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

  • Ion Grama
  • Sebastian Mentemeier
  • Hui Xiao
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Abstract

We study the behavior at infinity of the invariant Radon measure for the multidimensional affine stochastic recursion Vn = An Vn−1 + Bn, where (An)n ≥ 1 are positive random matrices, (Bn)n ≥ 1 are random vectors with nonnegative entries, and (An, Bn)n ≥ 1 are independent and identically distributed. In the critical regime where the top Lyapunov exponent of the random matrix products An ⋯ A1 is zero, Brofferio, Peigné 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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Cite

BibTeXRIS
BibTeX
@misc{grama2026tails,
  title = {{On the tails of the invariant measure for multidimensional affine stochastic recursions in the critical case}},
  author = {Ion Grama and Sebastian Mentemeier and Hui Xiao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25944},
  eprint = {2609.25944},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25944},
}
RIS
TY  - UNPB
TI  - On the tails of the invariant measure for multidimensional affine stochastic recursions in the critical case
AU  - Ion Grama
AU  - Sebastian Mentemeier
AU  - Hui Xiao
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.25944
UR  - https://arxiv.org/abs/2609.25944
AB  - 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 ≥ 1 are positive random matrices, (B_n)_n ≥ 1 are random vectors with nonnegative entries, and (A_n, B_n)_n ≥ 1 are independent and identically distributed. In the critical regime where the top Lyapunov exponent of the random matrix products A_n ⋯ A_1 is zero, Brofferio, Peigné 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.
ER  -