Diffusion Approximation of Branching Processes in Semi-Markov Environment
- PDF / 243,950 Bytes
- 8 Pages / 439.642 x 666.49 pts Page_size
- 41 Downloads / 203 Views
Diffusion Approximation of Branching Processes in Semi-Markov Environment Nikolaos Limnios1 · Elena Yarovaya2 Received: 19 February 2019 / Revised: 4 May 2020 / Accepted: 27 July 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We consider continuous-time Markov branching processes in semi-Markov random environment and obtain diffusion approximation results for the near critical case. The problem of semi-Markov environment, presented here, is new and more interesting than the Markov case, since it includes many particular interesting cases: Markov, renewal, etc. The particular case of the Markov random environment of continuous-time branching process diffusion approximation results are obtained. Keywords Branching process · Semi-Markov environment · Diffusion approximation · Near critical case · Random environment Mathematics Subject Classification (2010) 60J80 · 60K15 · 60K37 · 60J60
1 Introduction In the present work we study diffusion approximation of near critical Markov branching processes in continuous-time in semi-Markov random environment. For standard references on branching processes reader can be refereed to Harris (1963), Athreya and Ney (1972), Jagers (1971), Gikhman and Skorokhod (1974), Haccou et al. (2005), Kersting and Vatutin (2016), see also Iosifescu et al. (2010). Results on diffusion approximation given by Feller (1951), and then by Jiˇrina (1969), for Bienaym´e-Galton-Watson (BGW) processes, together with results given by Jagers (1971), for the continuous-time Markov branching processes, are now classical. See also Aliev and Shurenkov (1983), Borovkov (1983, 2006) and Wei and Winnicki (1989). Methods used to obtain such diffusion approximation results are based mostly on transform methods, as generating functions or Laplace transforms. Branching processes in random environment are an important topic, where many works are dedicated. The first result concerning diffusion approximation of branching processes in random environment, especially for BGW processes in Markov chain environment was Nikolaos Limnios
[email protected] 1
Universit´e de Technologie de Compi`egne, Sorbonne University Alliance, Compi`egne, Paris, France
2
Lomonosov Moscow State University, Moscow, Russia
Methodology and Computing in Applied Probability
given in Kurtz (1978). Additional results on random environment for branching processes, the reader can find in Dyakonova (2014), B¨oingoff and Hutzenthaler (2012), Bansaye and Simatos (2015), and Bansaye et al. (2019). In Limnios and Yarovaya (2018) we present results concerning Markov branching processes in Markov random environment in continuous-time. The main results presented in this paper concern the semi-Markov random environment in general state space in continuous-time. In this case we don’t have the semigroup property, as in the Markov case and we present a different method to obtain diffusion approximation. We compensating operator of extended Markov renewal processes and the semimartingale tightness. In particular,
Data Loading...
