Approximating the identity of convolution with random mean and random variance

TL;DR

This paper establishes conditions under which a convolution with random mean and variance approximates the identity operator for functions in L^p spaces, extending classical approximation results to stochastic settings.

Contribution

It introduces sufficient conditions on the profile, random variances, and means ensuring convergence of the stochastic convolution to the identity.

Findings

Convergence holds for almost every point in R^n.

Conditions on the profile and random variables are sufficient for approximation.

The result applies to functions in L^p spaces with 1 ≤ p ≤ ∞.

Abstract

We provide sufficient conditions on the profile $\varphi$, on the sequence of random variables $\varepsilon_j>0$ and on the sequence of random vectors $y_j\in\mathbb{R}^n$ such that $\mathscr{E}\left(\frac{1}{\varepsilon_j^n(\omega)}\int_{z\in\mathbb{R}^n}\varphi\left(\frac{|x-z-y_j(\omega)|}{\varepsilon_j(\omega)}\right)f(z) dz\right)\longrightarrow f(x)$ when $j\to\infty$ for almost every $x\in\mathbb{R}^n$, $f\in L^p(\mathbb{R}^n)$, $1\leq p\leq\infty$, where $\mathscr{E}$ denotes the expectation, $\varepsilon_j$ tends to $0\in\mathbb{R}$ in law and $y_j$ tends to $\mathbf{0}\in\mathbb{R}^n$ in law.