Approximating the identity of convolution with random mean and random\n variance

2021/01/19 by Hugo Aimar, Ivana Gómez, Aimar, Hugo +1
Decision Sciences · Mathematics · #42B25 #44A35 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Probability (math.PR) #Probability and Risk Models

paper · pdf · doi:10.48550/arxiv.2101.07867

openalex publication_date 2021/01/19 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We provide sufficient conditions on the profile \φ, on the sequence of\nrandom variables \ε_j>0 and on the sequence of random vectors\ny_j\∈\ℝⁿ such that\n mathscrE\(\(1)/(\ε_jⁿ(\ω))\∫_z\∈\ℝⁿ\φ\(\(|x-z-y_j(\ω)|)/(\ε_j(\ω))\)f(z)\ndz\) longrightarrow f(x) when j\→\∞ for almost every\nx\∈\ℝⁿ, f\∈ L^p(\ℝⁿ), 1\≤ p\≤\∞, where\n mathscrE denotes the expectation, \ε_j tends to\n0\∈\ℝ in law and y_j tends to \0\∈\ℝⁿ in law.\n

Related

• On the Fourier transform of random Bernoulli convolutions
• Extreme expectations of Bernoulli convolutions given their first few moments are attained at shifted convolutions of as few binomials
• Convolution powers of complex functions on ℤ^d
• Recent progress on Bernoulli convolutions
• Convolution powers of complex functions on Z