Sebastian Mentemeier

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

Given a sequence of i.i.d. random variables with generic copy such that M is a regular matrix and Q takes values in , we consider the random difference equation Under suitable assumptions stated below, this equation has a unique stationary solution R such that for some and some finite positive and continuous function K on , holds true. A rather long proof of this result, originally stated by Kesten [Acta Math. 131 (1973), pp. 207–248] at the end of his famous article, was given by Le Page [Séminaires de probabilités Rennes 1983, University of Rennes I, Rennes, 1983, p. 116]. The purpose of this article is to show how regeneration methods can be used to provide a much shorter argument (particularly for the positivity of K). It is based on a multidimensional extension of Goldies implicit renewal theory developed in Goldie [Ann. Appl. Probab. 1 (1991), pp. 126–166].

On multidimensional Mandelbrot cascades

Let Z be a random variable with values in a proper closed convex cone , A a random endomorphism of C and N a random integer. We assume that Z, A, N are independent. Given N independent copies of we define a new random variable . Let T be the corresponding transformation on the set of probability measures on C, i.e. T maps the law of Z to the law of . If the matrix has dominant eigenvalue 1, we study existence and properties of fixed points of T having finite non-zero expectation. Existing one-dimensional results concerning T are extended to higher dimensions. In particular we give conditions under which such fixed points of T have multidimensional regular variation in the sense of extreme value theory and we determine the index of regular variation.

HEAVY TAILED SOLUTIONS OF MULTIVARIATE SMOOTHING TRANSFORMS

Let N>1 be a fixed integer and (C1,…,CN,Q) a random element of M(d×d,R)N×Rd. We consider solutions of multivariate smoothing transforms, i.e. random variables R satisfying R=d∑i=1NCiRi+Q where =d denotes equality in distribution, and R,R1,…,RN are independent identically distributed Rd-valued random variables, and independent of (C1,…,CN,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 (C1,…,CN).

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

We consider real-valued random variables R satisfying the distributional equation

Precise large deviation results for products of random matrices

\displaystyle{ R\stackrel{d}{=}\sum _{k=1}^{N}T_{ k}R_{k} + Q, }

Solutions to complex smoothing equations

where \(R_{1},R_{2},\ldots\) are iid copies of R and independent of \(\mathbf{T} = (Q,(T_{k})_{k\geq 1})\). N is the number of nonzero weights T k and assumed to be a.s. finite. Its properties are governed by the function

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

\displaystyle{m(s):= \mathbb{E}\sum _{k=1}^{N}{\left \vert T_{ k}\right \vert }^{s}.}