Sebastian Mentemeier

activity

20102026

most citedHeavy tailed solutions of multivariate smoothing transforms

24 citations · 83 across the 18 of their papers we have counts for

collaborators

22 papers

math.PR2026

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

Ion Grama, Sebastian Mentemeier, Hui Xiao

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 positi…

math.PR2025

Hidden regular variation for stochastic recursions with diagonal matrices

Ewa Damek, Sebastian Mentemeier

We consider random vectors $X$ that satisfy the equation in law $X = A X + B$ , where $A$ is a given random diagonal matrix and $B$ a given random vector, both independent of $X$ . It is…

math.PR2025

Spinal decomposition, martingale convergence and the Seneta-Heyde scaling for matrix branching random walks

Ion Grama, Sebastian Mentemeier, Hui Xiao

We consider a matrix branching random walk on the semi-group of nonnegative matrices, where we are able to derive, under general assumptions, an analogue of Biggins' martingale con…

math.PR2025

A probabilistic study of the set of stationary solutions to spatial kinetic-type equations

Sebastian Mentemeier, Glib Verovkin

In this paper we study multivariate kinetic-type equations in a general setup, which includes in particular the spatially homogeneous Boltzmann equation with Maxwellian molecules,…

math.PR2024

Limit theorems for stochastic exponentials of matrix-valued Lévy processes

Anita Behme, Sebastian Mentemeier

We study the long-time behaviour of matrix-valued stochastic exponentials of Lévy processes, i.e. of multiplicative Lévy processes in the general linear group. In particular, we pr…

stat.ML2024

Analysing heavy-tail properties of Stochastic Gradient Descent by means of Stochastic Recurrence Equations

Ewa Damek, Sebastian Mentemeier

In recent works on the theory of machine learning, it has been observed that heavy tail properties of Stochastic Gradient Descent (SGD) can be studied in the probabilistic framewor…

math.PR2023

Limit theorems for first passage times of multivariate perpetuity sequences

Sebastian Mentemeier, Hui Xiao

We study the first passage time $τ_{u} = in f {n \geq 1 : ∣ V_{n} ∣ > u}$ for the multivariate perpetuity sequence $V_{n} = Q_{1} + M_{1} Q_{2} + \dots + (M_{1} \dots M_{n - 1}) Q_{n}$ , where $(M…

math.PR2022★ 1 cited

The extremal position of a branching random walk on the general linear group

Ion Grama, Sebastian Mentemeier, Hui Xiao

Consider a branching random walk $(G_{u})_{u \in T}$ on the general linear group $GL (V)$ of a finite dimensional space $V$ , where $T$ is the associated geneal…

math.PR2019★ 4 cited

Asymptotic Independence ex machina -- Extreme Value Theory for the Diagonal BEKK-ARCH(1) Model

Sebastian Mentemeier, Olivier Wintenberger

We consider multivariate stationary processes $(X_{t})$ satisfying a stochastic recurrence equation of the form $$ \boldsymbol{X}_t= \mathbb{ M}_t \boldsymbol{X}_{t-1} +…

math.PR2018

Absolute Continuity of Complex Martingales and of Solutions to Complex Smoothing Equations

Ewa Damek, Sebastian Mentemeier

Let $X$ be a $C$ -valued random variable with the property that $X has the same law as j \geq 1 \sum T_{j} X_{j}$ where $X_{j}$ are i.i.d.\ copies of $X$ , which are…

math.PR2016

Large excursions and conditioned laws for recursive sequences generated by random matrices

Jeffrey F. Collamore, Sebastian Mentemeier

We determine the large exceedance probabilities and large exceedance paths for the matrix recursive sequence $V_{n} = M_{n} V_{n - 1} + Q_{n}, n = 1, 2, \dots,$ where ${M_{n}}$ is an i.i.d…

math.PR2015★ 1 cited

Solutions to complex smoothing equations

Matthias Meiners, Sebastian Mentemeier

We consider smoothing equations of the form $X = law j \geq 1 \sum T_{j} X_{j} + C$ where $(C, T_{1}, T_{2}, \dots)$ is a given sequence of random variables and $X…

math.PR2015★ 1 cited

Precise Tail Asymptotics for Attracting Fixed Points of Multivariate Smoothing Transformations

Dariusz Buraczewski, Sebastian Mentemeier

Given $d \geq 1$ , let $(A_{i})_{i \geq 1}$ be a sequence of random $d \times d$ real matrices and $Q$ be a random vector in $R^{d}$ . We consider fixed points of multivariate smoot…

math.PR2014

Fixed Points of the Multivariate Smoothing Transform: The Critical Case

Konrad Kolesko, Sebastian Mentemeier

Given a sequence $(T_{1}, T_{2}, ...)$ of random $d \times d$ matrices with nonnegative entries, suppose there is a random vector $X$ with nonnegative entries, such that $ \sum_{i \ge…

math.PR2014

Precise Large Deviation Results for Products of Random Matrices

Dariusz Buraczewski, Sebastian Mentemeier

The theorem of Furstenberg and Kesten provides a strong law of large numbers for the norm of a product of random matrices. This can be extended under various assumptions, covering…

math.PR2013★ 15 cited

The Fixed Points of the Multivariate Smoothing Transform

Sebastian Mentemeier

Let $N, d > 1$ be fixed integers, let $(T_{1}, ..., T_{N})$ be random d-by-d matrices with nonnegative entries and $Q$ a random d-vector with nonnegative entries. This induces a mapping…

math.PR2013

On Kesten's Multivariate Choquet-Deny Lemma

Sebastian Mentemeier

Let $d > 1$ and $(A_{n})_{n \geq 1}$ be a sequence of independent identically distributed random matrices with nonnegative entries and no zero column. This induces a Markov chain $M_n…

math.PR2012

Precise tail index of fixed points of the two-sided smoothing transform

Gerold Alsmeyer, Ewa Damek, Sebastian Mentemeier

We consider real-valued random variables R satisfying the distributional equation R \eqdist \sum_{k=1}^{N}T_k R_k + Q, where R_1,R_2,... are iid copies of R and independent of T=(Q…

math.PR2012★ 24 cited

Heavy tailed solutions of multivariate smoothing transforms

Dariusz Buraczewski, Ewa Damek, Sebastian Mentemeier +1

Let $N > 1$ be a fixed integer and $(C_{1}, ..., C_{N}, Q)$ a random element of $G L (d, R)^{N} x R^{d}$ . We consider solutions of multivariate smoothing transforms, i.e. random variables $R…

math.PR2011★ 7 cited

On multidimensional Mandelbrot's cascades

Dariusz Buraczewski, Ewa Damek, Yves Guivarc'h +1

Let $Z$ be a random variable with values in a proper closed convex cone $C \subset R^{d}$ , $A$ a random endomorphism of $C$ and $N$ a random integer. We assume that $Z$ , $A$ ,…

math.PR2011★ 6 cited

Convergence to stable laws for multidimensional stochastic recursions: the case of regular matrices

Ewa Damek, Sebastian Mentemeier, Mariusz Mirek +1

Given a sequence $(M_{n}, Q_{n})_{n \geq 1}$ of i.i.d.\ random variables with generic copy $(M, Q) \in G L (d, R) \times R^{d}$ , we consider the random difference equation (RDE) $$ R_{n}…

math.PR2010★ 24 cited

Tail behavior of stationary solutions of random difference equations: the case of regular matrices

Gerold Alsmeyer, Sebastian Mentemeier

Given a sequence $(M_{n}, Q_{n})_{n \geq 1}$ of i.i.d. random variables with generic copy $(M, Q)$ such that $M$ is a regular $d \times d$ matrix and $Q$ takes values in $\mathbb{R}^{d}…