A PHASE TRANSITION FOR LARGE VALUES OF BIFURCATING AUTOREGRESSIVE MODELS

Autor: Vincent Bansaye, S. Valère Bitseki Penda
Přispěvatelé: Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Chair 'Modelisation Mathematique et Biodiversite' of VEOLIA Environnement-Ecole Polytechnique-MNHN-F.X French National Research Agency (ANR)ANR-16-CE40-0001, ANR-16-CE40-0001,ABIM,Approximations et comportement de modèles aléatoires individu-centrés(2016), Bitseki Penda, Siméon Valère
Jazyk: angličtina
Rok vydání: 2019
Předmět:
Statistics and Probability
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR]
Phase transition
random environment
General Mathematics
media_common.quotation_subject
moderate deviations
limit-theorems
markov-chains
Statistics::Other Statistics
Branching process
deviation inequalities
92D25
01 natural sciences
Asymmetry
010104 statistics & probability
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST]
Convergence (routing)
[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]
Applied mathematics
60C05
[MATH]Mathematics [math]
0101 mathematics
autoregressive process
60J20
law
Mathematics
media_common
Event (probability theory)
parameters
convergence
Markov chain
010102 general mathematics
[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
Large deviations
large deviations Mathematics Subject Classification (2010): 60J80
60K37
Autoregressive model
cells
Large deviations theory
Statistics
Probability and Uncertainty
asymmetry
60F10
Zdroj: Journal of Theoretical Probability
Journal of Theoretical Probability, Springer, 2021, ⟨10.1007/s10959-020-01033-w⟩
ISSN: 0894-9840
1572-9230
DOI: 10.1007/s10959-020-01033-w⟩
Popis: We describe the asymptotic behavior of the number $$Z_n[a_n,\infty )$$ of individuals with a large value in a stable bifurcating autoregressive process, where $$a_n\rightarrow \infty $$ . The study of the associated first moment is equivalent to the annealed large deviation problem of an autoregressive process in a random environment. The trajectorial behavior of $$Z_n[a_n,\infty )$$ is obtained by the study of the ancestral paths corresponding to the large deviation event together with the environment of the process. This study of large deviations of autoregressive processes in random environment is of independent interest and achieved first. The estimates for bifurcating autoregressive process involve then a law of large numbers for non-homogenous trees. Two regimes appear in the stable case, depending on whether one of the autoregressive parameters is greater than 1 or not. It yields different asymptotic behaviors for large local densities and maximal value of the bifurcating autoregressive process.
Databáze: OpenAIRE
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