Approximating the identity of convolution with random mean and random variance

Abstract

We provide sufficient conditions on the function φ, on the sequence of random variables εj>0 and on the sequence of random vectors yj∈Rn such that E([Formula presented]∫z∈Rnφ([Formula presented])f(z)dz)⟶j→∞f(x) for almost every x∈Rn, f∈Lp(Rn), 1≤p≤∞, where E denotes the expectation, εj tends to 0∈R in law and yj tends to 0∈Rn in law.