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On the tails of the invariant measure for multidimensional affine stochastic recursions in the critical case

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Ion Grama, Sebastian Mentemeier, Hui Xiao

Probability · Submitted 22 Sep 2026

abstract compiled · 2026-09-24 · no language-model review

The authors' main claim Our goal is to show that this slowly varying function is actually bounded.

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arXiv ID2609.25944 (version 1)
Pith ID7HXBXAAK, Pith's permanent record of this paper and its review
CitationHui Xiao, Ion Grama, Sebastian Mentemeier. "On the tails of the invariant measure for multidimensional affine stochastic recursions in the critical case." arXiv:2609.25944 (2026). https://pith.science/paper/7HXBXAAK
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-24: 1 request, 4,435 input tokens.

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We study the behavior at infinity of the invariant Radon measure for the multidimensional affine stochastic recursion $V_n = A_n V_{n-1} + B_n,$ where $(A_n)_{n \geq 1}$ are positive random matrices, $(B_n)_{n \geq 1}$ are random vectors with nonnegative entries, and $(A_n, B_n)_{n \geq 1}$ are independent and identically distributed. In the critical regime where the top Lyapunov exponent of the random matrix products $A_n \cdots A_1$ is zero, Brofferio, Peign\'e and Pham [6] recently established the existence and uniqueness, up to multiplication by a constant, of an invariant Radon measure with infinite total mass. They proved that the tail behavior of this measure when applied to radial sets is governed by a slowly varying function. Our goal is to show that this slowly varying function is actually bounded. Moreover, we investigate directional tail behavior.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study the behavior at infinity of the invariant Radon measure for the multidimensional affine stochastic recursion $V_n = A_n V_{n-1} + B_n,$ where $(A_n)_{n \geq 1}$ are positive random matrices, $(B_n)_{n \geq 1}$ are random vectors with nonnegative entries, and $(A_n, B_n)_{n \geq 1}$ are independent and identically distributed. background 0.59 0.48 0.03 asserted 0.94
s1 In the critical regime where the top Lyapunov exponent of the random matrix products $A_n \cdots A_1$ is zero, Brofferio, Peign\'e and Pham [6] recently established the existence and uniqueness, up to multiplication by a constant, of an invariant Radon measure with infinite total mass. background 1.00 0.03 0.05 asserted 1.00
s2 They proved that the tail behavior of this measure when applied to radial sets is governed by a slowly varying function. background 0.98 0.03 0.05 asserted 1.00
s3 Our goal is to show that this slowly varying function is actually bounded. contribution 0.84 0.36 0.62 asserted 0.99
s4 Moreover, we investigate directional tail behavior. contribution 0.98 0.60 0.04 asserted 0.62

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

We study the behavior at infinity of the invariant Radon measure for the multidimensional affine stochastic recursion $V_n = A_n V_{n-1} + B_n,$ where $(A_n)_{n \geq 1}$ are positive random matrices, $(B_n)_{n \geq 1}$ are random vectors with nonnegative entries, and $(A_n, B_n)_{n \geq 1}$ are independent and identically distributed. In the critical regime where the top Lyapunov exponent of the random matrix products $A_n \cdots A_1$ is zero, Brofferio, Peign\'e and Pham [6] recently established the existence and uniqueness, up to multiplication by a constant, of an invariant Radon measure with infinite total mass. They proved that the tail behavior of this measure when applied to radial sets is governed by a slowly varying function. Our goal is to show that this slowly varying function is actually bounded. Moreover, we investigate directional tail behavior.

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