Abstract
Linear fractional Galton–Watson branching processes in i.i.d. random environment are, on the quenched level, intimately connected to random difference equations by the evolution of the random parameters of their linear fractional marginals. On the other hand, any random difference equation defines an autoregressive Markov chain (a random affine recursion) which can be positive recurrent, null recurrent and transient and which, as the forward iterations of an iterated function system, has an a.s. convergent counterpart in the positive recurrent case given by the corresponding backward iterations. The present expository article aims to provide an explicit view at how these aspects of random difference equations and their stationary limits, called perpetuities, enter into the results and the analysis, especially in quenched regime. Although most of the results presented here are known, we hope that the offered perspective will be welcomed by some readers.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: EXC 2044–390685587
Funding statement: Work partially funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure.
References
[1] V. I. Afanas’ev, A limit theorem for a critical branching process in a random environment, Diskret. Mat. 5 (1993), no. 1, 45–58. 10.1515/dma.2001.11.6.587Suche in Google Scholar
[2] V. I. Afanas’ev, A new limit theorem for a critical branching process in a random environment, Discrete Math. Appl. 7 (1997), no. 5, 497–513. 10.1515/dma.1997.7.5.497Suche in Google Scholar
[3] G. Alsmeyer, D. Buraczewski and A. Iksanov, Null recurrence and transience of random difference equations in the contractive case, J. Appl. Probab. 54 (2017), no. 4, 1089–1110. 10.1017/jpr.2017.54Suche in Google Scholar
[4] G. Alsmeyer and A. Iksanov, A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks, Electron. J. Probab. 14 (2009), no. 10, 289–312. 10.1214/EJP.v14-596Suche in Google Scholar
[5] G. Alsmeyer and A. Iksanov, Recurrence and transience of random difference equations in the critical case, preprint (2021), http://arxiv.org/abs/2105.04994. Suche in Google Scholar
[6] G. Alsmeyer, A. Iksanov and U. Rösler, On distributional properties of perpetuities, J. Theoret. Probab. 22 (2009), no. 3, 666–682. 10.1007/s10959-008-0156-8Suche in Google Scholar
[7] K. B. Athreya and S. Karlin, On branching processes with random environments. I. Extinction probabilities, Ann. Math. Statist. 42 (1971), 1499–1520. 10.1214/aoms/1177693150Suche in Google Scholar
[8] K. B. Athreya and P. E. Ney, Branching Processes, Grundlehren Math. Wiss. 196, Springer, New York, 1972. 10.1007/978-3-642-65371-1Suche in Google Scholar
[9]
M. Babillot, P. Bougerol and L. Elie,
The random difference equation
[10] K. A. Borovkov and V. A. Vatutin, Reduced critical branching processes in random environment, Stochastic Process. Appl. 71 (1997), no. 2, 225–240. 10.1016/S0304-4149(97)00074-4Suche in Google Scholar
[11] D. Buraczewski, E. Damek and T. Mikosch, Stochastic Models with Power-Law Tails, Springer Ser. Oper. Res. Financ. Eng., Springer, Cham, 2016. 10.1007/978-3-319-29679-1Suche in Google Scholar
[12] Y. S. Chow and H. Teicher, Probability Theory. Independence, Interchangeability, Martingales, 3rd ed., Springer Texts Statist., Springer, New York, 1997. 10.1007/978-1-4612-1950-7Suche in Google Scholar
[13] K. Fleischmann and V. A. Vatutin, Reduced subcritical Galton–Watson processes in a random environment, Adv. in Appl. Probab. 31 (1999), no. 1, 88–111. 10.1239/aap/1029954268Suche in Google Scholar
[14] J. Geiger and G. Kersting, The survival probability of a critical branching process in random environment, Theory Probab. Appl. 45 (2000), no. 3, 518–526. 10.4213/tvp491Suche in Google Scholar
[15] J. Geiger, G. Kersting and V. A. Vatutin, Limit theorems for subcritical branching processes in random environment, Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 4, 593–620. 10.1016/S0246-0203(02)00020-1Suche in Google Scholar
[16] C. M. Goldie and R. A. Maller, Stability of perpetuities, Ann. Probab. 28 (2000), no. 3, 1195–1218. 10.1214/aop/1019160331Suche in Google Scholar
[17] A. Iksanov, Renewal Theory for Perturbed Random Walks and Similar Processes, Probab. Appl., Birkhäuser/Springer, Cham, 2016. 10.1007/978-3-319-49113-4Suche in Google Scholar
[18] G. Kersting and V. Vatutin, Discrete Time Branching Processes in Random Environment, Math. Stat. Ser., Wiley-ISTE, London, 2017. 10.1002/9781119452898Suche in Google Scholar
[19] H. Kesten, Random difference equations and renewal theory for products of random matrices, Acta Math. 131 (1973), 207–248. 10.1007/BF02392040Suche in Google Scholar
[20] F. C. Klebaner, U. Rösler and S. Sagitov, Transformations of Galton–Watson processes and linear fractional reproduction, Adv. in Appl. Probab. 39 (2007), no. 4, 1036–1053. 10.1239/aap/1198177238Suche in Google Scholar
[21] M. V. Kozlov, The asymptotic behavior of the probability of non-extinction of critical branching processes in a random environment, Theory Probab. Appl. 21 (1976), no. 4, 791–804. 10.1137/1121091Suche in Google Scholar
[22] A. Lindo, Some computational aspects of Markov processes, PhD thesis, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden, 2016. Suche in Google Scholar
[23] S. Sagitov and A. Lindo, A special family of Galton–Watson processes with explosions, Branching Processes and Their Applications, Lect. Notes Stat. 219, Springer, Cham (2016), 237–254. 10.1007/978-3-319-31641-3_14Suche in Google Scholar
[24] V. Vatutin and E. Dyakonova, Reduced branching processes in random environment, Mathematics and Computer Science. II (Versailles 2002), Trends Math., Birkhäuser, Basel (2002), 455–467. 10.1137/S0040585X97983845Suche in Google Scholar
[25] V. A. Vatutin and E. E. Dyakonova, Critical branching processes in a random environment: probabilities of extinction at a fixed time, Diskret. Mat. 9 (1997), no. 4, 100–126. Suche in Google Scholar
[26] W. Vervaat, On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables, Adv. in Appl. Probab. 11 (1979), no. 4, 750–783. 10.2307/1426858Suche in Google Scholar
[27] M. P. W. Zerner, Recurrence and transience of contractive autoregressive processes and related Markov chains, Electron. J. Probab. 23 (2018), Paper No. 27. 10.1214/18-EJP152Suche in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- 5th International Workshop on Branching Processes and Their Applications (IWBPA 2021)
- Critical Galton–Watson Processes with Overlapping Generations
- Linear Fractional Galton–Watson Processes in Random Environment and Perpetuities
- Limit Theorems for a Strongly Supercritical Branching Process with Immigration in Random Environment
- Asymptotic Properties of a Supercritical Branching Process with Immigration in a Random Environment
- Branching Process Modelling of COVID-19 Pandemic Including Immunity and Vaccination
- Homogeneous Branching Processes with Non-Homogeneous Immigration
Artikel in diesem Heft
- Frontmatter
- 5th International Workshop on Branching Processes and Their Applications (IWBPA 2021)
- Critical Galton–Watson Processes with Overlapping Generations
- Linear Fractional Galton–Watson Processes in Random Environment and Perpetuities
- Limit Theorems for a Strongly Supercritical Branching Process with Immigration in Random Environment
- Asymptotic Properties of a Supercritical Branching Process with Immigration in a Random Environment
- Branching Process Modelling of COVID-19 Pandemic Including Immunity and Vaccination
- Homogeneous Branching Processes with Non-Homogeneous Immigration