Polynomial Rate Research Articles

Polynomial decay rate for a wave equation with general acoustic boundary feedback laws

We consider the stabilization of the wave equation by a general class of acoustic boundary feedback laws at one extremity. This system is not uniformly stable but we furnish sufficient conditions that guarantee a polynomial stability. Our method combines the use of an observability inequality for the associated undamped problem obtained via sharp spectral results with regularity results of the solution of the undamped problem with a specific right-hand side. In some particular cases, the optimality of the decay is shown again with the help of precise spectral results of the operator associated with the damped problem. We illustrate our general results by some examples.

Convergence of option rewards for multivariate price processes

American type options with general payoff functions possessing polynomial rate of growth are considered for multivariate Markov price processes.Convergence results areobtained for optimal reward fu ...

Nonparametric estimation of a renewal reward process from discrete data

We study the nonparametric estimation of the jump density of a renewal reward process from one discretely observed sample path over [0,T]. We consider regimes where the sampling rate goes to 0 as T tends to infinity. We propose an adaptive wavelet threshold density estimator and study its performance for the Lp loss, over Besov spaces. We achieve minimax rates of convergence for sampling rates that vanish with T at arbitrary polynomial rate. In the same spirit as Buchmann and Grubel (2003) the estimation procedure is based on the inversion of the compounding operator. The inverse has no closed form expression and is approached with a fixed point technique.

Frontier estimation with kernel regression on high order moments

We present a new method for estimating the frontier of a multidimensional sample. The estimator is based on a kernel regression on high order moments. It is assumed that the order of the moments goes to infinity while the bandwidth of the kernel goes to zero. The consistency of the estimator is proved under mild conditions on these two parameters. The asymptotic normality is also established when the conditional distribution function decreases at a polynomial rate to zero in the neighborhood of the frontier. The good performance of the estimator is illustrated in some finite sample situations.

Minimax optimal estimation of general bandable covariance matrices

Cai et al. (2010) [4] have studied the minimax optimal estimation of a collection of large bandable covariance matrices whose off-diagonal entries decay to zero at a polynomial rate. They have shown that the minimax optimal procedures are fundamentally different under Frobenius and spectral norms, regardless of the rate of polynomial decay. To gain more insight into this interesting problem, we study minimax estimation of large bandable covariance matrices over a parameter space characterized by a general positive decay function. We obtain explicit results to show how the decay function determines the minimax rates of convergence and the optimal procedures. From the general minimax analysis we find that for certain decay functions there is a tapering estimator that simultaneously attains the minimax optimal rates of convergence under the two norms. Moreover, we show that under the ultra-high dimension scenario it is possible to achieve adaptive minimax optimal estimation under the spectral norm. These new findings complement previous work.

On max-stable processes and the functional D-norm

We introduce a functional domain of attraction approach for stochastic processes, which is more general than the usual one based on weak convergence. The distribution function G of a continuous max-stable process on [0,1] is introduced and it is shown that G can be represented via a norm on functional space, called D-norm. This is in complete accordance with the multivariate case and leads to the definition of functional generalized Pareto distributions (GPD) W. These satisfy W=1+log(G) in their upper tails, again in complete accordance with the uni- or multivariate case. Applying this framework to copula processes we derive characterizations of the domain of attraction condition for copula processes in terms of tail equivalence with a functional GPD. \delta-neighborhoods of a functional GPD are introduced and it is shown that these are characterized by a polynomial rate of convergence of functional extremes, which is well-known in the multivariate case.

Nonuniform random geometric graphs with location-dependent radii

We propose a distribution-free approach to the study of random geometric graphs. The distribution of vertices follows a Poisson point process with intensity function $nf(\cdot)$, where $n\in \mathbb{N}$, and $f$ is a probability density function on $\mathbb{R}^{d}$. A vertex located at $x$ connects via directed edges to other vertices that are within a cut-off distance $r_{n}(x)$. We prove strong law results for (i) the critical cut-off function so that almost surely, the graph does not contain any node with out-degree zero for sufficiently large $n$ and (ii) the maximum and minimum vertex degrees. We also provide a characterization of the cut-off function for which the number of nodes with out-degree zero converges in distribution to a Poisson random variable. We illustrate this result for a class of densities with compact support that have at most polynomial rates of decay to zero. Finally, we state a sufficient condition for an enhanced version of the above graph to be almost surely connected eventually.

Subgeometric rates of convergence of the GI/G/1 queueing system

The article deals with the waiting time process of the GI/G/1 queueing system. We shall give that the rate of convergence to the stationary distribution and the decay of the stationary tail only depend on the tail of the service distribution, but not on the interarrival distribution. We shall also give explicit criteria for the rate of convergence and decay of stationary tail for three specific types of subgeometric cases (Case 1: the rate function r(n)=exp(sn11+α),α>0, s>0; Case 2: polynomial rate function r(n) = nα, α > 0; Case 3: logarithmic rate function r(n) = logαn, α > 0). Key words GI/G/1 queueing system; subgeometric rate of convergence; po

Pathwise estimation of stochastic functional Kolmogorov-type systems with infinite delay

In this paper, we study pathwise estimation of the global positive solutions for the stochastic functional Kolmogorov-type systems with infinite delay. Under some conditions, the growth rate of the solutions for such systems with general noise structures is less than a polynomial rate in the almost sure sense. To illustrate the applications of our theory more clearly, this paper also discusses various stochastic Lotka-Volterra-type systems as special cases. MSC:34K50, 60H10, 92D25, 93E03.

ON THE SPEED OF CONVERGENCE OF MULTIDIMENSIONAL DIFFUSIONS TO EQUILIBRIUM

We obtain new criteria for polynomial rates of convergence of ergodic multidimensional diffusions to equilibrium. For this, (1) a method is provided for adapting to continuous-time Markov processes coupling techniques for discrete parameter Harris processes, and (2) estimates of moments of return times to a ball are derived.

A note on Borel–Cantelli lemmas for non-uniformly hyperbolic dynamical systems

AbstractLet (Bi) be a sequence of measurable sets in a probability space (X,ℬ,μ) such that ∑ ∞n=1μ(Bi)=∞. The classical Borel–Cantelli lemma states that if the sets Bi are independent, then μ({x∈X:x∈Bi infinitely often})=1. Suppose (T,X,μ) is a dynamical system and (Bi) is a sequence of sets in X. We consider whether Tix∈Bi infinitely often for μ almost every x∈X and, if so, is there an asymptotic estimate on the rate of entry? If Tix∈Bi infinitely often for μ almost every x, we call the sequence (Bi) a Borel–Cantelli sequence. If the sets Bi :=B(p,ri) are nested balls of radius ri about a point p, then the question of whether Tix∈Bi infinitely often for μ almost every x is often called the shrinking target problem. We show, under certain assumptions on the measure μ, that for balls Bi if μ(Bi)≥i−γ, 0<γ<1, then a sufficiently high polynomial rate of decay of correlations for Lipschitz observables implies that the sequence is Borel–Cantelli. If μ(Bi)≥C1 /i, then exponential decay of correlations implies that the sequence is Borel–Cantelli. We give conditions in terms of return time statistics which quantify Borel–Cantelli results for sequences of balls such that μ(Bi)≥C/i. Corollaries of our results are that for planar dispersing billiards and Lozi maps, sequences of nested balls B(p,1/i) are Borel–Cantelli. We also give applications of these results to a variety of non-uniformly hyperbolic dynamical systems.

Some examples of $C^\infty$ extension by linear operators

For two kinds of sets X in \mathbb{R}^n , we prove the existence of linear continuous operators extending C^\infty functions on X to C^\infty functions on \mathbb{R}^n . The sets we consider are: (a) sequences of points in the real line converging to 0 at a polynomial rate, (b) flag-shaped sets in the plane, which are unions of half-lines with slopes as in (a).

Deterministic Designs with Deterministic Guarantees: Toeplitz Compressed Sensing Matrices, Sequence Design and System ID

In this paper we present a new family of discrete sequences having random like uniformly decaying auto-correlation properties. The new class of infinite length sequences are higher order chirps constructed using irrational numbers. Exploiting results from the theory of continued fractions and diophantine approximations, we show that the class of sequences so formed has the property that the worst-case auto-correlation coefficients for every finite length sequence decays at a polynomial rate. These sequences display doppler immunity as well. We also show that Toeplitz matrices formed from such sequences satisfy restricted-isometry-property (RIP), a concept that has played a central role recently in Compressed Sensing applications. Compressed sensing has conventionally dealt with sensing matrices with arbitrary components. Nevertheless, such arbitrary sensing matrices are not appropriate for linear system identification and one must employ Toeplitz structured sensing matrices. Linear system identification plays a central role in a wide variety of applications such as channel estimation for multipath wireless systems as well as control system applications. Toeplitz matrices are also desirable on account of their filtering structure, which allows for fast implementation together with reduced storage requirements.

On the wave operator for the generalized Boussinesq equation

We study the asymptotic behavior of solutions for the generalized Boussinesq equation in a singular framework. We construct a wave operator (inverse scattering) for large profiles ~h belonging to an infinite mass framework based on weak-L spaces. Our solutions converge towards the prescribed scattering state with a polynomial rate. 2000 MSC: 35Q35, 35Q55, 35P25, 35B40, 35L15, 42B35

Estimating an endpoint with high-order moments

We present a new method for estimating the endpoint of a unidimensional sample when the distribution function decreases at a polynomial rate to zero in the neighborhood of the endpoint. The estimator is based on the use of high-order moments of the variable of interest. It is assumed that the order of the moments goes to infinity, and we give conditions on its rate of divergence to get the asymptotic normality of the estimator. The good performance of the estimator is illustrated on some finite sample situations.

Scaling rate for semi-dispersing billiards with non-compact cusps

AbstractWe show that certain billiard tables with non-compact cusps are mixing with respect to the invariant infinite measure, in the sense of Krengel and Sucheston. Moreover, we show that the scaling rate is slower than a certain polynomial rate.

Dynamics of vertex-reinforced random walks

We generalize a result from Volkov [Ann. Probab. 29 (2001) 66--91] and prove that, on a large class of locally finite connected graphs of bounded degree $(G,\sim)$ and symmetric reinforcement matrices $a=(a_{i,j})_{i,j\in G}$, the vertex-reinforced random walk (VRRW) eventually localizes with positive probability on subsets which consist of a complete $d$-partite subgraph with possible loops plus its outer boundary. We first show that, in general, any stable equilibrium of a linear symmetric replicator dynamics with positive payoffs on a graph $G$ satisfies the property that its support is a complete $d$-partite subgraph of $G$ with possible loops, for some $d\ge1$. This result is used here for the study of VRRWs, but also applies to other contexts such as evolutionary models in population genetics and game theory. Next we generalize the result of Pemantle [Probab. Theory Related Fields 92 (1992) 117--136] and Bena\"{{\i}}m [Ann. Probab. 25 (1997) 361--392] relating the asymptotic behavior of the VRRW to replicator dynamics. This enables us to conclude that, given any neighborhood of a strictly stable equilibrium with support $S$, the following event occurs with positive probability: the walk localizes on $S\cup\partial S$ (where $\partial S$ is the outer boundary of $S$) and the density of occupation of the VRRW converges, with polynomial rate, to a strictly stable equilibrium in this neighborhood.

Conservative Interacting Particles System with Anomalous Rate of Ergodicity

We analyze certain conservative interacting particle system and establish ergodicity of the system for a family of invariant measures. Furthermore, we show that convergence rate to equilibrium is exponential. This result is of interest because it presents counterexample to the standard assumption of physicists that conservative system implies polynomial rate of convergence.

Constants in the estimates of the rate of convergence in von Neumann’s ergodic theorem with continuous time

Estimates for the rate of convergence in ergodic theorems are necessarily spectral. We find the equivalence constants relating the polynomial rate of convergence in von Neumann’s mean ergodic theorem with continuous time and the polynomial singularity at the origin of the spectral measure of the function averaged over the corresponding dynamical system. We also estimate the same rate of convergence with respect to the decrease rate of the correlation function. All results of this article have obvious exact analogs for the stochastic processes stationary in the wide sense.

On the distribution for sums of partial quotients in continued fraction expansions

Let x ∊ [0, 1) and [a1(x), a2(x), …] be the continued fraction expansion of x. For any n ⩾ 1, write . Khintchine (1935 Compos. Math. 1 361–82) proved that converges in measure to with respect to , where denotes the one dimensional Lebesgue measure. Philipp (1988 Monatsh. Math. 105 195–206) showed that there is not a reasonable normalizing sequence such that a strong law of large numbers is satisfied. In this paper, we show that for any α ⩾ 0, the set is of Hausdorff dimension 1. Furthermore, we prove that the Hausdorff dimension of the set consisting of reals whose sums of partial quotients grow at a given polynomial rate is 1.