Classical theorems of probability such as the central limit theorem or the Perron-Frobenius theorem seem to lean on positivity. It is natural to ask what happens without positivity. It turns out that there is a perfectly nice limit theory (some restrictions apply, as they say in the advertisements). This was started by Schoenberg and continued by Thomee and others. It has applications to data smoothing and numerical schemes for running PDE's. This is joint work with Laurent Saloff-Coste.