Asymptotics for relative frequency when population is driven by arbitrary evolution

Strongly consistent estimates are shown, via relative frequency, for the probability of "white balls" inside a dichotomous urn when such a probability is an arbitrary continuous time dependent function over a bounded time interval. The asymptotic behaviour of relative frequency is studied in a nonstationary context using a Riemann-Dini type theorem for SLLN of random variables with arbitrarily different expectations; furthermore the theoretical results concerning the SLLN can be applied for estimating the mean function of unknown form of a general nonstationary process.