Limit theorems for probability measures on totally disconnected groups (Q1279668)

An important problem in investigations of limit theorems on locally compact groups \(G\) is to find conditions, under which for probability measures \(\nu_n\) and natural numbers \(k_n\), the convergence of convolution products \(\nu_n^{k_n}\) to a probability measure \(\mu\), implies a so-called functional limit theorem (FLT). This means the convergence of the discrete convolution semigroups \((\nu_n^{[k_nt]})_{t\geq 0}\) to a continuous convolution semigroup \((\mu_t)_{t\geq 0}\). The paper provides, on arbitrary locally compact groups, sufficient conditions such that a FLT holds. This result is then refined on totally disconnected groups and especially on \(p\)-adic Lie groups. The author applies these results to characterize semistable laws in terms of domains of attraction on certain \(p\)-adic algebraic groups.