Related papers: Stopped processes and Doob's optional sampling the…
It is well known that if a submartingale $X$ is bounded then the increasing predictable process $Y$ and the martingale $M$ from the Doob decomposition $% X=Y+M$ can be unbounded. In this paper for some classes of increasing convex functions…
Motivated by problems from statistical analysis for discretely sampled SPDEs, first we derive central limit theorems for higher order finite differences applied to stochastic process with arbitrary finitely regular paths. These results are…
We explore two notions of stationary processes. The first is called a random-step Markov process in which the stationary process of states, $(X_i)_{i \in \mathbb{Z}}$ has a stationary coupling with an independent process on the positive…
We present a unified approach to Doob's $L^p$ maximal inequalities for $1\leq p<\infty$. The novelty of our method is that these martingale inequalities are obtained as consequences of elementary deterministic counterparts. The latter have…
We analyze an optimal stopping problem with a series of inequality-type and equality-type expectation constraints in a general non-Markovian framework. We show that the optimal stopping problem with expectation constraints (OSEC) in an…
The present paper is devoted to the second part of our project on asymmetric maximal inequalities, where we consider martingales in continuous time. Let $(\mathcal M,\tau)$ be a noncommutative probability space equipped with a continuous…
This work develops a convergence theory for H(div)-conforming finite element methods applied to the steady Oseen problem, focusing on cases where the exact finite element complex holds while the commuting diagram property may fail. The…
A system of uniform families on an infinite subset $M$ of $\nn$ is a collection $(\cca_{\xi})_{\xi<\omega_1}$ of families of finite subsets of $\nn$ (where, $\cca_k$ consists of all $k$--element subset of $M$, for $k\in \nn$) with the…
In this paper, we are concerned with the symmetric simple exclusion process (SSEP) on the regular tree $\mathcal{T}_d$. A central limit theorem and a moderate deviation principle of the additive functional of the process are proved, which…
We develop the self similarity argument known as sparse domination in an abstract martingale setting, using a continuous time parameter. With this method, we prove a sharp weighted L^p estimate for the maximal operator Y^* of Y with respect…
We give a theory of sublinear expectations and martingales in discrete time. Without assuming the existence of a dominating probability measure, we derive the extensions of classical results on uniform integrability, optional stopping of…
Four estimators of the directed information rate between a pair of jointly stationary ergodic finite-alphabet processes are proposed, based on universal probability assignments. The first one is a Shannon--McMillan--Breiman type estimator,…
Stationary ergodic processes with finite alphabets are estimated by finite memory processes from a sample, an n-length realization of the process, where the memory depth of the estimator process is also estimated from the sample using…
This paper consists of two independent parts. In the first one, we contribute to the study of the class $(\Sigma)$. For instance, we provide a new way to characterize stochastic processes of this class. We also present some new properties…
Assuming that a reflected Ornstein-Uhlenbeck state process is observed at discrete time instants, we propose generalized moment estimators to estimate all drift and diffusion parameters via the celebrated ergodic theorem. With the sampling…
We consider the process of partial sums of moving averages of finite order with a regular varying memory function, constructed from a stationary sequence, variance of the sum of which is a regularly varying function. We study the Gaussian…
Given $N\ge2$ closed subspaces $M_1,\dotsc, M_N$ of a Hilbert space $X$, let $P_k$ denote the orthogonal projection onto $M_k$, $1\le k\le N$. It is known that the sequence $(x_n)$, defined recursively by $x_0=x$ and $x_{n+1}=P_N\cdots…
In this short note, we will strengthen the classic Doob's $L^p$ inequality for sub-martingale processes. Because this inequality is of fundamental importance to the theory of stochastic process, we believe this generalization will find many…
The work [8] established memory loss in the time-dependent (non-random) case of uniformly expanding maps of the interval. Here we find conditions under which we have convergence to the normal distribution of the appropriately scaled…
Inspired by recent work of P.-L. Lions on conditional optimal control, we introduce a problem of optimal stopping under bounded rationality: the objective is the expected payoff at the time of stopping, conditioned on another event. For…