Related papers: Modulus of continuity for a martingale sequence

How sensitive should machine learning models be to input changes? We tackle the question of model smoothness and show that it is a useful inductive bias which aids generalization, adversarial robustness, generative modeling and…

Mathematical models of the real world are simplified representations of complex systems. A caveat to using mathematical models is that predicted causal effects and conditional independences may not be robust under model extensions, limiting…

The "Modularity Conjecture" is the assertion that the join of two nonmodular varieties is nonmodular. We establish the veracity of this conjecture for the case of linear idempotent varieties. We also establish analogous results concerning…

In this note, we show that the limiting spectral distribution of symmetric random matrices with stationary entries is absolutely continuous under some sufficient conditions. This result is applied to obtain sufficient conditions on a…

We investigate various boundary conditions in two dimensional turbulence systematically in the context of conformal field theory. Keeping the conformal invariance, we can either change the shape of boundaries through finite conformal…

We investigate sensitivity to cumulative perturbations for a few dynamical system classes of practical interest. A system is said to have bounded sensitivity to cumulative perturbations (bounded sensitivity, for short) if an additive…

The martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms of certain mixing coefficients. Along the…

We study branching mappings that satisfy some condition of distortion of the modulus of families of paths. In a situation where the definition domain of mappings is locally connected on its boundary, the mapped domain is regular, and the…

In this paper, we study the summability properties of double sequences of real constants which map sequences of random variables to sequences of random variables that are defined on the same probability sample space. We show that a regular…

The paper is devoted to the study of the boundary behavior of mappings. We consider mappings that satisfy inverse moduli inequalities of Poletskii type, under which the images of the domain under the mappings may change. It is proved that a…

We construct a class of nonnegative martingale processes that oscillate indefinitely with high probability. For these processes, we state a uniform rate of the number of oscillations and show that this rate is asymptotically close to the…

We establish an invariance principle for a general class of stationary random fields indexed by $\mathbb Z^d$, under Hannan's condition generalized to $\mathbb Z^d$. To do so we first establish a uniform integrability result for stationary…

The following fact is known for large classes of distributed control systems: when the target is regular, there exists a regular steering control. This fact is important to prove convergence estimates of numerical algorithms for the…

A proof of the continuous martingale convergence theorem is provided. It relies on a classical martingale inequality and the almost sure convergence of a uniformly bounded non-negative super-martingale, after a truncation argument.

Biggins [Uniform convergence of martingales in the branching random walk. {\em Ann. Probab.}, 20(1):137--151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex…

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…

Suppose we are given two probability measures on the set of one-way infinite finite-alphabet sequences and consider the question when one of the measures predicts the other, that is, when conditional probabilities converge (in a certain…

When working with (multi-parameter) persistence modules, one usually makes some type of tameness assumption in order to obtain better control over their algebraic behavior. One such notion is Ezra Millers notion of finite encodability,…

In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the…

The density hypothesis on random times becomes now a standard in modeling of risks. One of the basic reasons to introduce the density hypothesis is the desire to have a computable credit risk model. However, recent work shows that merely an…