Approximating the identity of convolution with random mean and random variance

Abstract

We provide sufficient conditions on the profile $\phi$, on the sequence of random variables ${\epsilon }_j>0$ and on the sequence of random vectors $y_j\in {\mathbb{R}}^n$ such that $\mathcal{E}\left(\frac{1}{{\epsilon }_j^n(\omega )}{\int }_{z\in {\mathbb{R}}^n}\phi \left(\frac{|x-z-y_j(\omega )|}{{\epsilon }_j(\omega )}\right)f(z)dz\right)⟶f(x)$ when $j\to \infty$ for almost every $x\in {\mathbb{R}}^n$, $f\in L^p({\mathbb{R}}^n)$, $1\le p\le \infty$, where $\mathcal{E}$ denotes the expectation, ${\epsilon }_j$ tends to $0\in \mathbb{R}$ in law and $y_j$ tends to $0\in {\mathbb{R}}^n$ in law.

12 pages