Ergodicity Coefficient Research Articles

MAXIMIZING THE GROWTH RATE UNDER RISK CONSTRAINTS

We investigate the ergodic problem of growth‐rate maximization under a class of risk constraints in the context of incomplete, Itô‐process models of financial markets with random ergodic coefficients. Including value‐at‐risk, tail‐value‐at‐risk, and limited expected loss, these constraints can be both wealth‐dependent (relative) and wealth‐independent (absolute). The optimal policy is shown to exist in an appropriate admissibility class, and can be obtained explicitly by uniform, state‐dependent scaling down of the unconstrained (Merton) optimal portfolio. This implies that the risk‐constrained wealth‐growth optimizer locally behaves like a constant relative risk aversion (CRRA) investor, with the relative risk‐aversion coefficient depending on the current values of the market coefficients.

The case of equality in the Dobrushin–Deutsch–Zenger bound

Suppose that A = ( a i , j ) is an n × n real matrix with constant row sums μ . Then the Dobrushin–Deutsch–Zenger (DDZ) bound on the eigenvalues of A other than μ is given by Z ( A ) : = 1 2 max 1 ⩽ s , t ⩽ n ∑ r = 1 n | a s , r - a t , r | . When A a transition matrix of a finite homogeneous Markov chain so that μ = 1 , Z ( A ) is called the coefficient of ergodicity of the chain as it bounds the asymptotic rate of convergence, namely, max { | λ | | λ ∈ σ ( A ) ⧹ { 1 } } , of the iteration x i T = x i - 1 T A , to the stationary distribution vector of the chain. In this paper we study the structure of real matrices for which the DDZ bound is sharp. We apply our results to the study of the class of graphs for which the transition matrix arising from a random walk on the graph attains the bound. We also characterize the eigenvalues λ of A for which | λ | = Z ( A ) for some stochastic matrix A .

Primitive digraphs with the largest scrambling index

The scrambling index of a primitive digraph D is the smallest positive integer k such that for every pair of vertices u and v, there is a vertex w such that we can get to w from u and v in D by directed walks of length k; it is denoted by k(D). In [M. Akelbek, S. Kirkland, Coefficients of ergodicity and the scrambling index, preprint] we gave the upper bound on k(D) in terms of the order and the girth of a primitive digraph D. In this paper, we characterize all the primitive digraphs such that the scrambling index is equal to the upper bound.

Coefficients of ergodicity and the scrambling index

For a primitive stochastic matrix S, upper bounds on the second largest modulus of an eigenvalue of S are very important, because they determine the asymptotic rate of convergence of the sequence of powers of the corresponding matrix. In this paper, we introduce the definition of the scrambling index for a primitive digraph. The scrambling index of a primitive digraph D is the smallest positive integer k such that for every pair of vertices u and v, there is a vertex w such that we can get to w from u and v in D by directed walks of length k; it is denoted by k(D). We investigate the scrambling index for primitive digraphs, and give an upper bound on the scrambling index of a primitive digraph in terms of the order and the girth of the digraph. By doing so we provide an attainable upper bound on the second largest modulus of eigenvalues of a primitive matrix that make use of the scrambling index.

On optimal condition numbers for Markov chains

Let T and $${\tilde{T}=T-E}$$be arbitrary nonnegative, irreducible, stochastic matrices corresponding to two ergodic Markov chains on n states. A function κ is called a condition number for Markov chains with respect to the (α, β)–norm pair if $${\|\pi-\tilde{\pi}\|_\alpha \leq \kappa(T)\|E\|_\beta}$$. Here π and $${\tilde \pi}$$are the stationary distribution vectors of the two chains, respectively. Various condition numbers, particularly with respect to the (1, ∞) and (∞, ∞)-norm pairs have been suggested in the literature. They were ranked according to their size by Cho and Meyer in a paper from 2001. In this paper we first of all show that what we call the generalized ergodicity coefficient $${\tau_p(A^\#)={\rm sup}_{y^{t}e=0} \frac{\|y^tA^\#\|_p}{\|y\|_1}}$$, where e is the n-vector of all 1’s and A # is the group generalized inverse of A = I − T, is the smallest condition number of Markov chains with respect to the (p, ∞)-norm pair. We use this result to identify the smallest condition number of Markov chains among the (∞, ∞) and (1, ∞)-norm pairs. These are, respectively, κ 3 and κ 6 in the Cho–Meyer list of 8 condition numbers. Kirkland has studied κ 3(T). He has shown that $${\kappa_3(T)\geq\frac{n-1}{2n}}$$and he has characterized transition matrices for which equality holds. We prove here again that 2κ 3(T) ≤ κ(6) which appears in the Cho–Meyer paper and we characterize the transition matrices T for which $${\kappa_6(T)=\frac{n-1}{n}}$$. There is actually only one such matrix: T = (J n − I)/(n − 1), where J n is the n × n matrix of all 1’s.

Itô equation model for dispersion of solutes in heterogeneous media

Transport processes in heterogeneous media such as ionized plasmas, natural porous media, and turbulent atmosphere are often modeled as diffusion processes in random velocity fields. Using the Itô formalism, we decompose the second spatial moments of the concentration and the equivalent effective dispersion coefficients in terms corresponding to various physical factors which influence the transport. We explicitly define "ergodic'' dispersion coefficients, independent of the initial conditions and completely determined by local dispersion coefficients and velocity correlations. Ergodic coefficients govern an upscaled process which describes the transport at large tine-space scales. The non-ergodic behavior at finite times shown by numerical experiments for large initial plumes is explained by "memory terms'' accounting for correlations between initial positions and velocity fluctuations on the trajectories of the solute molecules.

A note on a variation of Doeblin's condition for uniform ergodicity of Markov chains

We introduce a simple variation of Doeblin's condition, Condition D*, that assures the uniform ergodicity of a Markov chain. It is also shown that for non-homogeneous chains our conditions are equivalent to Dobrushin's weak ergodic coefficient.

Sensitivity and convergence of uniformly ergodic Markov chains

For uniformly ergodic Markov chains, we obtain new perturbation bounds which relate the sensitivity of the chain under perturbation to its rate of convergence to stationarity. In particular, we derive sensitivity bounds in terms of the ergodicity coefficient of the iterated transition kernel, which improve upon the bounds obtained by other authors. We discuss convergence bounds that hold in the case of finite state space, and consider numerical examples to compare the accuracy of different perturbation bounds.

Sensitivity and convergence of uniformly ergodic Markov chains

For uniformly ergodic Markov chains, we obtain new perturbation bounds which relate the sensitivity of the chain under perturbation to its rate of convergence to stationarity. In particular, we derive sensitivity bounds in terms of the ergodicity coefficient of the iterated transition kernel, which improve upon the bounds obtained by other authors. We discuss convergence bounds that hold in the case of finite state space, and consider numerical examples to compare the accuracy of different perturbation bounds.

Chains with Complete Connections: General Theory, Uniqueness, Loss of Memory and Mixing Properties

We introduce a statistical mechanical formalism for the study of discrete-time stochastic processes with which we prove: (i) General properties of extremal chains, including triviality on the tail σ-algebra, short-range correlations, realization via infinite-volume limits and ergodicity. (ii) Two new sufficient conditions for the uniqueness of the consistent chain. The first one is a transcription of a criterion due to Georgii for one-dimensional Gibbs measures, and the second one corresponds to the Dobrushin’s criterion in statistical mechanics. (iii) Results on loss of memory and mixing properties for chains in the Dobrushin regime. These results are complementary to those existing in the literature, and generalize the Markovian results based on the Dobrushin ergodic coefficient.

Ergodicity coefficient and perturbation bounds for continuous-time Markov chains

For the distribution of a finite, homogeneous, continuous-time Markov chain, we derive perturbation bounds in terms of the ergodicity coefficient of the transition probability matrix. Our perturbation bounds improve upon the known results. We give sensitivity bounds for the coefficient of ergodicity, providing a sufficient condition for the uniqueness of the stationary distribution of the perturbed Markov chain. These results are used to obtain estimates of the speed of convergence for singularly perturbed Markov chains. Mathematics subject classification (2000): 60J27, 60J35, 34D10, 34D15.

Markov chain analysis of succession in a rocky subtidal community.

We present a Markov chain model of succession in a rocky subtidal community based on a long-term (1986-1994) study of subtidal invertebrates (14 species) at Ammen Rock Pinnacle in the Gulf of Maine. The model describes successional processes (disturbance, colonization, species persistence, and replacement), the equilibrium (stationary) community, and the rate of convergence. We described successional dynamics by species turnover rates, recurrence times, and the entropy of the transition matrix. We used perturbation analysis to quantify the response of diversity to successional rates and species removals. The equilibrium community was dominated by an encrusting sponge (Hymedesmia) and a bryozoan (Crisia eburnea). The equilibrium structure explained 98% of the variance in observed species frequencies. Dominant species have low probabilities of disturbance and high rates of colonization and persistence. On average, species turn over every 3.4 years. Recurrence times varied among species (7-268 years); rare species had the longest recurrence times. The community converged to equilibrium quickly (9.5 years), as measured by Dobrushin's coefficient of ergodicity. The largest changes in evenness would result from removal of the dominant sponge Hymedesmia. Subdominant species appear to increase evenness by slowing the dominance of Hymedesmia. Comparison of the subtidal community with intertidal and coral reef communities revealed that disturbance rates are an order of magnitude higher in coral reef than in rocky intertidal and subtidal communities. Colonization rates and turnover times, however, are lowest and longest in coral reefs, highest and shortest in intertidal communities, and intermediate in subtidal communities.

On the stability of the computation of the stationary probabilities of Markov chains using Perron complements

AbstractFor an n‐state ergodic homogeneous Markov chain whose transition matrix is T∈ℝn,n, it has been shown by Meyer on the one hand and by Kirkland, Neumann, and Xu on the other hand that the stationary distribution vector and that the mean first passage matrix, respectively, can be computed by a divide and conquer parallel method from the Perron complements of T. This is possible due to the facts, shown by Meyer, that the Perron complements of T are themselves transition matrices for finite ergodic homogeneous Markov chains with fewer states and that their stationary distribution vectors are multiples of the corresponding subvectors of the stationary distribution vector of the entire chain.Here we examine various questions concerning the stability of computing the stationary distribution vectors of the Perron complements and compare them with the stability of computing the stationary distribution vector for the entire chain. In particular, we obtain that condition numbers which are related to the coefficients of ergodicity improve as we pass from the entire chain to the chains associated with its Perron complements. Copyright © 2003 John Wiley & Sons, Ltd.

On contraction properties of Markov kernels

We study Lipschitz contraction properties of general Markov kernels seen as operators on spaces of probability measures equipped with entropy-like ``distances''. Universal quantitative bounds on the associated ergodic constants are deduced from Dobrushin's ergodic coefficient. Strong contraction properties in Orlicz spaces for relative densities are proved under more restrictive mixing assumptions. We also describe contraction bounds in the entropy sense around arbitrary probability measures by introducing a suitable Dirichlet form and the corresponding modified logarithmic Sobolev constants. The interest in these bounds is illustrated on the example of inhomogeneous Gaussian chains. In particular, the existence of an invariant measure is not required in general.

The influence of the log-conductivity autocovariance structure on macrodispersion coefficients

Macrodispersion coefficients are derived for heterogeneous porous media under ergodic and nonergodic conditions. Influences of the log-conductivity autocovariance function on macrodispersion are investigated through six commonly used isotropic log-conductivity autocorrelation models. They are the exponential, Gaussian, spherical, linear, Whittle and Mizell A-type models. Analytical expressions for ergodic macrodispersion coefficients for each of these models are presented. The results for nonergodic macrodispersion coefficients are calculated numerically. The results show that the various autocovariance functions, which display slight differences in the preasympototic region, have little effect on the ultimate macrodispersion coefficient. The effect of nonergodicity is more significant than the log-conductivity autocovariance function for the aquifers exhibiting unimodal log-conductivity distribution.

The Local Coefficient of Ergodicity of a Nonnegative Matrix

The local coefficient of ergodicity $\tau(T,Y',w)$ of a nonnegative column-allowable matrix T at a fixed positive vector Y is defined as the supremum of d(X'T,Y'T)/d(X',Y') for X not colinear to Y and d(X',Y')\leq w$ (d is the projective distance in the positive quadrant). A near-closed-form expression is given for $\tau(T,Y',w)$. If T' is scrambling (i.e., no two rows of T' are orthogonal), then for any Y>0, $w < \infty$ we have $\tau(T,Y',w)<1$. When Y is a positive left eigenvector of T and Xo>0, these results can be used to prove the convergence in direction of X'oTp to Y'. Results are illustrated with a numerical example.

On Ergodicity Coefficients of Infinite Stochastic Matrices

A class of ergodicity coefficients for infinite stochastic matrices is introduced and investigated with respect to connections to the well-known \delta -coefficient. The theory yields results on the behaviour of infinite products of stochastic matrices, in particular on inhomogeneous Markov chains and Markov systems.

Bounds for sums of eigenvalues and applications

Let A be a matrix of order n × n with real spectrum λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n . Let 1 ≤ k ≤ n − 2. If λ n or λ 1 is known, then we find an upper bound (respectively, lower bound) for the sum of the k-largest (respectively, k-smallest) remaining eigenvalues of A. Then, we obtain a majorization vector for ( λ 1, λ 2,…, λ n−1 ) when λ n is known and a majorization vector for ( λ 2, λ 3,…, λ n ) when λ 1 is known. We apply these results to the eigenvalues of the Laplacian matrix of a graph and, in particular, a sufficient condition for a graph to be connected is given. Also, we derive an upper bound for the coefficient of ergodicity of a nonnegative matrix with real spectrum.

Nonlinear Matrix Iterative Processes and Generalized Coefficients of Ergodicity

A nonlinear matrix iterative process is a linear dynamical system for which there is a nonlinear feedback of the current vector on the entries of the matrix. Stability conditions for an asymptotically exponential solution are studied for such a process in the positive quadrant of $\mathbb{R}^{n}.$ The results hinge on a generalization of the coefficient of ergodicity of a nonnegative matrix.

ON THE ALMOST-SURE ASYMPTOTIC STABILITY OF SECOND ORDER LINEAR STOCHASTIC SYSTEM

A method for obtaining a sufficient almost-sure stability condition for second order linear systems with an ergodic stiffness coefficient is presented. In this method, a special Lyapunov function for achieving functional optimization is constructed and the probabilistic property of the derivative process of the stiffness is taken into account. A sufficient condition for almost-sure asymptotic stability is derived and numerical results are presented for the cases of Gaussian noise and periodic noise coefficient. The results obtained here are an improvement over previously available results for linear systems with stochastic stiffness.