On asymptotically automatic sequences

We study the notion of an asymptotically automatic sequence, which generalises the notion of an automatic sequence. While k-automatic sequences are characterised by finiteness of k-kernels, the k-kernels of asymptotically k-automatic sequences are only required to be finite up to equality almost everywhere. We prove basic closure properties and a linear bound on asymptotic subword complexity, show that results concerning frequencies of symbols are no longer true for the asymptotic analogue, and discuss some classification problems.