Related papers: On the functional central limit theorem via martin…
In this paper we study the functional central limit theorem for stationary Markov chains with self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n; and…
This paper extends classical probabilistic results to the broader class of demimartingales and demisubmartingales. We establish variants of Doob's-type optional sampling theorem under minimal structural conditions on stopping times, relying…
One problem of wide interest involves estimating expected crossing-times. Several tools have been developed to solve this problem beginning with the works of Wald and the theory of sequential analysis. An extension of his approach is…
A central limit theorem is established for a sum of random variables belonging to a sequence of random fields. The fields are assumed to have zero mean conditional on the past history and to satisfy certain conditional $\alpha$-mixing…
For a measure preserving transformation $T$ of a probability space $(X,\mathcal F,\mu)$ we investigate almost sure and distributional convergence of random variables of the form $$x \to \frac{1}{C_n} \sum_{i_1<n,...,i_d<n}…
We consider additive functionals of Markov processes in continuous time with general (metric) state spaces. We derive concentration bounds for their exponential moments and moments of finite order. Applications include diffusions,…
Comparison results for Markov processes w.r.t. function class induced (integral) stochastic orders have a long history. The most general results so far for this problem have been obtained based on the theory of evolution systems on Banach…
In this paper, we establish an almost sure central limit theorem for a general random sequence under a strong approximation condition. Additionally, we derive the law of the iterated logarithm for the center of mass corresponding to a…
In this paper, we give rates of convergence, for minimal distances and for the uniform distance, between the law of partial sums of martingale differences and thelimiting Gaussian distribution. More precisely, denoting by $P_{X}$ the law of…
We present normal approximation results at the process level for local functionals defined on dynamic Poisson processes in $\mathbb{R}^d$. The dynamics we study here are those of a Markov birth-death process. We prove functional limit…
We use the abstract method of (local) martingale problems in order to give criteria for convergence of stochastic processes. Extending previous notions, the formulation we use is neither restricted to Markov processes (or semimartingales),…
The typical central limit theorems in high-frequency asymptotics for semimartingales are results on stable convergence to a mixed normal limit with an unknown conditional variance. Estimating this conditional variance usually is a hard…
We prove a sequence of limiting results about weakly dependent stationary and regularly varying stochastic processes in discrete time. After deducing the limiting distribution for individual clusters of extremes, we present a new type of…
Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version…
This work shows how exponential concentration inequalities for additive functionals of stochastic processes over a finite time interval can be derived from concentration inequalities for martingales. The approach is entirely probabilistic…
Monotone processes, just like martingales, can often be recovered from their final values. Examples include running maxima of supermartingales, as well as running maxima, local times, and various integral functionals of sticky processes…
In this note we present a new sufficient condition which guarantees martingale approximation and central limit theorem a la Kipnis-Varadhan to hold for additive functionals of Markov processes. This condition which we call the relaxed…
First, sufficient conditions are given for a triangular array of random vectors such that the sequence of related random step functions converges towards a (not necessarily time homogeneous) diffusion process. These conditions are weaker…
The production of molecules in a chemical reaction network is modelled as a Poisson process with a Markov-modulated arrival rate and an exponential decay rate. We analyze the distributional properties of $M$, the number of molecules, under…
We establish central limit theorems for a large class of supercritical branching Markov processes in infinite dimension with spatially dependent and non-necessarily local branching mechanisms. This result relies on a fourth moment…