Related papers: Maximal Inequalities and Some Applications

Variational inequalities are a formalism that includes games, minimization, saddle point, and equilibrium problems as special cases. Methods for variational inequalities are therefore universal approaches for many applied tasks, including…

Integral representations for expectations of functions of a stable L\'evy process $X$ and its supremum $\bar X$ are derived. As examples, cumulative probability distribution functions (cpdf) of $X_T, \barX_T$, the joint cpdf of $X_T$ and…

This article is a survey of results concerning an inequality, which may be seen as a versatile tool to solve problems in the domain of Applied Probability. The inequality, which we call BRS-inequality, gives a convenient upper bound for the…

We investigate maxima of linear processes with i.i.d. heavy-tailed innovations and random coefficients. Using the point process approach we derive functional convergence of the partial maxima stochastic process in the space of…

In the paper we pursue the analysis from the section 5 of the Talagrand's paper "Sample boundedness of stochastic processes under increment conditions." Ann. Probab. 18, No. 1, 1-49. In particular we give the proof of some Sobolev…

This is a survey article about recent developments in dimension-free estimates for maximal functions corresponding to the Hardy--Littlewood averaging operators associated with convex symmetric bodies in $\mathbb R^d$ and $\mathbb Z^d$.

A broad set of sufficient conditions that guarantees the existence of the maximum entropy (maxent) distribution consistent with specified bounds on certain generalized moments is derived. Most results in the literature are either focused on…

This paper considers a controlled It\^o-L\'evy process where the information available to the controller is possibly less than the overall information. All the system coefficients and the objective performance functional are allowed to be…

Stochastic processes find applications in modelling systems in a variety of disciplines. A large number of stochastic models considered are Markovian in nature. It is often observed that higher order Markov processes can model the data…

In this article we first establish a complete characterization of Hardy's inequalities in $\mathbb{R}^n$ involving distances to different codimension subspaces. In particular the corresponding potentials have strong interior singularities.…

We study the maximal correlation coefficient $R(X,Y)$ between two stochastic processes $X$ and $Y$. In the case when $(X,Y)$ is a random walk, we find $R(X,Y)$ using the Cs\'{a}ki-Fischer identity and the lower semicontinuity of the map…

We develop almost-orthogonality principles for maximal functions associated with averages over line segments and directional singular integrals. Using them, we obtain sharp $L^2$-bounds for these maximal functions when the underlying…

Being the max-analogue of $\alpha$-stable stochastic processes, max-stable processes form one of the fundamental classes of stochastic processes. With the arrival of sufficient computational capabilities, they have become a benchmark in the…

We derive an upper bound for the mean of the supremum of the empirical process indexed by a class of functions that are known to have variance bounded by a small constant $\delta$. The bound is expressed in the uniform entropy integral of…

We derive an equality for non-equilibrium statistical mechanics in finite-dimensional quantum systems. The equality concerns the worst-case work output of a time-dependent Hamiltonian protocol in the presence of a Markovian heat bath. It…

We find a maximum principle for general non-Markovian semi-martingales. We do so by describing the adjoint processes with non-anticipating stochastic derivatives in a martingale random field setting. In the case of the L\'evy processes this…

We consider the Hardy-Littlewood maximal function associated with ball averages on spaces with exponential volume growth. We focus on discrete groups with balls defined by invariant metrics associated with a variety of length functions.…

For a L\'evy process $X$ on a finite time interval consider the probability that it exceeds some fixed threshold $x>0$ while staying below $x$ at the points of a regular grid. We establish exact asymptotic behavior of this probability as…

Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several…

The $\epsilon$-machine is a stochastic process' optimal model -- maximally predictive and minimal in size. It often happens that to optimally predict even simply-defined processes, probabilistic models -- including the $\epsilon$-machine --…