Harmonic moments and large deviations for a supercritical branching process in a random environment

Grama, Ion; Liu, Quansheng; Miqueu, Eric

doi:10.1214/17-ejp71

Cited by 12 publications

(9 citation statements)

References 10 publications

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“…This is indeed the case, see [7], [12] and [14]. However, in the case of a random environment, the equation ( 6) is not exactly of the form in (8) and so a different approach had to be elaborated.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…Most of that was done for the i.i.d environment, because then properties of the so-called "associated random walks" could be applied, but some results hold also in a stationary and ergodic environment. For a sample of results see [3,5,4,8,9,13] and references therein.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…However, except of [10] the local regularity of the law of W has not been studied. Due to the basic equation (6) satisfied by W it is closely related to the local regularity for stationary solutions to affine type equations (8) which is partly our motivation as it is explained in the end of the introduction. We stress that our arguments are valid for stationary and ergodic environment.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Absolute continuity of the martingale limit in branching processes in random environment

Damek¹,

Gantert²,

Kolesko³

2019

Electron. Commun. Probab.

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We consider a supercritical branching process Z n in a stationary and ergodic random environment ξ = (ξ n ) n≥0 . Due to the martingale convergence theorem, it is known that the normalized population size W n = Z n /(E(Z n |ξ)) converges almost surely to a random variable W . We prove that if W is not concentrated at 0 or 1 then for almost every environment ξ the law of W conditioned on the environment ξ is absolutely continuous with a possible atom at 0. The result generalizes considerably the main result of [10], and of course it covers the well-known case of the martingale limit of a Galton-Watson process. Our proof combines analytical arguments with the recursive description of W .

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Absolute continuity of the martingale limit in branching processes in random environment

Damek¹,

Gantert²,

Kolesko³

2019

Electron. Commun. Probab.

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“…An important field of applications of large deviation asymptotics for the coefficients of type (2.6) is the study of asymptotic behaviors of multi-type branching processes in random environment. For results in the case of singletype branching processes we refer to [36,37] and for the relation between the coefficients of products of random matrices and the multi-type branching processes we refer to [17].…”

Section: Resultsmentioning

confidence: 99%

Large deviation expansions for the coefficients of random walks on the general linear group

Xiao¹,

Grama²,

Liu³

2020

Preprint

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Let (gn) n 1 be a sequence of independent and identically distributed elements of the general linear group GL(d, R). Consider the random walk Gn := gn . . . g1. Under suitable conditions, we establish Bahadur-Rao-Petrov type large deviation expansion for the coefficients f, Gnv , where f ∈ (R d ) * and v ∈ R d . In particular, our result implies the large deviation principle with an explicit rate function, thus improving significantly the large deviation bounds established earlier. Moreover, we establish Bahadur-Rao-Petrov type large deviation expansion for the coefficients f, Gnv under the changed measure. Toward this end we prove the Hölder regularity of the stationary measure corresponding to the Markov chain Gnv/|Gnv| under the changed measure, which is of independent interest. In addition, we also prove local limit theorems with large deviations for the coefficients of Gn.

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“…Proof. The proof idea is inspired by [12,10] by working on the quenched Laplace transform of W and constructing the recursive relationship. We think that p ∈ (1, 2], otherwise we use min{p, 2} to replace p. Set φ ξ (t) = E ξ e −t W .…”

Section: Moments Of Log W Nmentioning

confidence: 99%

Exact convergence rate in the central limit theorem for a branching process with immigration in a random environment

Huang¹,

Zhang²,

Gao³

2022

Preprint

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Harmonic moments and large deviations for a supercritical branching process in a random environment

Cited by 12 publications

References 10 publications

Absolute continuity of the martingale limit in branching processes in random environment

Absolute continuity of the martingale limit in branching processes in random environment

Large deviation expansions for the coefficients of random walks on the general linear group

Exact convergence rate in the central limit theorem for a branching process with immigration in a random environment

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