Esa Nummelin

A Splitting Technique for Harris Recurrent Markov Chains

SummaryA technique is presented, which enables the state space of a Harris recurrent Markov chain to be “split” in a way, which introduces into the split state space an “atom”. Hence the full force of renewal theory can be used in the analysis of Markov chains on a general state space. As a first illustration of the method we show how Dermans construction for the invariant measure works in the general state space. The Splitting Technique is also applied to the study of sums of transition probabilities.

Geometric ergodicity of Harris recurrent Marcov chains with applications to renewal theory

Let (Xn) be a positive recurrent Harris chain on a general state space, with invariant probability measure [pi]. We give necessary and sufficient conditions for the geometric convergence of [lambda]Pnf towards its limit [pi](f), and show that when such convergence happens it is, in fact, uniform over f and in L1([pi])-norm. As a corollary we obtain that, when (Xn) is geometrically ergodic, [is proportional to] [pi](dx)||Pn(x,·)-[pi]|| converges to zero geometrically fast. We also characterize the geometric ergodicity of (Xn) in terms of hitting time distributions. We show that here the so-called small sets act like individual points of a countable state space chain. We give a test function criterion for geometric ergodicity and apply it to random walks on the positive half line. We apply these results to non-singular renewal processes on [0,[infinity]) providing a probabilistic approach to the exponencial convergence of renewal measures.

The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory

We derive sufficient conditions for [is proportional to] [lambda] (dx)||Pn(x, ·) - [pi]|| to be of order o([psi](n)-1), where Pn (x, A) are the transition probabilities of an aperiodic Harris recurrent Markov chain, [pi] is the invariant probability measure, [lambda] an initial distribution and [psi] belongs to a suitable class of non-decreasing sequences. The basic condition involved is the ergodicity of order [psi], which in a countable state space is equivalent to [Sigma] [psi](n)Pi{[tau]i[greater-or-equal, slanted]n} 0 and [is proportional to] [psi](t)(1- F(t))dt

The kink of cellular automaton rule 18 performs a random walk

We give an exact characterization of the movement of a single kink in the elementary cellular automaton Rule 18. It is a random walk with independent increments as well as independent delay times. Its statistical parameters are computed to confirm the earlier simulation results by Grassberger.

On non-singular renewal kernels with an application to a semigroup of transition kernels

We study the non-singularity and limit properties of the renewal kernel R=[summation operator]K*n associated with a positive convolution kernel K(x,dyxdt) defined on a general measurable space (E, ). The principal tool is the use of embedded renewal measures. As an application we consider continuous parameter semigroups (Rt(x,dy);t[greater-or-equal, slanted]0) of transition kernels on (E, ).

Some limit theorems for Markov additive processes

At the Semi-Markov Symposium we presented some new results on Markov-additive processes which will be published in Ney and Nummelin (1984), Ney and Nummelin (1985). In their proofs we used regeneration constructions similar to those in several previous papers (Athreya and Ney, 1978; Nummelin, 1978; Iscoe, Ney and Nummelin, 1984). The proof of the particular regeneration used in Ney and Nummelin (1984) was omitted, and we will now provide it here. We also prove a slight extension of the results in Iscoe, Ney and Nummelin (1984); Ney and Nummelin (1984), and summarize the results we announced at the symposium.

On the concepts of [alpha]-recurrence and [alpha]-transience for Markov renewal process

Let I be a denumerable set and let Q = (Qij)i,j[set membership, variant]l be an irreducible semi-Markov kernel. The main results of the paper are: 1. (i) Q is [alpha]-recurrent (resp. [alpha]-transient, [alpha]-positive recurrent, [alpha]-null recurrent) if and only if it can be written in the form 73, where 0

Large deviations for functionals of stationary processes

SummaryLet (Sn) be a sequence ofRd-valued random variables adapted to the internal history of a stationary sequence of random elements (Xn). We formulate conditions under which the principle of large deviations holds true for the sequence (Sn).

Regeneration for chains with infinite memory

SummaryA regeneration structure is established for chains with infinite memory. The memory is required to decay only along a single recurrent path. When there are many recurrent paths (e.g. under conservativity) the construction yields a decomposition into regenerative recurrent classes.

Renewal representations for Markov operators

Abstract As is known, due to the existence of an embedded renewal structure, the iterates of a Harris recurrent Markov operator can be represented as a (delayed) renewal sequence. We show that these kind of representations also exist for a larger class of Markov operators, provided only that certain “filling schemes” are “successful.” As applications of the theory we study the co-Feller operators and Markov operators which “contract the variation.”