Convergence of the derivative martingale for the branching random walk in time-inhomogeneous random environment

被引：0

作者：

Hong, Wenming ^{[1
,2
]}

Liang, Shengli ^{[3
]}

机构：

[1] Beijing Normal Univ, Sch Math Sci, Beijing 100875, Peoples R China

[2] Beijing Normal Univ, Lab Math & Complex Syst, Beijing 100875, Peoples R China

[3] Southern Univ Sci & Technol, Shenzhen Int Ctr Math, Shenzhen 518055, Guangdong, Peoples R China

来源：

ADVANCES IN APPLIED PROBABILITY | 2024年

基金：

中国国家自然科学基金;

关键词：

Branching random walk; random environment; derivative martingale; quenched harmonic function; random walk conditioned to stay non-negative; Tanaka's decomposition; LIMIT-THEOREM; FIXED-POINTS; SUFFICIENT CONDITION; EQUATION; CRITICALITY;

D O I：

10.1017/apr.2024.55

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

摘要：

Consider a branching random walk on the real line with a random environment in time(BRWRE). A necessary and sufficient condition for the non-triviality of the limit of thederivative martingale is formulated. To this end, we investigate the random walk in atime-inhomogeneous random environment (RWRE), which is related to the BRWRE bythe many-to-one formula. The key step is to figure out Tanaka's decomposition for theRWRE conditioned to stay non-negative (or above a line), which is interesting in itself.

引用

页码：642 / 676

页数：35

共 33 条

[1] Criticality for branching processes in random environment [J].

Afanasyev, VI ;

Geiger, J ;

Kersting, G ;

Vatutin, VA .

ANNALS OF PROBABILITY, 2005, 33 (02) :645-673

[2] THE SENETA-HEYDE SCALING FOR THE BRANCHING RANDOM WALK [J].

Aidekon, Elie ;

Shi, Zhan .

ANNALS OF PROBABILITY, 2014, 42 (03) :959-993

[3] CONVERGENCE IN LAW OF THE MINIMUM OF A BRANCHING RANDOM WALK [J].

Aidekon, Elie .

ANNALS OF PROBABILITY, 2013, 41 (3A) :1362-1426

[4] Change of measures for Markov chains and the LlogL theorem for branching processes [J].

Athreya, KB .

BERNOULLI, 2000, 6 (02) :323-338

[5] Fixed points of the smoothing transform: the boundary case [J].

Biggins, JD ;

Kyprianou, AE .

ELECTRONIC JOURNAL OF PROBABILITY, 2005, 10 :609-631

[6] Measure change in multitype branching [J].

Biggins, JD ;

Kyprianou, AE .

ADVANCES IN APPLIED PROBABILITY, 2004, 36 (02) :544-581

[7] Random walk conditioned to stay positive [J].

Biggins, JD .

JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 2003, 67 :259-272

[8] MARTINGALE CONVERGENCE IN BRANCHING RANDOM-WALK [J].

BIGGINS, JD .

JOURNAL OF APPLIED PROBABILITY, 1977, 14 (01) :25-37

[9] A NECESSARY AND SUFFICIENT CONDITION FOR THE NONTRIVIAL LIMIT OF THE DERIVATIVE MARTINGALE IN A BRANCHING RANDOM WALK [J].

Chen, Xinxin .

ADVANCES IN APPLIED PROBABILITY, 2015, 47 (03) :741-760

[10] FIXED-POINTS OF THE SMOOTHING TRANSFORMATION [J].

DURRETT, R ;

LIGGETT, TM .

ZEITSCHRIFT FUR WAHRSCHEINLICHKEITSTHEORIE UND VERWANDTE GEBIETE, 1983, 64 (03) :275-301

← 1 2 3 4 →