On the multidimensional stochastic equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msub><mml:mi>Y</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:msub><mml:mi>Y</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math>

Saporta, Benoı̂te de; Guivarc’h, Yves; Page, Émile Le

doi:10.1016/j.crma.2004.07.024

Cited by 23 publications

(28 citation statements)

References 8 publications

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“…For information on products of random matrices, we refer to [2], [11], [12] and [13]. For an account of our main result in a special case, see [6]. The present exposition is on results which extend the results in the joint work [6].…”

Section: The Recursionsupporting

confidence: 47%

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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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Section: The Recursionsupporting

confidence: 47%

“…For an account of our main result in a special case, see [6]. The present exposition is on results which extend the results in the joint work [6]. It is also an improved version of part of the unpublished work [18].…”

Section: The Recursionmentioning

confidence: 58%

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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“…Even though we are concerned by the one-dimensional case in this paper, let us mention that a significant generalization of Kesten's result, in the multi-dimensional case, was recently achieved by de Saporta, Guivarc'h and Le Page [3], who relaxed the assumption of positivity on A i .…”

Section: Introductionmentioning

confidence: 99%

A probabilistic representation of constants in Kesten’s renewal theorem

Enriquez

Sabot

Zindy

2008

Probab. Theory Relat. Fields

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“…We observe that conditions (H 0 ), (H 1 ) and (H 2 ) are analogous to those used in [16,Theorem 2.5] in order to get a homogeneous behavior at infinity of the potential measure inṼ associated with µ, hence also of the Γ-orbits at infinity inṼ . (See also [10] for the case of affine actions. )…”

Section: Contains a Proximal And Expanding Element And The γ-Action Omentioning

confidence: 99%

Semigroup actions on tori and stationary measures on projective spaces

Guivarc’h

Urban

2005

Studia Math.

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Abstract. Let Γ be a sub-semigroup of G = GL(d, R), d > 1. We assume that the action of Γ on R d is strongly irreducible and that Γ contains a proximal and expanding element. We describe contraction properties of the dynamics of Γ on R d at infinity. This amounts to the consideration of the action of Γ on some compact homogeneous spaces of G, which are extensions of the projective space P d−1 . In the case where Γ is a sub-semigroup of GL(d, R)∩M (d, Z) and Γ has the above properties, we deduce that the Γ-orbits on

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On the multidimensional stochastic equation Yn+1=AnYn+Bn

Cited by 23 publications

References 8 publications

Heavy tail properties of stationary solutions of multidimensional stochastic recursions

Heavy tail properties of stationary solutions of multidimensional stochastic recursions

A probabilistic representation of constants in Kesten’s renewal theorem

Semigroup actions on tori and stationary measures on projective spaces

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