### Citations

104 |
Martingale convergence in the branching random walk,
- Biggins
- 1977
(Show Context)
Citation Context ...t∈R+0 , is given by Mt(p) := ∑ k∈N |Πk(t)| 1+peΦ(p)t (4) for every p ∈ (p, p̄) and t ∈ R+0 . Let us point out that M(p) is the analogue of Biggins’ additive martingale for branching random walks (cf. =-=[Big77]-=-). These martingales appear frequently in the literature on branching processes and fragmentation processes. In addition, we shall need a second martingale which is defined in terms of (λη)η∈(0,1] and... |

103 |
Random fragmentation and coagulation processes,
- Bertoin
- 2006
(Show Context)
Citation Context ...which were 2 introduced in [Ber01] in the homogeneous case and were extended to the self-similar setting in [Ber02b]. For a comprehensive treatise on fragmentation processes we refer to the monograph =-=[Ber06]-=-. Let F := (Ft)η∈(0,1] be the filtration generated by Π. It is known from [Ber02b] that the distribution of Π is determined by some α ∈ R (the index of self-similarity; α = 0 corresponding to the homo... |

71 |
On the convergence of supercritical general (C-M-J) branching processes.
- Nerman
- 1981
(Show Context)
Citation Context ...mentioned extensions to the setting with immigration lies at the heart of Theorem 4.2 in [Olo96], where Olofsson extends an L 1-convergence result that was obtained for Crump-Mode-Jagers processes in =-=[Ner81]-=- to such processes with immigration. We also refer to [AH77] for a result on supercritical immigration-branching processes in the spirit of this paper. The final result of this paper is concerned with... |

38 |
Measure and integration theory. de Gruyter
- Bauer
- 2001
(Show Context)
Citation Context ...∈J}v 1+p∗j j ) E ( 〈ρ(j)η ,1〉 ) = E ∑ j∈J v 1+p∗j j instead of Proposition 9, we infer that lim η↓0 E ( 〈ρIη , f I〉 ) = ∑ j∈J v 1+p∗j j 〈ρ (j), f (j)〉, which by means of (32) and Lemma 21.6 in =-=[Bau01]-=- proves that (17) also holds in L 1(P). Remark 10 Assume that the Π(j), j ∈ J , are i.i.d. and thus we omit the index j for terms which are determined by the distribution of Π(j). Then (32) can be p... |

32 |
The asymptotic behavior of fragmentation processes
- Bertoin
- 2003
(Show Context)
Citation Context ... is dissipative, then there exists a p∗ ∈ (p, 0) such that Φ(p∗) = 0. If Π is conservative, that is if ν (∑ n∈N sn < 1 ) = 0, then Φ(0) = 0 and thus we set p∗ := 0 in that case. In view of Lemma 1 in =-=[Ber03]-=- let p̄ be the unique solution to (1 + p)Φ′(p) = Φ(p) 3 on (p,∞), where Φ′ denotes the derivative of Φ. Notice that it follows from Lemma 1 in [Ber03] that p ≥ p̄ if and only if (1+p)Φ′(p) ≤ Φ(p). Sin... |

28 | Discretization methods for homogeneous fragmentations
- Bertoin, Rouault
- 2005
(Show Context)
Citation Context ...22) onto the sub-filtration G , which is generated by the asymptotic frequencies of Π, results in dP(p) dP ∣∣∣∣∣ Gt = Mt(p) for every t ∈ R+0 . According to Theorem 1 of [BR03] (cf. also Theorem 4 of =-=[BR05]-=- for the conservative case) the unit-mean martingale M(p) is uniformly integrable. Hence, E(M∞(p)) = 1 and thus P(p) is a probability measure on G∞ := ⋃ t∈R+0 Gt. Moreover, using that E(M∞(p)) > 0 one... |

24 |
Homogeneous fragmentation processes, Probab. Theory Relat
- Bertoin
(Show Context)
Citation Context ...) := ∞. This paper is concerned with a P-valued fragmentation process Π := (Π(t))η∈(0,1], where Π(t) = (Πn(t))n∈N. P-valued fragmentations are exchangeable Markov processes which were 2 introduced in =-=[Ber01]-=- in the homogeneous case and were extended to the self-similar setting in [Ber02b]. For a comprehensive treatise on fragmentation processes we refer to the monograph [Ber06]. Let F := (Ft)η∈(0,1] be t... |

23 |
Strong limit theorems for general supercritical branching processes with applications to branching diffusions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36
- Asmussen, Hering
- 1976
(Show Context)
Citation Context ...sition 1 follows a series of strong laws of large numbers that were obtained for different classes of stochastic processes with a branching structure. Related classical strong laws were considered in =-=[AH76]-=- and [AH77] for spatial branching processes. Nerman [Ner81] proved a more general strong law of large numbers in the setting of general branching processes, from which the conclusion of Proposition 1 ... |

23 | Strong law of large numbers for branching diffusions
- Engländer, Harris, et al.
(Show Context)
Citation Context ...not as clear as for general branching processes, see Theorem 4 of [Kno12] in this regard. In addition, strong laws of large numbers in the spirit of Proposition 1 were obtained in [CS07], [CRW08] and =-=[EHK10]-=- for branching diffusions and in [EW06] as well as [Eng09] in the setting of superdiffusions. For related results, in the sense of L 2-convergence, in the context of conservative fragmentation process... |

20 | Ranked fragmentations
- Berestycki
(Show Context)
Citation Context ...Π(t)| ↓ n)n∈N denotes the decreasing reordering of the asymptotic frequencies of the blocks of Π(t). In addition, let G := (Gt)t∈R+0 be the filtration generated by λ. In this respect we also refer to =-=[Ber02a]-=-, where mass fragmentation processes (also known as ranked fragmentations) were considered. The main property of fragmentation processes is the (strong) fragmentation property, which is the analogue o... |

15 |
Renewal theory and level passage by subordinators
- Bertoin, Harn, et al.
- 1999
(Show Context)
Citation Context ...ingale Λ(p∗), up to a multiplicative function 〈ρ, ·〉 on B+b given by 〈ρ, f〉 := ∫ (0,1) fdρ = 1 Φ′(p∗) ∫ (0,1) f(t) (∫ S1 ∑ n∈N 1{sn<t}s 1+p∗ n ν(ds) ) dt t for all f ∈ B+b . Resorting to Theorem 1 of =-=[BHS99]-=- and applying Lemma 2 of [HKK10] one can show that 〈ρ, f〉 = lim η↓0 E(〈ρη, f〉). (7) The goal of this paper is to extend the abovmentioned result that was proven in [HKK10]. In this respect see also Th... |

13 |
Strong limit theorems for supercritical immigration-branching processes
- Asmussen, Hering
- 1977
(Show Context)
Citation Context ...the heart of Theorem 4.2 in [Olo96], where Olofsson extends an L 1-convergence result that was obtained for Crump-Mode-Jagers processes in [Ner81] to such processes with immigration. We also refer to =-=[AH77]-=- for a result on supercritical immigration-branching processes in the spirit of this paper. The final result of this paper is concerned with the asymptotic decay of the largest block in the process wi... |

11 | Law of large numbers for superdiffusions: the non-ergodic case
- Engländer
(Show Context)
Citation Context ...em 4 of [Kno12] in this regard. In addition, strong laws of large numbers in the spirit of Proposition 1 were obtained in [CS07], [CRW08] and [EHK10] for branching diffusions and in [EW06] as well as =-=[Eng09]-=- in the setting of superdiffusions. For related results, in the sense of L 2-convergence, in the context of conservative fragmentation processes we refer to [BM05] as well as [HK11], where the latter ... |

10 |
Limit theorems for branching Markov processes
- Chen, Shiozawa
(Show Context)
Citation Context ...th Proposition 1 is not as clear as for general branching processes, see Theorem 4 of [Kno12] in this regard. In addition, strong laws of large numbers in the spirit of Proposition 1 were obtained in =-=[CS07]-=-, [CRW08] and [EHK10] for branching diffusions and in [EW06] as well as [Eng09] in the setting of superdiffusions. For related results, in the sense of L 2-convergence, in the context of conservative ... |

8 |
Fragmentation energy
- Bertoin, Martinez
- 2005
(Show Context)
Citation Context ... immigration which is stopped at the stopping line of first passage times below a given block-size. In the setting of fragmentation chains (without immigration) this kind of problem was considered in =-=[BM05]-=- as well as [HK11] and we shall resort to [HKK10] where the corresponding strong law of large numbers was obtained for general fragmentation processes without immigration. Moreover, we prove the conve... |

8 | An almost sure scaling limit theorem for Dawson-Watanabe superprocesses
- Chen, Ren, et al.
(Show Context)
Citation Context ...sition 1 is not as clear as for general branching processes, see Theorem 4 of [Kno12] in this regard. In addition, strong laws of large numbers in the spirit of Proposition 1 were obtained in [CS07], =-=[CRW08]-=- and [EHK10] for branching diffusions and in [EW06] as well as [Eng09] in the setting of superdiffusions. For related results, in the sense of L 2-convergence, in the context of conservative fragmenta... |

8 |
Strong law of large numbers for fragmentation processes
- Harris, Knobloch, et al.
- 2010
(Show Context)
Citation Context ...onstant. ∗Institut für Mathematik, FB 12, Goethe-Universität Frankfurt am Main, 60054 Frankfurt am Main, Germany e-mail: knobloch@math.uni-frankfurt.de 1 In this paper we extend the main results of =-=[HKK10]-=- and [Kno12] to fragmentation processes with immigration. That is to say, the goal we are concerned with is tantamount to showing that in a certain manner the limiting behaviour of the whole system is... |

7 |
Additive martingales and probability tilting for homogeneous fragmentations
- Bertoin, Rouault
- 2003
(Show Context)
Citation Context ... of the change of measure in (22) onto the sub-filtration G , which is generated by the asymptotic frequencies of Π, results in dP(p) dP ∣∣∣∣∣ Gt = Mt(p) for every t ∈ R+0 . According to Theorem 1 of =-=[BR03]-=- (cf. also Theorem 4 of [BR05] for the conservative case) the unit-mean martingale M(p) is uniformly integrable. Hence, E(M∞(p)) = 1 and thus P(p) is a probability measure on G∞ := ⋃ t∈R+0 Gt. Moreove... |

6 | General branching processes with immigration.
- OLOFSSON
- 1996
(Show Context)
Citation Context ... convergence results for the fragmentation energy model were established. A similar motivation as for the abovementioned extensions to the setting with immigration lies at the heart of Theorem 4.2 in =-=[Olo96]-=-, where Olofsson extends an L 1-convergence result that was obtained for Crump-Mode-Jagers processes in [Ner81] to such processes with immigration. We also refer to [AH77] for a result on supercritica... |

4 |
of large numbers for a class of superdiffusions
- Law
(Show Context)
Citation Context ...ocesses, see Theorem 4 of [Kno12] in this regard. In addition, strong laws of large numbers in the spirit of Proposition 1 were obtained in [CS07], [CRW08] and [EHK10] for branching diffusions and in =-=[EW06]-=- as well as [Eng09] in the setting of superdiffusions. For related results, in the sense of L 2-convergence, in the context of conservative fragmentation processes we refer to [BM05] as well as [HK11]... |

4 | Statistical analysis of self-similar conservative fragmentation chains. http://hal.archives-ouvertes.fr/hal-00261291/fr
- Hoffmann, Krell
- 2008
(Show Context)
Citation Context ...[EW06] as well as [Eng09] in the setting of superdiffusions. For related results, in the sense of L 2-convergence, in the context of conservative fragmentation processes we refer to [BM05] as well as =-=[HK11]-=-, where the latter is concerned with conservative fragmentation chains. An extension of Proposition 1 to certain random functions f was given in Theorem 4 of [Kno12] and thus our forthcoming Theorem 1... |

3 | Energy efficiency of consecutive fragmentation processes
- Fontbona, Krell, et al.
- 2010
(Show Context)
Citation Context ...ks of size less than some given η ∈ (0, 1]. Convergence results for the process counted with this particular characteristic were obtained in [BM05] as well as [Kno12]. In this regard we also refer to =-=[FKM10]-=-. Notice that up to the deterministic constants the random object in the limits of (8) and (11) coincide. In fact, this is no coincidence, since in principle the convergence in (8) could be considered... |

3 |
Equilibrium for fragmentation with immigration, Ann
- Haas
(Show Context)
Citation Context ...ng law of large numbers. 1 Introduction The present paper deals with some asymptotic properties of fragmentation processes with immigration. Random fragmentations with immigrations were introduced in =-=[Haa05]-=-, where also the deterministic setting was considered. The model this paper is concerned with consists of the interplay of two independent random actions with opposing effects. On the one hand there i... |

3 | Asymptotic properties of fragmentation processes
- Knobloch
- 2011
(Show Context)
Citation Context ...s well as Hypothesis 2. On this note let us mention that Theorem 1 of [HKK10] was proven under an additional third assumption (cf. (A3) in [HKK10]). However, it follows by means of Proposition 3.5 in =-=[Kno11]-=- (cf. also Theorem 2 in [Ber03] for the conservative case) that also without this additional assumption the proof provided in [HKK10] works. Proposition 1 follows a series of strong laws of large numb... |

3 | Survival of homogenous fragmentation processes with killing
- Knobloch, Kyprianou
- 2011
(Show Context)
Citation Context ...f of Theorem 3. In view of the martingale arguments used to prove similar results for fragmentation processes (cf. Corollary 1.4 of [Ber06]) and fragmentation processes with killing (see Theorem 4 of =-=[KK12]-=-) we aim at using the submartingale convergence theorem to prove Theorem 3. As in the previous two sections we consider throughout this section, unless stated otherwise, a deterministic vector (vj)j∈J... |

1 |
Asymptotic properties of the process counted with a random characteristic in the context of fragmentation processes, submitted, 2012, arXiv: 1203.4212
- Knobloch
(Show Context)
Citation Context ...stitut für Mathematik, FB 12, Goethe-Universität Frankfurt am Main, 60054 Frankfurt am Main, Germany e-mail: knobloch@math.uni-frankfurt.de 1 In this paper we extend the main results of [HKK10] and =-=[Kno12]-=- to fragmentation processes with immigration. That is to say, the goal we are concerned with is tantamount to showing that in a certain manner the limiting behaviour of the whole system is the same as... |