Ergodic sequence

Abstract: A homeomorphism U of a compact metric space onto itself is said to be uniquely ergodic if it possesses a unique invariant Borel probability measure /U. For an introduction to the theory of unique ergodicity, we refer the reader to J. Oxtoby [11]. A point x in the shift space (z, where Q is a finite state space, is called a uniquely ergodic sequence if the shift S, (Sx)i = xi+1, where i E Z, x e z, is a uniquely ergodic homeomorphism of the orbit closure Qx = {S'x: i E Z} ofx. We denote the shift invariant probability measure ofa uniquely ergodic sequence x by p., S. Kakutani [7], M. Keane [8], and K. Jacobs and M. Keane [5] have constructed a variety of uniquely ergodic sequences and investigated their measure theoretic properties. The first examples of weakly mixing uniquely ergodic systems were given by Jacobs [3]. F. Hahn and Y. Katznelson [2] constructed uniquely ergodic sequences with arbitrarily high entropy and Ch. Grillenberger [1] produced uniquely ergodic sequences in flZ whose entropy is arbitrarily close to log IQI Further constructions of uniquely ergodic sequences were given by W. Veech (Section 3 of [12]). We shall prove in Section 3 that for every ergodic shift invariant measure ju on lz whose entropy h(t) is less than log IQI there exists a uniquely ergodic sequence x e QZ such that the systems (flZ, p, S) and (C., p., S) are isomorphic and such that py. is in any given weak neighborhood of it. This result and the finite generator theorem for ergodic measure preserving transformations (see [9] and [10]) imply that every ergodic measure preserving invertible transformation T of a Lebesgue measure space with finite entropy h(T) is isomorphic to a system (0,, ui,, S), where x is a uniquely ergodic sequence in Qz and exp {h(T)} < IQI _ exp {h(T)} + 1. In Section 4, we shall show that every ergodic invertible measure preserving transformation T of a Lebesgue measure space is isomorphic to a system (U, C, hu), where U is a uniquely ergodic homeomorphism of the Cantor discontinuum C. This was recently established by R. Jewett [6] under the additional assumption that T be weakly mixing, and conjectured by him to hold in the ergodic case. Our method of proof combines the basic idea of Jewett with the methods that were developed for the proof of the finite generator theorem for ergodic measure preserving transformations (see [9] and [10]). We require some tools that we develop in Section 2. Jacobs has recently shown that every weakly mixing flow on a Lebesgue measure space is isomorphic to a flow of homeomorphisms of a compact metric space together with a unique invariant Borel measure [4].

Abstract: When solving a convex optimization problem through a Lagrangian dual reformulation subgradient optimization methods are favorably utilized, since they often find near-optimal dual solutions quickly. However, an optimal primal solution is generally not obtained directly through such a subgradient approach unless the Lagrangian dual function is differentiable at an optimal solution. We construct a sequence of convex combinations of primal subproblem solutions, a so called ergodic sequence, which is shown to converge to an optimal primal solution when the convexity weights are appropriately chosen. We generalize previous convergence results from linear to convex optimization and present a new set of rules for constructing the convexity weights that define the ergodic sequence of primal solutions. In contrast to previously proposed rules, they exploit more information from later subproblem solutions than from earlier ones. We evaluate the proposed rules on a set of nonlinear multicommodity flow problems and demonstrate that they clearly outperform the ones previously proposed.

Abstract: We consider recursions of the form x/sub n+1/=/spl phi//sub n/[x/sub n/], where {/spl phi//sub n/, n/spl ges/0} is a stationary ergodic sequence of maps from a Polish space (E, /spl epsiv/) into itself, and {x/sub n/, n/spl ges/0} are random variables taking values in (E, /spl epsiv/). The question of when stationary solutions exist for such recursions, whether they are unique, and whether there is convergence to a stationary solution starting from arbitrary initial conditions is of considerable interest in discrete event system applications. Currently available techniques can only answer such questions under strong simplifying assumptions on the statistics of {/spl phi//sub n/}/sub n/, (such as Markov assumptions), or on the nature of these maps (such as monotonicity), In this paper we introduce a new technique for studying stochastic recursions without such simplifying assumptions. To do so, we weaken the solution concept: rather than constructing a pathwise solution we construct a probability measure on another sample space and families of random variables on this space whose law gives a stationary solution to the recursion. The problem of existence of a stationary solution is then translated into the problem of establishing tightness of a sequence of probability distributions, and uniqueness questions can be addressed using techniques familiar from the ergodic theory of positive Markov operators on spaces of continuous functions. >