Related papers: Invariance principle for additive functionals of M…
In this paper we propose an efficient variance reduction approach for additive functionals of Markov chains relying on a novel discrete time martingale representation. Our approach is fully non-asymptotic and does not require the knowledge…
The notion of a successful coupling of Markov processes, based on the idea that both components of the coupled system ``intersect'' in finite time with probability one, is extended to cover situations when the coupling is unnecessarily…
The theory of dependency graphs is a powerful toolbox to prove asymptotic normality of sums of random variables. In this article, we introduce a more general notion of weighted dependency graphs and give normality criteria in this context.…
The objective of this paper is to prove a functional weak invariance principle for a local time of a process of the form $X_{n}=\varphi\circ T^{n}$ where $\left(X,\mathcal{B},T,m\right)$ is a measure preserving system with a transfer…
Predictive constructions are a powerful way of characterizing the probability law of stochastic processes with certain forms of invariance, such as exchangeability or Markov exchangeability. When de Finetti-like representation theorems are…
We introduce a definition of pressure for almost-additive sequences of continuous functions defined over (non-compact) countable Markov shifts. The variational principle is proved. Under certain assumptions we prove the existence of Gibbs…
Based on deleting-item central limit theory, the classical Donsker's theorem of partial-sum process of independent and identically distributed (i.i.d.) random variables is extended to incomplete partial-sum process. The incomplete…
We study weak convergence of empirical processes of dependent data $(X_i)_{i\geq0}$, indexed by classes of functions. Our results are especially suitable for data arising from dynamical systems and Markov chains, where the central limit…
Motivated by the theory of inhomogeneous Markov chains, we determine a sufficient condition for the convergence to 0 of a general product formed from a sequence of real or complex matrices. When the matrices have a common invariant subspace…
The "fundamental theorem of Vassiliev invariants" says that every weight system can be integrated to a knot invariant. We discuss four different approaches to the proof of this theorem: a topological/combinatorial approach following M.…
We prove the Local Limit Theorems for bounded additive functionals of uniformly elliptic inhomogeneous Markov arrays. As an application we obtain the precise asymptotics in the large deviation regime for bounded additive functionals of…
We derive the necessary and sufficient condition for almost sure convergence of the sequence of measurable functions, and consider some applications in the theory of Fourier series and in the theory of random fields.
We study the limit behaviour of a generally non-linear ordinary differential equation whose solution is a superadditive generalisation of a stochastic matrix, and provide necessary and sufficient conditions for this solution to be ergodic,…
The paper presents a generalization of the local limit theorem on the convergence of inhomogeneous Markov chains to the diffusion limit for the case where the corresponding process coefficients satisfy weak regularity conditions and…
We generalize Lindeberg's proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables. The result is applied to get a similar theorem for smooth functions…
We establish convergence to an invariant measure as time tends to infinity, for a large class of (possibly non-Markovian) stochastic volatility models. Our arguments are based on a novel coupling idea for Markov chains which also extends to…
Markov chains arising from random iteration of functions $S_{\theta}:X\to X$, $\theta \in \Theta$, where $X$ is a Polish space and $\Theta$ is arbitrary set of indices are considerd. At $x\in X$, $\theta$ is sampled from distribution…
In this paper we survey some recent results on the central limit theorem and its weak invariance principle for stationary sequences. We also describe several maximal inequalities that are the main tool for obtaining the invariance…
We investigate absorption, i.e., almost sure convergence to an absorbing state, in time-varying (non-homogeneous) discrete-time Markov chains with finite state space. We consider systems that can switch among a finite set of transition…
We consider a Markov chain $\{X_n\}_{n=0}^\8$ on $\R^d$ defined by the stochastic recursion $X_{n}=M_n X_{n-1}+Q_n$, where $(Q_n,M_n)$ are i.i.d. random variables taking values in the affine group $H=\R^d\rtimes {\rm GL}(\R^d)$. Assume that…