Semigroup actions on tori and stationary measures on projective spaces

Guivarc’h, Yves; Urban, Roman

doi:10.4064/sm171-1-3

Cited by 18 publications

(29 citation statements)

References 25 publications

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“…d) The fact that the stability group here is R * + , if α belongs to [0, 2] instead of a more complex one as in [5], is a consequence of the following property depending on condition i-p and d ≥ 2 (see [23,24]): the closed subsemigroup of R * + generated by the moduli of the dominant eigenvalues for the proximal elements in [suppμ] is equal to R * + . This can be compared with the situation of [5] where semi-stable laws in the sense of [36, p.204] appear as limits.…”

Section: /Nmentioning

confidence: 99%

Stable laws and spectral gap properties for affine random walks

Gao¹,

Guivarc’h²,

Page³

2015

Ann. Inst. H. Poincaré Probab. Statist.

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We consider a general multidimensional affine recursion with corresponding Markov operator P and a unique P -stationary measure. We show spectral gap properties on Hölder spaces for the corresponding Fourier operators and we deduce convergence to stable laws for the Birkhoff sums along the recursion. The parameters of the stable laws are expressed in terms of basic quantities depending essentially on the matricial multiplicative part of P . Spectral gap properties of P and homogeneity at infinity of the P -stationary measure play an important role in the proofs.

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Section: /Nmentioning

confidence: 99%

Stable laws and spectral gap properties for affine random walks

Gao¹,

Guivarc’h²,

Page³

2015

Ann. Inst. H. Poincaré Probab. Statist.

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“…In the proofs, we will rely strongly on the analytic tools of [18], which have also been essential in [17]. We will also use the dynamical aspects of linear group actions on real vector spaces (see the recent surveys [13], [14]). For information on products of random matrices, we refer to [2], [11], [12] and [13].…”

Section: The Recursionmentioning

confidence: 99%

“…which guarantees the χ-homogeneity of λ at infinity is given by the following (see [11], [14]). Together with Proposition 5.6, it is one of the main algebraic facts which play a role in Theorem 6.2 below.…”

Section: Y Guivarc'hmentioning

confidence: 99%

“…In general, and in contrast to supp ν, it has a transversal Cantor structure given by supp ν ∞ (see [13,14]). …”

Section: Remarksmentioning

confidence: 99%

“…for T is equivalent to condition i-p for the Zariski closure of T [13]. This Zariski closure is a Lie group with a finite number of components with a special structure described in [14], Lemma 2.7.…”

Section: Notationmentioning

confidence: 99%

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Heavy tail properties of stationary solutions of multidimensional stochastic recursions

Guivarc’h¹

2006

Institute of Mathematical Statistics Lecture Notes - Monograph Series

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We consider the following recurrence relation with random i.i.d. coefficients (an, bn):x n+1 = a n+1 xn + b n+1 where an ∈ GL(d, R), bn ∈ R d . Under natural conditions on (an, bn) this equation has a unique stationary solution, and its support is non-compact. We show that, in general, its law has a heavy tail behavior and we study the corresponding directions. This provides a natural construction of laws with heavy tails in great generality. Our main result extends to the general case the results previously obtained by H. Kesten in [16] under positivity or density assumptions, and the results recently developed in [17] in a special framework.1. Notation and problem General notationWe consider the d-dimensional vector space V = R d , endowed with the scalar product Let G = GL(V ) be the general linear group, and H = Af f (V ) the affine group. An element h ∈ H can be written in the formwhere a ∈ G, b ∈ V . In reduced form we write h = (b, a) and we observe that the projection map (b, a) → a is a homomorphism from H to G. We consider also the projection (b, a) → b of H on V and we observe that H can be written as a semi-direct product H = G ⋉ V where V denotes also the translation group of V .For a locally compact second countable (l.c.s.c.) space X, we denote by M 1 (X) the convex set of probability measures on X and we endow M 1 (X) with the weak

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On fixed points of a generalized multidimensional affine recursion

Mirek

2012

Probab. Theory Relat. Fields

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Let G be a multiplicative subsemigroup of the general linear group Gl(R d ) which consists of matrices with positive entries such that every column and every row contains a strictly positive element. Given a G-valued random matrix A, we consider the following generalized multidimensional affine equationwhere N ≥ 2 is a fixed natural number, A 1 , . . . , A N are independent copies of A, B ∈ R d is a random vector with positive entries, and R 1 , . . . , R N are independent copies of R ∈ R d , which have also positive entries. Moreover, all of them are mutually independent and D

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Semigroup actions on tori and stationary measures on projective spaces

Cited by 18 publications

References 25 publications

Stable laws and spectral gap properties for affine random walks

Stable laws and spectral gap properties for affine random walks

Heavy tail properties of stationary solutions of multidimensional stochastic recursions

On fixed points of a generalized multidimensional affine recursion

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