Do you want to publish a course? Click here

Regularity of linear and polynomial images of Skorohod differentiable measures

78   0   0.0 ( 0 )
 Added by Egor Kosov
 Publication date 2019
  fields
and research's language is English
 Authors Egor D. Kosov




Ask ChatGPT about the research

In this paper we study the regularity properties of linear and polynomial images of Skorohod differentiable measures. Firstly, we obtain estimates for the Skorohod derivative norm of a projection of a product of Scorohod differentiable measures. In the second part of the paper we prove Nikolskii--Besov regularity of a polynomial image of a Skorohod differentiable measure on $mathbb{R}^n$.



rate research

Read More

292 - Jian Song , Samy Tindel 2021
Given a continuous Gaussian process $x$ which gives rise to a $p$-geometric rough path for $pin (2,3)$, and a general continuous process $y$ controlled by $x$, under proper conditions we establish the relationship between the Skorohod integral $int_0^t y_s {mathrm{d}}^diamond x_s$ and the Stratonovich integral $int_0^t y_s {mathrm{d}} {mathbf x}_s$. Our strategy is to employ the tools from rough paths theory and Malliavin calculus to analyze discrete sums of the integrals.
We propose a decomposition method to prove non-asymptotic bound for the convergence of empirical measures in various dual norms. The main point is to show that if one measures convergence in duality with sufficiently regular observables, the convergence is much faster than for, say, merely Lipschitz observables. Actually, assuming $s$ derivatives with $s < d/2$ ($d$ the dimension) ensures an optimal rate of convergence of $1/sqrt{n}$ ($n$ the number of samples). The method is flexible enough to apply to Markov chains which satisfy a geometric contraction hypothesis, assuming neither stationarity nor reversibility, with the same convergence speed up to a power of logarithm factor. Our results are stated as controls of the expected distance between the empirical measure and its limit, but we explain briefly how the classical method of bounded difference can be used to deduce concentration estimates.
286 - Giovanni Peccati 2008
This survey provides a unified discussion of multiple integrals, moments, cumulants and diagram formulae associated with functionals of completely random measures. Our approach is combinatorial, as it is based on the algebraic formalism of partition lattices and Mobius functions. Gaussian and Poisson measures are treated in great detail. We also present several combinatorial interpretations of some recent CLTs involving sequences of random variables belonging to a fixed Wiener chaos.
We show that the problem of finding the measure supported on a compact subset K of the complex plane such that the variance of the least squares predictor by polynomials of degree at most n at a point exterior to K is a minimum, is equivalent to the problem of finding the polynomial of degree at most n, bounded by 1 on K with extremal growth at this external point. We use this to find the polynomials of extremal growth for the interval [-1,1] at a purely imaginary point. The related problem on the extremal growth of real polynomials was studied by ErdH{o}s in 1947.
We consider Gaussian measures $mu, tilde{mu}$ on a separable Hilbert space, with fractional-order covariance operators $A^{-2beta}$ resp. $tilde{A}^{-2tilde{beta}}$, and derive necessary and sufficient conditions on $A, tilde{A}$ and $beta, tilde{beta} > 0$ for I. equivalence of the measures $mu$ and $tilde{mu}$, and II. uniform asymptotic optimality of linear predictions for $mu$ based on the misspecified measure $tilde{mu}$. These results hold, e.g., for Gaussian processes on compact metric spaces. As an important special case, we consider the class of generalized Whittle-Matern Gaussian random fields, where $A$ and $tilde{A}$ are elliptic second-order differential operators, formulated on a bounded Euclidean domain $mathcal{D}subsetmathbb{R}^d$ and augmented with homogeneous Dirichlet boundary conditions. Our outcomes explain why the predictive performances of stationary and non-stationary models in spatial statistics often are comparable, and provide a crucial first step in deriving consistency results for parameter estimation of generalized Whittle-Matern fields.
comments
Fetching comments Fetching comments
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا