discrete filtering using branching and interacting particle systems. Markov Processes and Related Fields, 5(3), 293–318.
 Mcke03: Eddie McKenzie (2003) Discrete variate time series. In Handbook of Statistics (Vol. 21, pp. 573–606). Elsevier
 EFBS04: U Eden, L Frank, R Barbieri, V Solo, E Brown (2004) Dynamic Analysis of Neural Encoding by Point Process Adaptive Filtering. Neural Computation, 16(5), 971–998. DOI
 DeSo05: Fabrice Deschâtres, Didier Sornette (2005) Dynamics of book sales: Endogenous versus exogenous shocks in complex networks. Physical Review E, 72(1), 016112. DOI
 FiWS15: Vladimir Filimonov, Spencer Wheatley, Didier Sornette (2015) Effective measure of endogeneity for the Autoregressive Conditional Duration point processes via mapping to the selfexcited Hawkes process. Communications in Nonlinear Science and Numerical Simulation, 22(1–3), 23–37. DOI
 DrAW09: Feike C. Drost, Ramon van den Akker, Bas J. M. Werker (2009) Efficient estimation of autoregression parameters and innovation distributions for semiparametric integervalued AR(p) models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 71(2), 467–485. DOI
 Sorn06: Didier Sornette (2006) Endogenous versus exogenous origins of crises. In Extreme events in nature and society (pp. 95–119). Springer
 SDGA04: Didier Sornette, Fabrice Deschâtres, Thomas Gilbert, Yann Ageon (2004) Endogenous versus exogenous shocks in complex networks: An empirical test using book sale rankings. Physical Review Letters, 93(22), 228701. DOI
 SoHe03: D Sornette, A Helmstetter (2003) Endogenous versus exogenous shocks in systems with memory. Physica A: Statistical Mechanics and Its Applications, 318(3–4), 577–591. DOI
 Over98: Ludger Overbeck (1998) Estimation for Continuous Branching Processes. Scandinavian Journal of Statistics, 25(1), 111–126. DOI
 BaJM14: Emmanuel Bacry, Thibault Jaisson, JeanFrancois Muzy (2014) Estimation of slowly decreasing Hawkes kernels: Application to high frequency order book modelling. ArXiv:1412.7096 [qFin, Stat].
 VeSc08: Alejandro Veen, 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), 614–624. DOI
 WeWi90: C. Z. Wei, J. Winnicki (1990) Estimation of the Means in the Branching Process with Immigration. The Annals of Statistics, 18(4), 1757–1773. DOI
 Winn91: J. Winnicki (1991) Estimation of the variances in the branching process with immigration. Probability Theory and Related Fields, 88(1), 77–106. DOI
 AtKe77: K. B. Athreya, Niels Keiding (1977) Estimation Theory for ContinuousTime Branching Processes. Sankhyā: The Indian Journal of Statistics, Series A (19612002), 39(2), 101–123.
 HeSe10: C. C. Heyde, E. Seneta (2010) Estimation Theory for Growth and Immigration Rates in a Multiplicative Process. In Selected Works of C.C. Heyde (pp. 214–235). Springer New York
 Çinl75: Erhan Çinlar (1975) Exceptional Paper—Markov Renewal Theory: A Survey. Management Science, 21(7), 727–752. DOI
 Lato98: Alain Latour (1998) Existence and Stochastic Structure of a Nonnegative Integervalued Autoregressive Process. Journal of Time Series Analysis, 19(4), 439–455. DOI
 RXSC17: MarianAndrei Rizoiu, Lexing Xie, Scott Sanner, Manuel Cebrian, Honglin Yu, Pascal Van Hentenryck (2017) Expecting to be HIP: Hawkes Intensity Processes for Social Media Popularity. In World Wide Web 2017, International Conference on (pp. 1–9). Perth, Australia: International World Wide Web Conferences Steering Committee DOI
 MaLe08: David Marsan, Olivier Lengliné (2008) Extending earthquakes’ reach through cascading. Science, 319(5866), 1076–1079. DOI
 LSTB15: Karthik C. Lakshmanan, Patrick T. Sadtler, Elizabeth C. TylerKabara, Aaron P. Batista, Byron M. Yu (2015) Extracting LowDimensional Latent Structure from Time Series in the Presence of Delays. Neural Computation, 27(9), 1825–1856. DOI
 HaSC09: Andreia Hall, Manuel Scotto, João Cruz (2009) Extremes of integervalued moving average sequences. TEST, 19(2), 359–374. DOI
 MiRX16: Swapnil Mishra, MarianAndrei Rizoiu, 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 (pp. 1069–1078). New York, NY, USA: ACM DOI
 AlAl92: Mohamed A. AlOsh, EmadEldin A. A. Aly (1992) First order autoregressive time series with negative binomial and geometric marginals. Communications in Statistics  Theory and Methods, 21(9), 2483–2492. DOI
 AlAl88: A. Alzaid, M. AlOsh (1988) FirstOrder IntegerValued Autoregressive (INAR (1)) Process: Distributional and Regression Properties. Statistica Neerlandica, 42(1), 53–61. DOI
 AlAl87: M. A. AlOsh, A. A. Alzaid (1987) FirstOrder IntegerValued Autoregressive (INAR(1)) Process. Journal of Time Series Analysis, 8(3), 261–275. DOI
 ZhBa08: Haitao Zheng, Ishwar V. Basawa (2008) Firstorder observationdriven integervalued autoregressive processes. Statistics & Probability Letters, 78(1), 1–9. DOI
 ZhBD07: Haitao Zheng, Ishwar V. Basawa, Somnath Datta (2007) Firstorder random coefficient integervalued autoregressive processes. Journal of Statistical Planning and Inference, 137(1), 212–229. DOI
 JáMe50: L. Jánossy, H. Messel (1950) Fluctuations of the ElectronPhoton Cascade  Moments of the Distribution. Proceedings of the Physical Society. Section A, 63(10), 1101. DOI
 ZhSi13: Zhizhen Zhao, Amit Singer (2013) Fourier–Bessel rotational invariant eigenimages. Journal of the Optical Society of America A, 30(5), 871. DOI
 KrPa14: Andrea Kraus, Victor M. Panaretos (2014) Frequentist estimation of an epidemic’s spreading potential when observations are scarce. Biometrika, 101(1), 141–154. DOI
 RaWi06: Carl Edward Rasmussen, Christopher K. I. Williams (2006) Gaussian processes for machine learning. Cambridge, Mass: MIT Press
 CoFa92: P. C. Consul, Felix Famoye (1992) Generalized poisson regression model. Communications in Statistics  Theory and Methods, 21(1), 89–109. DOI
 SaSo11a: A. Saichev, D. Sornette (2011a) Generating functions and stability study of multivariate selfexcited epidemic processes. ArXiv:1101.5564 [CondMat, Physics:Physics].
 SaSo10: A. I. Saichev, D. Sornette (2010) Generationbygeneration dissection of the response function in long memory epidemic processes. The European Physical Journal B, 75(3), 343–355. DOI
 RRGT14: Patricia ReynaudBouret, Vincent Rivoirard, Franck Grammont, Christine TuleauMalot (2014) GoodnessofFit Tests and Nonparametric Adaptive Estimation for Spike Train Analysis. The Journal of Mathematical Neuroscience, 4(1), 3. DOI
 EiDD16: Michael Eichler, Rainer Dahlhaus, Johannes Dueck (2016) Graphical Modeling for Multivariate Hawkes Processes with Nonparametric Link Functions. Journal of Time Series Analysis, n/an/a. DOI
 BaMu14a: Emmanuel Bacry, JeanFrançois Muzy (2014a) Hawkes model for price and trades highfrequency dynamics. Quantitative Finance, 14(7), 1147–1166. DOI
 LaTP15: Patrick J. Laub, Thomas Taimre, Philip K. Pollett (2015) Hawkes Processes. ArXiv:1507.02822 [Math, qFin, Stat].
 SaSo11b: A. Saichev, D. Sornette (2011b) Hierarchy of temporal responses of multivariate selfexcited epidemic processes. ArXiv:1101.1611 [CondMat, Physics:Physics].
 DuPo15: Moritz Duembgen, Mark Podolskij (2015) Highfrequency asymptotics for pathdependent functionals of Itô semimartingales. Stochastic Processes and Their Applications, 125(4), 1195–1217. DOI
 SaEb94: J. Sandkühler, A. A. EblenZajjur (1994) Identification and characterization of rhythmic nociceptive and nonnociceptive spinal dorsal horn neurons in the rat. Neuroscience, 61(4), 991–1006. DOI
 LaDG09: Catherine Laredo, Olivier David, Aurélie Garnier (2009) Inference for Partially Observed Multitype Branching Processes and Ecological Applications. ArXiv:0902.4520 [Stat].
 BrHe75: B. M. Brown, J. I. Hewitt (1975) Inference for the Diffusion Branching Process. Journal of Applied Probability, 12(3), 588–594. DOI
 MoSP12: Magda Monteiro, Manuel G. Scotto, Isabel Pereira (2012) IntegerValued SelfExciting Threshold Autoregressive Processes. Communications in Statistics  Theory and Methods, 41(15), 2717–2737. DOI
 SMSG10: Vishal Sood, Myléne Mathieu, Amer Shreim, Peter Grassberger, Maya Paczuski (2010) Interacting branching process as a simple model of innovation. Physical Review Letters, 105(17), 178701. DOI
 JáMe51: L. Jánossy, H. Messel (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.
 Cons14: P. C. Consul (2014) Lagrange and Related Probability Distributions. In Wiley StatsRef: Statistics Reference Online. John Wiley & Sons, Ltd
 CoFa06: P. C. Consul, Felix Famoye (2006) Lagrangian probability distributions. Boston: Birkhäuser
 PaSa17: Etienne Pardoux, Brice SamegniKepgnou (2017) Large deviation principle for epidemic models. Journal of Applied Probability, 54(3), 905–920. DOI
 PaSa16: Etienne Pardoux, Brice SamegniKepgnou (2016) Large deviation principle for Poisson driven SDEs in epidemic models. ArXiv:1606.01619 [Math].
 AtVi97: K. B. Athreya, A. N. Vidyashankar (1997) Large Deviation Rates for Supercritical and Critical Branching Processes. In Classical and Modern Branching Processes (pp. 1–18). Springer New York
 KrPa16: Peter Kratz, Etienne Pardoux (2016) Large deviations for infectious diseases models. ArXiv:1602.02803 [Math].
 HaRR15: Niels Richard Hansen, Patricia ReynaudBouret, Vincent Rivoirard (2015) Lasso and probabilistic inequalities for multivariate point processes. Bernoulli, 21(1), 83–143. DOI
 ZhZS13: Ke Zhou, Hongyuan Zha, Le Song (2013) Learning triggering kernels for multidimensional Hawkes processes. In Proceedings of the 30th International Conference on Machine Learning (ICML13) (pp. 1301–1309).
 Appl09: David Applebaum (2009) Lévy processes and stochastic calculus. Cambridge ; New York: Cambridge University Press
 Appl04: David Applebaum (2004) Lévy processesfrom probability to finance and quantum groups. Notices of the AMS, 51(11), 1336–1347.
 JaPV10: Jean Jacod, Mark Podolskij, Mathias Vetter (2010) Limit theorems for moving averages of discretized processes plus noise. The Annals of Statistics, 38(3), 1478–1545. DOI
 SoUt09: D. Sornette, 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. DOI
 GSMP16: Boris I. Godoy, Victor Solo, Jason Min, Syed Ahmed Pasha (2016) Local likelihood estimation of timevariant Hawkes models. (pp. 4199–4203). IEEE DOI
 Muta95: Ljuben Mutafchiev (1995) Local limit approximations for Lagrangian distributions. Aequationes Mathematicae, 49(1), 57–85. DOI
 ZeQa88: Scott L. Zeger, Bahjat Qaqish (1988) Markov Regression Models for Time Series: A QuasiLikelihood Approach. Biometrics, 44(4), 1019–1031. DOI
 BiSø95: Bo Martin Bibby, Michael Sørensen (1995) Martingale Estimation Functions for Discretely Observed Diffusion Processes. Bernoulli, 1(1/2), 17–39. DOI
 Nola01: John P. Nolan (2001) Maximum Likelihood Estimation and Diagnostics for Stable Distributions. In Lévy Processes (pp. 379–400). Birkhäuser Boston DOI
 BhAd81: B. R. Bhat, S. R. Adke (1981) Maximum Likelihood Estimation for Branching Processes with Immigration. Advances in Applied Probability, 13(3), 498–509. DOI
 Feig76: Paul David Feigin (1976) Maximum Likelihood Estimation for ContinuousTime Stochastic Processes. Advances in Applied Probability, 8(4), 712–736. DOI
 CoSh84: P.C. Consul, M. M. Shoukri (1984) Maximum likelihood estimation for the generalized poisson distribution. Communications in Statistics  Theory and Methods, 13(12), 1533–1547. DOI
 Ozak79: T. Ozaki (1979) Maximum likelihood estimation of Hawkes’ selfexciting point processes. Annals of the Institute of Statistical Mathematics, 31(1), 145–155. DOI
 YNRS08: G. Yaari, A. Nowak, K. Rakocy, S. Solomon (2008) Microscopic study reveals the singular origins of growth. The European Physical Journal B, 62(4), 505–513. DOI
 CoFe89: P. C. Consul, Famoye Felix (1989) Minimum variance unbiased estimation for the lagrange power series distributions. Statistics, 20(3), 407–415. DOI
 Böck98: Ulf Böckenholt (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
 YaZh13: ShuangHong Yang, Hongyuan Zha (2013) Mixture of Mutually Exciting Processes for Viral Diffusion. In Proceedings of The 30th International Conference on Machine Learning (Vol. 28, pp. 1–9).
 PiCh14: Julio Cesar Louzada Pinto, Tijani Chahed (2014) Modeling Multitopic Information Diffusion in Social Networks Using Latent Dirichlet Allocation and Hawkes Processes. In Proceedings of the 2014 Tenth International Conference on SignalImage Technology and InternetBased Systems (pp. 339–346). Washington, DC, USA: IEEE Computer Society DOI
 HaBo13: Peter F. Halpin, Paul De Boeck (2013) Modelling dyadic Interaction with Hawkes Processes. Psychometrika, 78(4), 793–814. DOI
 BDHM13a: E. Bacry, S. Delattre, M. Hoffmann, J. F. Muzy (2013a) Modelling microstructure noise with mutually exciting point processes. Quantitative Finance, 13(1), 65–77. DOI
 Bows07: Clive G. Bowsher (2007) Modelling security market events in continuous time: Intensity based, multivariate point process models. Journal of Econometrics, 141(2), 876–912. DOI
 TuSB14: Kamil Feridun Turkman, Manuel González Scotto, Patrícia de Zea Bermudez (2014) Models for IntegerValued Time Series. In NonLinear Time Series (pp. 199–244). Springer International Publishing
 Lini09: Thomas Josef Liniger (2009) Multivariate Hawkes processes. Diss., Eidgenössische Technische Hochschule ETH Zürich, Nr. 18403, 2009
 EmLL11: Paul Embrechts, Thomas Liniger, Lu Lin (2011) Multivariate Hawkes processes: an application to financial data. Journal of Applied Probability, 48A, 367–378. DOI
 ChHa16: Feng Chen, Peter Hall (2016) Nonparametric Estimation for SelfExciting Point Processes—A Parsimonious Approach. Journal of Computational and Graphical Statistics, 25(1), 209–224. DOI
 BaDM12: E. Bacry, K. Dayri, J. F. Muzy (2012) Nonparametric kernel estimation for symmetric Hawkes processes Application to high frequency financial data. The European Physical Journal B, 85(5), 1–12. DOI
 Pazs87: I. Pazsit (1987) Note on the calculation of the variance in linear collision cascades. Journal of Physics D: Applied Physics, 20(2), 151. DOI
 MePo52: H. Messel, R. B. Potts (1952) Note on the Fluctuation Problem in Cascade Theory. Proceedings of the Physical Society. Section A, 65(10), 854. DOI
 Pake71a: A. G. Pakes (1971a) On a Theorem of Quine and Seneta for the GaltonWatson Process With Immigration. Australian Journal of Statistics, 13(3), 159–164. DOI
 Jaco97: Jean Jacod (1997) On continuous conditional Gaussian martingales and stable convergence in law. In Séminaire de Probabilités XXXI (pp. 232–246). Springer Berlin Heidelberg
 OgAk82: Yosihiko Ogata, Hirotugu Akaike (1982) On linear intensity models for mixed doubly stochastic Poisson and selfexciting point processes. Journal of the Royal Statistical Society, Series B, 44, 269–274. DOI
 BeYO01: Jean Bertoin, Marc Yor, others (2001) On subordinators, selfsimilar Markov processes and some factorizations of the exponential variable. Electron. Comm. Probab, 6(95), 106.
 Pake71b: A. G. Pakes (1971b) On the critical GaltonWatson process with immigration. Journal of the Australian Mathematical Society, 12(4), 476–482. DOI
 LiFL10: S Li, F Famoye, C Lee (2010) On the generalized Lagrangian probability distributions. Journal of Probability and Statistical Science, 8(1), 113–123.
 DeSp97: F. M. Dekking, E. R. Speer (1997) On the Shape of the Wavefront of Branching Random Walk. In Classical and Modern Branching Processes (pp. 73–88). Springer New York
 FaTe12: Neil Falkner, Gerald Teschl (2012) On the substitution rule for Lebesgue–Stieltjes integrals. Expositiones Mathematicae, 30(4), 412–418. DOI
 DoKy06: R. A. Doney, A. E. Kyprianou (2006) Overshoots and undershoots of Lévy processes. The Annals of Applied Probability, 16(1), 91–106. DOI
 Li14: Zenghu Li (2014) Pathvalued branching processes and nonlocal branching superprocesses. The Annals of Probability, 42(1), 41–79. DOI
 CrSS10: Riley Crane, Frank Schweitzer, Didier Sornette (2010) Power law signature of media exposure in human response waiting time distributions. Physical Review E, 81(5), 056101. DOI
 SaHS05: A. Saichev, A. Helmstetter, D. Sornette (2005) Powerlaw Distributions of Offspring and Generation Numbers in Branching Models of Earthquake Triggering. Pure and Applied Geophysics, 162(6–7), 1113–1134. DOI
 Evan08: Steven N. Evans (2008) Probability and real trees (Vol. 1920). Berlin: Springer
 Olof05: Peter Olofsson (2005) Probability, statistics, and stochastic processes. Hoboken, N.J: Hoboken, N.J. : WileyInterscience
 CaLB09: MariaEmilia Caballero, Amaury Lambert, Geronimo Uribe Bravo (2009) Proof(s) of the Lamperti representation of ContinuousState Branching Processes. Probability Surveys, 6, 62–89. DOI
 Imot16: Tomoaki Imoto (2016) Properties of Lagrangian distributions. Communications in Statistics  Theory and Methods, 45(3), 712–721. DOI
 FBMS14: Vladimir Filimonov, David Bicchetti, Nicolas Maystre, Didier Sornette (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
 Whea13: Spencer Wheatley (2013, July) Quantifying endogeneity in market prices with point processes: methods & applications. Masters Thesis, ETH Zürich
 Kest73: Harry Kesten (1973) Random difference equations and Renewal theory for products of random matrices. Acta Mathematica, 131(1), 207–248. DOI
 Lega05: JeanFrançois Le Gall (2005) Random trees and applications. Probability Surveys, 2, 245–311. DOI
 Lyon90: Russell Lyons (1990) Random Walks and Percolation on Trees. The Annals of Probability, 18(3), 931–958. DOI
 KeFo02: Benjamin Kedem, Konstantinos Fokianos (2002) Regression models for time series analysis. Chichester; Hoboken, NJ: John Wiley & Sons
 Houd02: Christian Houdré (2002) Remarks on deviation inequalities for functions of infinitely divisible random vectors. The Annals of Probability, 30(3), 1223–1237. DOI
 Seva68: B. A. Sevast’yanov (1968) Renewal equations and moments of branching processes. Mathematical Notes of the Academy of Sciences of the USSR, 3(1), 3–10. DOI
 Neut78: Marcel F. Neuts (1978) Renewal processes of phase type. Naval Research Logistics Quarterly, 25(3), 445–454. DOI
 Jage69: Peter Jagers (1969) Renewal theory and the almost sure convergence of branching processes. Arkiv För Matematik, 7(6), 495–504. DOI
 LeMi12: JeanFrançois Le Gall, Grégory Miermont (2012) Scaling limits of random trees and planar maps. Probability and Statistical Physics in Two and More Dimensions, 15, 155–211.
 BaMu14b: Emmanuel Bacry, JeanFrancois Muzy (2014b) Second order statistics characterization of Hawkes processes and nonparametric estimation. ArXiv:1401.0903 [Physics, qFin, Stat].
 Ogat99: Y. Ogata (1999) Seismicity analysis through pointprocess modeling: a review. Pure and Applied Geophysics, 155(2–4), 471–507. DOI
 HaSV82: K. van Harn, F. W. Steutel, W. Vervaat (1982) Selfdecomposable discrete distributions and branching processes. Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete, 61(1), 97–118. DOI
 MSBS11: G. O. Mohler, M. B. Short, P. J. Brantingham, F. P. Schoenberg, G. E. Tita (2011) Selfexciting point process modeling of crime. Journal of the American Statistical Association, 106(493), 100–108. DOI
 Mcke88: Ed McKenzie (1988) Some ARMA Models for Dependent Sequences of Poisson Counts. Advances in Applied Probability, 20(4), 822–835. DOI
 ShCo87: M. M. Shoukri, P. C. Consul (1987) Some Chance Mechanisms Generating the Generalized Poisson Probability Models. In Biostatistics (pp. 259–268). Dordrecht: Springer Netherlands
 CoSh88: P.C. Consul, 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. DOI
 CoSh73: P. C. Consul, L. R. Shenton (1973) Some interesting properties of Lagrangian distributions. Communications in Statistics, 2(3), 263–272. DOI
 BDHM13b: E. Bacry, S. Delattre, M. Hoffmann, J. F. Muzy (2013b) Some limit theorems for Hawkes processes and application to financial statistics. Stochastic Processes and Their Applications, 123(7), 2475–2499. DOI
 ReRo07: Patricia ReynaudBouret, Emmanuel Roy (2007) Some non asymptotic tail estimates for Hawkes processes. Bulletin of the Belgian Mathematical Society  Simon Stevin, 13(5), 883–896.
 Foki11: Konstantinos Fokianos (2011) Some recent progress in count time series. Statistics, 45(1), 49–58. DOI
 Badd07: Adrian Baddeley (2007) Spatial Point Processes and their Applications. In Stochastic Geometry (pp. 1–75). Springer Berlin Heidelberg
 Hawk71: Alan G. Hawkes (1971) Spectra of some selfexciting and mutually exciting point processes. Biometrika, 58(1), 83–90. DOI
 HaSt93: K. van Harn, 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), 209–230. DOI
 AlBo05: EmadEldin A. A. Aly, Nadjib Bouzar (2005) Stationary solutions for integervalued autoregressive processes. International Journal of Mathematics and Mathematical Sciences, 2005(1), 1–18. DOI
 Gutt91: Peter Guttorp (1991) Statistical inference for branching processes. New York: Wiley
 Ogat88: Yosihiko Ogata (1988) Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association, 83(401), 9–27. DOI
 BaHa06: Luc Bauwens, Nikolaus Hautsch (2006) Stochastic Conditional Intensity Processes. Journal of Financial Econometrics, 4(3), 450–493. DOI
 Cohn97: Harry Cohn (1997) Stochastic Monotonicity and Branching Processes. In Classical and Modern Branching Processes (pp. 51–56). Springer New York
 PrMW09: Viola Priesemann, Matthias HJ Munk, Michael Wibral (2009) Subsampling effects in neuronal avalanche distributions recorded in vivo. BMC Neuroscience, 10(1), 40. DOI
 Dynk04: E. B. Dynkin (2004) Superdiffusions and positive solutions of nonlinear partial differential equations. Providence, R.I: American Mathematical Society
 LeHe13: Anna Levina, J. Michael Herrmann (2013) The Abelian distribution. Stochastics and Dynamics, 14(03), 1450001. DOI
 Ogat78: Yoshiko Ogata (1978) The asymptotic behaviour of maximum likelihood estimators for stationary point processes. Annals of the Institute of Statistical Mathematics, 30(1), 243–261. DOI
 CuLe13: Nicolas Curien, JeanFrançois Le Gall (2013) The Brownian Plane. Journal of Theoretical Probability, 27(4), 1249–1291. DOI
 Aldo91: David Aldous (1991) The Continuum Random Tree I. The Annals of Probability, 19(1), 1–28. DOI
 Aldo93: David Aldous (1993) The Continuum Random Tree III. The Annals of Probability, 21(1), 248–289. DOI
 BoSh89: K.O. Bowman, 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), 867–893. DOI
 Lamp67b: John Lamperti (1967b) The Limit of a Sequence of Branching Processes. Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete, 7(4), 271–288. DOI
 Oake75: David Oakes (1975) The Markovian selfexciting process. Journal of Applied Probability, 12(1), 69. DOI
 Otte49: Richard Otter (1949) The Multiplicative Process. The Annals of Mathematical Statistics, 20(2), 206–224. DOI
 Otte48: Richard Otter (1948) The Number of Trees. Annals of Mathematics, 49(3), 583–599. DOI
 GeHW06: Peter V. Gehler, Alex D. Holub, Max Welling (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
 GeKa04: Jochen Geiger, Lars Kauffmann (2004) The Shape of Large GaltonWatson Trees with Possibly Infinite Variance. Random Struct. Algorithms, 25(3), 311–335. DOI
 Mess52: H. Messel (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
 Bhat87: M. C. Bhattacharjee (1987) The Time to Extinction of Branching Processes and LogConvexity: I. Probability in the Engineering and Informational Sciences, 1(03), 265–278. DOI
 Dwas69: Meyer Dwass (1969) The Total Progeny in a Branching Process and a Related Random Walk. Journal of Applied Probability, 6(3), 682–686. DOI
 SaMS08: A. Saichev, Y. Malevergne, 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, qFin].
 Weiß08: Christian H. Weiß (2008) Thinning operations for modeling time series of counts—a survey. Advances in Statistical Analysis, 92(3), 319–341. DOI
 Jage97: Peter Jagers (1997) Towards Dependence in General Branching Processes. In Classical and Modern Branching Processes (pp. 127–139). Springer New York
 AlPi98: David Aldous, Jim Pitman (1998) Treevalued Markov chains derived from GaltonWatson processes. Annales de l’Institut Henri Poincare (B) Probability and Statistics, 34(5), 637–686. DOI
 PoVe10: Mark Podolskij, Mathias Vetter (2010) Understanding limit theorems for semimartingales: a short survey: Limit theorems for semimartingales. Statistica Neerlandica, 64(3), 329–351. DOI
 Lega13: JeanFrançois Le Gall (2013) Uniqueness and universality of the Brownian map. The Annals of Probability, 41(4), 2880–2960. DOI
 CoSh72: P. Consul, L. Shenton (1972) Use of Lagrange Expansion for Generating Discrete Generalized Probability Distributions. SIAM Journal on Applied Mathematics, 23(2), 239–248. DOI
 SoMM04: D. Sornette, Y. Malevergne, J.F. Muzy (2004) Volatility fingerprints of large shocks: endogenous versus exogenous. In The Application of Econophysics (pp. 91–102). Springer Japan
 Mein09: Matthias Meiners (2009) Weighted branching and a pathwise renewal equation. Stochastic Processes and Their Applications, 119(8), 2579–2597. DOI