On the invariance principle for U-statistics

Sure Invariance Principle for Multivariate Kolmogorov - Smirnov Statistics

Invariance Principle on the Slice

The non-linear invariance principle of Mossel, O’Donnell and Oleszkiewicz establishes that if fpx1, . . . , xnq is a multilinear low-degree polynomial with low influences then the distribution of fpB1, . . . ,Bnq is close (in various senses) to the distribution of fpG1, . . . ,Gnq, where Bi PR t ́1, 1u are independent Bernoulli random variables and Gi „ Np0, 1q are independent standard Gaussians...

Invariance principle for the Random Conductance Model

We study a continuous time random walk X in an environment of i.i.d. random conductances μe ∈ [0,∞) in Zd. We assume that P(μe > 0) > pc, so that the bonds with strictly positive conductances percolate, but make no other assumptions on the law of the μe. We prove a quenched invariance principle for X, and obtain Green’s functions bounds and an elliptic Harnack inequality.

On the Noether Invariance Principle for Constrained Optimal Control Problems

Noether’s invariance principle is one of the most helpful and fundamental results of physics. It describes the universal fact that “invariance with respect to some family of parameter transformations gives rise to the existence of certain conserved quantities.” Such relation is used to explain everything from the fusion of hydrogen to the motion of planets orbiting the sun [6]. For a modern acc...

On weighted U-statistics for stationary processes

A weighted U -statistic based on a random sample X1, . . . , Xn has the form Un = ∑ 1≤i,j≤n wi−jK(Xi, Xj) where K is fixed symmetric measurable function and the wi are symmetric weights. A large class of statistics can be expressed as weighted U -statistics or variations thereof. This paper establishes the asymptotic normality of Un when the sample observations come from a non-linear time serie...