Search

In this paper, we study the limiting behavior of the perimeter and diameter functionals of the convex hull spanned by the first $n$ steps of two planar random walks. As the main results, we obtain the strong law of large numbers and the central limit theorem for the perimeter and diameter of these random sets., Comment: 36 pages, 14 figures

In this article, basing upon probabilistic methods, we discuss periodic homogenization of a class of weakly coupled systems of linear elliptic and parabolic partial differential equations. Under the assumption that the systems have rapidly periodically oscillating coefficients, we first prove that the appropriately centered and scaled continuous component of the associated regime switching diffusion process converges weakly to a Brownian motion with covariance matrix given in terms of the coefficients of the systems. The homogenization results then follow by employing probabilistic representation of the solutions to the systems and the continuous mapping theorem. The presented results generalize the well-known results related to periodic homogenization of the classical elliptic boundary-value problem and the classical parabolic initial-value problem for a single equation., Comment: arXiv admin note: text overlap with arXiv:2102.04935

We establish laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is different, were obtained by Khoshnevisan. Our starting point is a version of Strassen's functional law of the iterated logarithm for random walks with drift. For the special case of the area of a planar random walk with drift, we compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the classical Dido problem. For general intrinsic volumes and dimensions, our proof exploits a novel zero--one law for functionals of convex hulls of walks with drift, of some independent interest. As another application of our approach, we obtain iterated-logarithm laws for intrinsic volumes of the convex hull of the centre of mass (running average) process associated to the random walk., Comment: Accepted for publication in TAMS

In this article, we discuss subgeometric ergodicity of a class of regime-switching diffusion processes. We derive conditions on the drift and diffusion coefficients, and the switching mechanism which result in subgeometric ergodicity of the corresponding semigroup with respect to the total variation distance as well as a class of Wasserstein distances. At the end, subgeometric ergodicity of certain classes of regime-switching Markov processes with jumps is also discussed., Comment: arXiv admin note: text overlap with arXiv:2006.01567

We consider convex hulls of random walks whose steps belong to the domain of attraction of a stable law in $\mathbb{R}^d$. We prove convergence of the convex hull in the space of all convex and compact subsets of $\mathbb{R}^d$, equipped with the Hausdorff distance, towards the convex hull spanned by a path of the limit stable L\'{e}vy process. As an application, we establish convergence of (expected) intrinsic volumes under some mild moment/structure assumptions posed on the random walk., Comment: 28 pages

In this article, we establish Hoeffding's inequality for bounded Lipschitz functions of a class of not necessarily irreducible Markov models. The result complements the existing literature on this topic where Hoeffding's inequality for bounded measurable functions of a class of irreducible Markov models has been considered. Our approach is based on the assumption of uniform ergodicity of the underlying Markov model in $L^1$-Wasserstein space., Comment: 9 pages

Mostof the existing literature on supervised machine learning problems focuses on the case when the training data set is drawn from an i.i.d. sample. However, many practical problems are characterized by temporal dependence and strong correlation between the marginals of the data-generating process, suggesting that the i.i.d. assumption is not always justified. This problem has been already considered in the context of Markov chains satisfying the Doeblin condition. This condition, among other things, implies that the chain is not singular in its behavior, i.e. it is irreducible. In this article, we focus on the case when the training data set is drawn from a not necessarily irreducible Markov chain. Under the assumption that the chain is uniformly ergodic with respect to the $\mathrm{L}^1$-Wasserstein distance, and certain regularity assumptions on the hypothesis class and the state space of the chain, we first obtain a uniform convergence result for the corresponding sample error, and then we conclude learnability of the approximate sample error minimization algorithm and find its generalization bounds. At the end, a relative uniform convergence result for the sample error is also discussed.

In this paper, we discuss asymptotic behavior of the capacity of the range of symmetric simple random walks on finitely generated groups. We show the corresponding strong law of large numbers and central limit theorem., Comment: 18 pages

In this article, we obtain a functional CLT for a class of degenerate diffusion processes with periodic coefficients, thus generalizing the already classical results in the context of uniformly elliptic diffusions. As an application, we also discuss periodic homogenization of a class of linear degenerate elliptic and parabolic PDEs.

In this article, we discuss ergodicity properties of a diffusion process given through an It\^{o} stochastic differential equation. We identify conditions on the drift and diffusion coefficients which result in sub-geometric ergodicity of the corresponding semigroup with respect to the total variation distance. We also prove sub-geometric contractivity and ergodicity of the semigroup under a class of Wasserstein distances. Finally, we discuss sub-geometric ergodicity of two classes of Markov processes with jumps.

In this article, we study fluctuations of the volume of a stable sausage defined via a $d$-dimensional rotationally invariant $\alpha$-stable process. As the main results, we establish a functional central limit theorem (in the case when $d/\alpha>3 /2$) with a standard one-dimensional Brownian motion in the limit, and Khintchine's and Chung's laws of the iterated logarithm (in the case when $d/\alpha>9 /5$)., Comment: Due to a problem in the proof, almost sure invariance principle has been removed and the laws of the iterated logarithms are proved for the case when $d/\alpha>9/5.$

In this article, we consider the problem of periodic homogenization of a Feller process generated by a pseudo-differential operator, the so-called L\'evy-type process. Under the assumptions that the generator has rapidly periodically oscillating coefficients, and that it admits "small jumps" only (that is, the jump kernel has finite second moment), we prove that the appropriately centered and scaled process converges weakly to a Brownian motion with covariance matrix given in terms of the coefficients of the generator. The presented results generalize the classical and well-known results related to periodic homogenization of a diffusion process.

We prove an almost sure invariance principle for the capacity and the cardinality of the range of a class of $\alpha$-stable random walks on the integer lattice $\mathbb{Z}^d$ with $d/\alpha > 5/2$, and $d/\alpha >3/2$, respectively. As a direct consequence, we conclude Khintchine's and Chung's laws of the iterated logarithm for both processes., Comment: Accepted for publication in Stochastic Processes and their Applications

In this note, we establish a functional central limit theorem for the capacity of the range for a class of $\alpha$-stable random walks on the integer lattice $\mathbb{Z}^d$ with $d > 5\alpha/2$. Using similar methods, we also prove an analogous result for the cardinality of the range when $d > 3\alpha / 2$.

In this article, relying on Foster-Lyapunov drift conditions, we establish subexponential upper and lower bounds on the rate of convergence in the $\mathrm{L}^p$-Wasserstein distance for a class of irreducible and aperiodic Markov processes. We further discuss these results in the context of Markov L\'evy-type processes. In the lack of irreducibility and/or aperiodicity properties, we obtain exponential ergodicity in the $\mathrm{L}^p$-Wasserstein distance for a class of It\^{o} processes under an asymptotic flatness (uniform dissipativity) assumption. Lastly, applications of these results to specific processes are presented, including Langevin tempered diffusion processes, piecewise Ornstein-Uhlenbeck processes with jumps under constant and stationary Markov controls, and backward recurrence time chains, for which we provide a sharp characterization of the rate of convergence via matching upper and lower bounds., Comment: 32 pages

In this article, we establish a central limit theorem for the capacity of the range process for a class of $d$-dimensional symmetric $\alpha$-stable random walks with the index satisfying $d > 5\alpha /2$. Our approach is based on controlling the limit behavior of the variance of the capacity of the range process which then allows us to apply the Lindeberg-Feller theorem.

We study the ergodic properties of a class of controlled stochastic differential equations (SDEs) driven by $\alpha$-stable processes which arise as the limiting equations of multiclass queueing models in the Halfin-Whitt regime that have heavy-tailed arrival processes. When the safety staffing parameter is positive, we show that the SDEs are uniformly ergodic and enjoy a polynomial rate of convergence to the invariant probability measure in total variation, which is uniform over all stationary Markov controls resulting in a locally Lipschitz continuous drift. We also derive a matching lower bound on the rate of convergence (under no abandonment). On the other hand, when all abandonment rates are positive, we show that the SDEs are exponentially ergodic uniformly over the above-mentioned class of controls. Analogous results are obtained for L\'evy-driven SDEs arising from multiclass many-server queues under asymptotically negligible service interruptions. For these equations, we show that the aforementioned ergodic properties are uniform over all stationary Markov controls. We also extend a key functional central limit theorem concerning diffusion approximations so as to make it applicable to the models studied here.

In this article, we discuss stability of the one-dimensional overdamped Lange\-vin equation in double-well potential. We determine unstable and stable equilibria, and discuss the rate of convergence to stable ones. Also, we derive conditions for stability of general diffusion processes which generalize the classical and well-known results of Khasminskii 2012.

We study the ergodic properties of a class of multidimensional piecewise Ornstein-Uhlenbeck processes with jumps, which contains the limit of the queueing processes arising in multiclass many-server queues with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin-Whitt regime as special cases. In these queueing models, the It\^o equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump L\'evy process, or (2) an anisotropic L\'evy process with independent one-dimensional symmetric $\alpha$-stable components, or (3) an anisotropic L\'evy process as in (2) and a pure-jump L\'evy process. We also study the class of models driven by a subordinate Brownian motion, which contains an isotropic (or rotationally invariant) $\alpha$-stable L\'evy process as a special case. We identify conditions on the parameters in the drift, the L\'evy measure and/or covariance function which result in subexponential and/or exponential ergodicity. We show that these assumptions are sharp, and we identify some key necessary conditions for the process to be ergodic. In addition, we show that for the queueing models described above with no abandonment, the rate of convergence is polynomial, and we provide a sharp quantitative characterization of the rate via matching upper and lower bounds., Comment: 42 pages

The classical Birkhoff ergodic theorem states that for an ergodic Markov process the limiting behaviour of the time average of a function (having finite $p$-th moment, $p\ge1$, with respect to the invariant measure) along the trajectories of the process, starting from the invariant measure, is a.s. and in the $p$-th mean constant and equals to the space average of the function with respect to the invariant measure. The crucial assumption here is that the process starts from the invariant measure, which is not always the case. In this paper, under the assumptions that the underlying process is a Markov process on Polish space, that it admits an invariant probability measure and that its marginal distributions converge to the invariant measure in the $L^{1}$-Wasserstein metric, we show that the assertion of the Birkhoff ergodic theorem holds in the $p$-th mean, $p\geq1$, for any bounded Lipschitz function and any initial distribution of the process.

In this paper we discuss weak convergence of continuous-time Markov chains to a non-symmetric pure jump process. We approach this problem using Dirichlet forms as well as semimartingales. As an application, we discuss how to approximate a given Markov process by Markov chains.

In this paper, we study weak and strong transience of a class of Feller processes associated with pseudo-differential operators, the so-called L\'evy-type processes. As a main result, we derive Chung-Fuchs type conditions (in terms of the symbol of the corresponding pseudo-differential operator) for these properties, which are sharp for L\'evy processes. Also, as a consequence, we discuss the weak and strong transience with respect to the dimension of the state space and Pruitt indices, thus generalizing some well-known results related to elliptic diffusion and stable L\'evy processes. Finally, in the case when the symbol is radial (in the co-variable) we provide conditions for the weak and strong transience in terms of the L\'evy measures.

In this paper, conditions for transience, recurrence, ergodicity and strong, subexponential (polynomial) and exponential ergodicity of a class of Feller processes are derived. The conditions are given in terms of the coefficients of the corresponding infinitesimal generator. As a consequence, mixing properties of these processes are also discussed.

In this paper, we study the homogenization of a diffusion process with jumps, that is, Feller process generated by an integro-differential operator. This problem is closely related to the problem of homogenization of boundary value problems arising in studying the behavior of heterogeneous media. Under the assumptions that the corresponding generator has vanishing drift coefficient, rapidly periodically oscillating diffusion and jump coefficients, that it admits only "small jumps" (that is, the jump kernel has finite second moment) and under certain additional regularity conditions, we prove that the homogenized process is a Brownian motion. The presented results generalize the classical and well-known results related to the homogenization of a diffusion process., Comment: arXiv admin note: text overlap with arXiv:1502.04440

In this paper, we study recurrence and transience of L\'evy-type processes, that is, Feller processes associated with pseudo-differential operators. Since the recurrence property of L\'evy-type processes in dimensions greater than two is vacuous and the recurrence and transience of one-dimensional L\'evy-type processes have been very well investigated, in this paper we are focused on the two-dimensional case only. In particular, we study perturbations of two-dimensional L\'evy-type processes which do not affect their recurrence and transience properties, we derive sufficient conditions for their recurrence and transience in terms of the corresponding L\'evy measure and we provide some comparison conditions for the recurrence and transience also in terms of the L\'evy measures., Comment: arXiv admin note: text overlap with arXiv:1401.2646

In this paper, we study the long-time behavior of a fluid particle immersed in a turbulent fluid driven by a diffusion with jumps, that is, a Feller process associated with a non-local operator. We derive the law of large numbers and central limit theorem for the evolution process of the tracked fluid particle in the cases when the driving process: (i) has periodic coefficients, (ii) is ergodic or (iii) is a class of L\'evy processes. The presented results generalize the classical and well-known results for fluid flows driven by elliptic diffusion processes.

A stable-like Markov chain is a time-homogeneous Markov chain on the real line with the transition kernel $p(x,dy)=f_x(y-x)dy$, where the density functions $f_x(y)$, for large $|y|$, have a power-law decay with exponent $\alpha(x)+1$, where $\alpha(x)\in(0,2)$. In this paper, under a certain uniformity condition on the density functions $f_x(y)$ and additional mild drift conditions, we give sufficient conditions for recurrence in the case when $0<\liminf_{|x|\longrightarrow\infty}\alpha(x)$, sufficient conditions for transience in the case when $\limsup_{|x|\longrightarrow\infty}\alpha(x)<2$ and sufficient conditions for ergodicity in the case when $0<\inf\{\alpha(x):x\in\mathbb{R}\}$. As a special case of these results, we give a new proof for the recurrence and transience property of a symmetric $\alpha$-stable random walk on $\mathbb{R}$ with the index of stability $\alpha\neq1.$, Comment: arXiv admin note: text overlap with arXiv:1203.0447

In this paper, as a main result, we derive a Chung-Fuchs type condition for the recurrence of Feller processes associated with pseudo-differential operators. In the L\'evy process case, this condition reduces to the classical and well-known Chung-Fuchs condition. Further, we also discuss the recurrence and transience of Feller processes with respect to the dimension of the state space and Pruitt indices and the recurrence and transience of Feller-Dynkin diffusions and stable-like processes. Finally, in the one-dimensional symmetric case, we study perturbations of Feller processes which do not affect their recurrence and transience properties, and we derive sufficient conditions for their recurrence and transience in terms of the corresponding L\'evy measure. In addition, some comparison conditions for recurrence and transience also in terms of the L\'evy measures are obtained.

In this paper, we give a sufficient condition for transience for a class of one-dimensional symmetric L\'evy processes. More precisely, we prove that a one-dimensional symmetric L\'evy process with the L\'evy measure $\nu(dy)=f(y)dy$ or $\nu(\{n\})=p_n$, where the density function $f(y)$ is such that $f(y)>0$ a.e. and the sequence $\{p_n\}_{n\geq1}$ is such that $p_n>0$ for all $n\geq1$, is transient if $$\int_1^{\infty}\frac{dy}{y^{3}f(y)}<\infty\quad\textrm{or}\quad \sum_{n=1}^{\infty}\frac{1}{n^{3}p_n}<\infty.$$ Similarly, we derive an analogous transience condition for one-dimensional symmetric random walks with continuous and discrete jumps., Comment: 13 pages

In this paper, we consider a long-time behavior of stable-like processes. A stable-like process is a Feller process given by the symbol $p(x,\xi)=-i\beta(x)\xi+\gamma(x)|\xi|^{\alpha(x)},$ where $\alpha(x)\in(0,2)$, $\beta(x)\in\R$ and $\gamma(x)\in(0,\infty)$. More precisely, we give sufficient conditions for recurrence, transience and ergodicity of stable-like processes in terms of the stability function $\alpha(x)$, the drift function $\beta(x)$ and the scaling function $\gamma(x)$. Further, as a special case of these results we give a new proof for the recurrence and transience property of one-dimensional symmetric stable L\'{e}vy processes with the index of stability $\alpha\neq1.$, Comment: To appear in: Stochastic Processes and their Applications

We give recurrence and transience criteria for two cases of time-homogeneous Markov chains on the real line with transition kernel $p(x,dy)=f_x(y-x)dy$, where $f_x(y)$ are probability densities of symmetric distributions and, for large $|y|$, have a power-law decay with exponent $\alpha(x)+1$, with $\alpha(x)\in(0,2)$. If $f_x(y)$ is the density of a symmetric $\alpha$-stable distribution for negative $x$ and the density of a symmetric $\beta$-stable distribution for non-negative $x$, where $\alpha,\beta\in(0,2)$, then the chain is recurrent if and only if $\alpha+\beta\geq2.$ If the function $x\longmapsto f_x$ is periodic and if the set $\{x:\alpha(x)=\alpha_0:=\inf_{x\in\R}\alpha(x)\}$ has positive Lebesgue measure, then, under a uniformity condition on the densities $f_x(y)$ and some mild technical conditions, the chain is recurrent if and only if $\alpha_0\geq1.$

We consider the recurrence and transience problem for a time-homogeneous Markov chain on the real line with transition kernel $p(x,\mathrm{d}y)=f_x(y-x)\,\mathrm{d}y$, where the density functions $f_x(y)$, for large $|y|$, have a power-law decay with exponent $\alpha(x)+1$, where $\alpha(x)\in(0,2)$. In this paper, under a uniformity condition on the density functions $f_x(y)$ and an additional mild drift condition, we prove that when $\lim\inf_{|x|\longrightarrow\infty}\alpha(x)>1$, the chain is recurrent. Similarly, under the same uniformity condition on the density functions $f_x(y)$ and some mild technical conditions, we prove that when $\lim\sup_{|x|\longrightarrow\infty}\alpha(x)<1$, the chain is transient. As a special case of these results, we give a new proof for the recurrence and transience property of a symmetric $\alpha$-stable random walk on $\mathbb {R}$ with the index of stability $\alpha\in(0,1)\cup(1,2).$, Comment: Published in at http://dx.doi.org/10.3150/12-BEJ448 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)