arXiv subjects

Sebastian Mentemeier

Publications and source records attributed to Sebastian Mentemeier.

22 records · Page 2Linked to original sources

arXiv2013

Heavy tailed solutions of multivariate smoothing transforms

Let $N > 1$ be a fixed integer and $(C_1,..., C_N,Q)$ a random element of $GL(d, \R)^N x \R^d$. We consider solutions of multivariate smoothing transforms, i.e. random variables $R$ satisfying $$R \eqdist \sum_{i=1}^N C_i R_i +Q $$ where $\eqdist$ denotes equality in distribution, and $R, R_1,..., R_N$ are independent identically distributed $\R^d$-valued random variables, and independent of $(C_1,..., C_N, Q)$. We briefly review conditions for the existence of solutions, and then study their asymptotic behaviour. We show that under natural conditions, these solutions exhibit heavy tails. Our results also cover the case of complex valued weights $(C_1,..., C_N)$.

math.PR↗

arXiv2012

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

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, (T_k)_{k \ge 1}). N is the number of nonzero weights T_k and assumed to be a.s. finite. Its properties are governed by the function m(s) := \E \sum_{k=1}^N |T_k|^s . There are at most two values α< βsuch that m(α)=m(β)=1. We consider solutions R with finite moment of order s > α. We review results about existence and uniqueness. Assuming the existence of βand an additional mild moment condition on the T_{k}, our main result asserts that \lim_{t \to \infty} t^βP(|R| > t) = K > 0, the main contribution being that K is indeed positive and therefore βthe precise tail index of |R|, for the convergence was recently shown by Jelenkovic and Olvera-Cravioto (arXiv:1012.2165).

math.PR↗

arXiv2012

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

Given a sequence $(M_{n},Q_{n})_{n\ge 1}$ of i.i.d.\ random variables with generic copy $(M,Q) \in GL(d, \R) \times \R^d$, we consider the random difference equation (RDE) $$ R_{n}=M_{n}R_{n-1}+Q_{n}, $$ $n\ge 1$, and assume the existence of $κ>0$ such that $$ \lim_{n \to \infty}(\E{\norm{M_1 ... M_n}^κ})^{\frac{1}{n}} = 1 .$$ We prove, under suitable assumptions, that the sequence $S_n = R_1 + ... + R_n$, appropriately normalized, converges in law to a multidimensional stable distribution with index $κ$. As a by-product, we show that the unique stationary solution $R$ of the RDE is regularly varying with index $κ$, and give a precise description of its tail measure. This extends the prior work http://arxiv.org/abs/1009.1728v3 .

math.PR↗

arXiv2011

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

Given a sequence $(M_{n},Q_{n})_{n\ge 1}$ 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 $\mathbb{R}^{d}$, we consider the random difference equation (RDE) $R_{n}=M_{n}R_{n-1}+Q_{n}$, $n\ge 1$. Under suitable assumptions, this equation has a unique stationary solution $R$ such that, for some $κ>0$ and some finite positive and continuous function $K$ on $S^{d-1}:=\{x \in \mathbb{R}^{d}:|x|=1\}$, $ \lim_{t \to \infty} t^κ P(xR>t)=K(x)$ for all $x \in S^{d-1} $ holds true. This result is originally due to Kesten and Le Page. The purpose of this article is to show how regeneration methods can be used to provide a much shorter argument (in particular for the positivity of K). It is based on a multidimensional extension of Goldie's implicit renewal theory.

math.PR↗