On Maximal Inequalities for Stable Stochastic Integrals

Sharp maximal inequalities in large and small range are derived for stable stochastic integrals. In order to control the tail of a stable process, we introduce a truncation level in the support of its Lévy measure: we show that the contribution of the compound Poisson stochastic integral is negligible as the truncation level is large, so that the study is reduced to establish maximal inequalities for the martingale part with a suitable choice of truncation level. The main problem addressed in this paper is to give upper bounds which remain bounded as the parameter of stability of the underlying stable process goes to 2. Applications to estimates of first passage times of symmetric stable processes above positive continuous curves complete this work.      

随机过程增量不等式的推广

本文将推广在［３］中由Ｅ．Ｃｓａｋｉ及Ｍ．Ｃｓｏｒｇｏ所引入的关于随机过程不等式，并把它应用到某些随机过程中，从而得到这些随机过程的一些极限定理．     

Probability Inequalities for Generalized L-Statistics

Siberian Mathematical Journal -     

Harnack Inequalities for Jump Processes

We consider a class of pure jump Markov processes in R ^d whose jump kernels are comparable to those of symmetric stable processes. We establish a Harnack inequality for nonnegative functions that are harmonic with respect to these processes. We also establish regularity for the solutions to certain integral equations.     

Regularizations for Stochastic Linear Variational Inequalities

This paper applies the Moreau–Yosida regularization to a convex expected residual minimization (ERM) formulation for a class of stochastic linear variational inequalities. To have the convexity of the corresponding sample average approximation (SAA) problem, we adopt the Tikhonov regularization. We show that any cluster point of minimizers of the Tikhonov regularization for the SAA problem is a minimizer of the ERM formulation with probability one as the sample size goes to infinity and the Tikhonov regularization parameter goes to zero. Moreover, we prove that the minimizer is the least \(l_2\) -norm solution of the ERM formulation. We also prove the semismoothness of the gradient of the Moreau–Yosida and Tikhonov regularizations for the SAA problem.     

Optimal Stopping and Maximal Inequalities for Linear Diffusions

Given a linear diffusion the solution is found to the optimal stopping problem where the gain is given by the maximum of the process and the cost is proportional to the duration of time. The optimal stopping boundary is shown to be the maximal solution of a nonlinear differential equation expressed in terms of the scale function and the speed measure. Applications to maximal inequalities are indicated.     

Maximal Inequalities for Martingales and Their Differential Subordinates

We introduce a method of proving maximal inequalities for Hilbert- space-valued differentially subordinate local martingales. As an application, we prove that if $X=(X_t)_{t\ge 0},\, Y=(Y_t)_{t\ge 0}$ are local martingales such that $Y$ is differentially subordinate to $X$ , then $$\begin{aligned} ||Y||_1\le \beta ||\sup _{t\ge 0}|X_t|\;||_1, \end{aligned}$$ where $\beta =2.585\ldots $ is the best possible.     

Two-Weight Norm Inequalities for the Local Maximal Function

For a local maximal function defined on a certain family of cubes lying “well inside” of \(\Omega \), a proper open subset of \({\mathbb {R}}^n\), we characterize the couple of weights (u, v) for which it is bounded from \(L^p(v)\) on \(L^q(u)\).     

Inequalities for Stationary Poisson Cuboid Processes

A cuboid is a rectangular parallelepipedon. By the notion “stationary Poisson cuboid process” we understand a stationary Poisson hyperplane process which divides the Euclidean space ?^d into cuboids. It is equivalent to speak of a stationary Poisson cuboid tessellation. The distributions of volume and total edge length of the typical cuboid and the origin-cuboid of a stationary Poisson cuboid process are considered. It is shown that these distributions become minimal, in the sense of a specific order relation, in the case of quasi-isotropy. A possible connection to a more general problem, treated in [6], is also discussed.     

混合型随机微分方程的传输不等式

该文探讨一类由Wiener过程和Hurst参数1/2＜H＜1分数布朗运动驱动的混合型随机微分方程.通过使用一些变换技巧和逼近方法,这类方程的强解在d2度量和一致度量d∞下的二次传输不等式被建立.     

Stochastic Inequalities for Single-Server Loss Queueing Systems

Abstract

The present article provides some new stochastic inequalities for the characteristics of the M/GI/1/n and GI/M/1/n loss queueing systems. These stochastic inequalities are based on substantially deepen up- and down-crossings analysis, and they are stronger than the known stochastic inequalities obtained earlier. Specifically, for a class of GI/M/1/n queueing system, two-side stochastic inequalities are obtained.     

Sharp Weak Type Inequalities for the Dyadic Maximal Operator

We obtain sharp estimates for the localized distribution function of $\mathcal{M}\phi $ , when ? belongs to L ^p,∞ where $\mathcal{M}$ is the dyadic maximal operator. We obtain these estimates given the L ¹ and L ^q norm, q<p and certain weak-L ^p conditions.In this way we refine the known weak (1,1) type inequality for the dyadic maximal operator. As a consequence we prove that the inequality 0.1 is sharp allowing every possible value for the L ¹ and the L ^q norm for a fixed q such that 1<q<p, where ∥?∥ _p,∞ is the usual quasi norm on L ^p,∞.     

Statistical Inference for Stochastic Processes

Table of Contents

Statistical Inference for Stochastic Processes     

Weighted Inequalities for Certain Maximal Functions in Orlicz Spaces

Let M_g be the maximal operator defined by

Weighted Inequalities for Certain Maximal Functions in Orlicz Spaces

Let M_g be the maximal operator defined by $$M_g f\left( x \right) = \sup \frac{{\int_a^b {f\left( y \right)g\left( y \right){\text{d}}y} }}{{\int_a^b {g\left( y \right){\text{d}}y} }}$$ , where g is a positive locally integrable function on R and the supremum is taken over all intervals [a,b] such that 0≤a≤x≤b/η(b?a), here η is a non-increasing function such that η (0) = 1 and $\mathop {{\text{lim}}}\limits_{t \to {\text{ + }}\infty } \eta \left( t \right) = 0$ η (t) = 0. This maximal function was introduced by H. Aimar and L. L. Forzani [AF]. Let Φ be an N - function such that Φ and its complementary N - function satisfy Δ₂. It gives an A′_Φ(g) type characterization for the pairs of weights (u,v) such that the weak type inequality $$u\left( {\left\{ {x \in {\text{R}}\left| {M_g f\left( x \right) >\lambda } \right.} \right\}} \right) \leqslant \frac{C}{{\Phi \left( \lambda \right)}}\int_{\text{R}} {\Phi \left( {\left| f \right|v} \right)} $$ holds for every f in the Orlicz space L_Φ(v). And, there are no (nontrivial) weights w for which (w,w) satisfies the condition A′_Φ(g).     

Weighted Norm Inequalities for the Local Sharp Maximal Function

A weighted norm inequality for the local sharp maximal function M^# f is proved. Our main result along with the extrapolation theorem by D. Cruz-Uribe and C. Perez is applied to obtaining several new weighted norm inequalities for maximal functions and singular integrals. Several open problems are given.