Gerold Alsmeyer

On the Markov renewal theorem

Let (S, £) be a measurable space with countably generated [sigma]-field £ and (Mn, Xn)n[greater-or-equal, slanted]0 a Markov chain with state space S x and transition kernel :S x ( [circle times operator] )-->[0, 1]. Then (Mn,Sn)n[greater-or-equal, slanted]0, where Sn = X0+...+Xn for n[greater-or-equal, slanted]0, is called the associated Markov random walk. Markov renewal theory deals with the asymptotic behavior of suitable functionals of (Mn,Sn)n[greater-or-equal, slanted]0 like the Markov renewal measure [Sigma]n[greater-or-equal, slanted]0P((Mn,Sn)[epsilon]Ax (t+B)) as t-->[infinity] where A[epsilon] and B denotes a Borel subset of . It is shown that the Markov renewal theorem as well as a related ergodic theorem for semi-Markov processes hold true if only Harris recurrence of (Mn)n[greater-or-equal, slanted]0 is assumed. This was proved by purely analytical methods by Shurenkov [15] in the one-sided case where (x,Sx[0,[infinity])) = 1 for all x[epsilon]S. Our proof uses probabilistic arguments, notably the construction of regeneration epochs for (Mn)n[greater-or-equal, slanted]0 such that (Mn,Xn)n[greater-or-equal, slanted]0 is at least nearly regenerative and an extension of Blackwells renewal theorem to certain random walks with stationary, 1-dependent increments.

The functional equation of the smoothing transform

Given a sequence T = (Ti)i� 1 of non-negative random variables, a function f on the positive halfline can be transformed to E Q i� 1 f(tTi). We study the fixed points of this transform within the class of decreasing functions. By exploiting the intimate relationship with general branching processes, a full description of the set of solutions is established without the moment conditions that figure in earlier studies. Since the class of functions under consideration contains all Laplace transforms of probability distributions on [0,1 ), the results provide the full description of the set of solutions to the fixed-point equation of the smoothing transform, X d

Fixed Points of Inhomogeneous Smoothing Transforms

We consider the inhomogeneous version of the fixed-point equation of the smoothing transformation, that is, the equation , where means equality in distribution, is a given sequence of non-negative random variables and is a sequence of i.i.d. copies of the non-negative random variable X independent of . In this situation, X (or, more precisely, the distribution of X) is said to be a fixed point of the (inhomogeneous) smoothing transform. In the present paper, we give a necessary and sufficient condition for the existence of a fixed point. Furthermore, we establish an explicit one-to-one correspondence with the solutions to the corresponding homogeneous equation with C = 0. Using this correspondence and the known theory on the homogeneous equation, we present a full characterization of the set of fixed points under mild assumptions.

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].

Some relations between harmonic renewal measures and certain first passage times

Let X1, X2,... be i.i.d. random variables with common mean [mu] [greater-or-equal, slanted] 0 and associated random walk S0 = 0, Sn = X1 + ... + Xn, n [greater-or-equal, slanted] 1. Let U(t) = [Sigma]n [greater-or-equal, slanted] 1(1/n)P(Sn [less-than-or-equals, slant] t) be the harmonic renewal function of (Sn)n [greater-or-equal, slanted] 0 and [tau](t) = inf{itn [greater-or-equal, slanted] 1: Sn > t}. It is shown that U(t) = E[Psi]([tau](t)) + [gamma] for all t [greater-or-equal, slanted] 0, where [Psi](t) denotes Eulers psi function and [gamma] Eulers constant. This identity is further used to derive a number of interesting global and asymptotic properties of U(t). Some extensions to so-called generalized renewal measures are discussed in the final section.

SUPERPOSED CONTINUOUS RENEWAL PROCESSES : A MARKOV RENEWAL APPROACH

Lam and Lehoczky (1991) have recently given a number of extensions of classical renewal theorems to superpositions of p independent renewal processes. In this article we want to advertise an approach that more explicitly uses a Markov renewal theoretic framework and thus leads to a simplified derivation of their main results together with a number of new ones. Those include a Stone-type decomposition for the resulting Markov renewal measure and a number of convergence rate results which extend the corresponding results for single renewal processes.

On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results

AbstractA result by Elton(6) states that an iterated function system

Limit theorems for iterated random functions by regenerative methods

M_n = F_n (M_{n - 1} ),{\text{ }}n \geqslant 1,