Estimation of the mean of functional time series and a two sample   problem
  This paper is concerned with inference based on the mean function of a functional time series, which is defined as a collection of curves obtained by splitting a continuous time record, e.g. into daily or annual curves. We develop a normal approximation for the functional sample mean, and then focus on the estimation of the asymptotic variance kernel. Using these results, we develop and asymptotically justify a testing procedure for the equality of means in two functional samples exhibiting temporal dependence. Evaluated by means of a simulations study and application to real data sets, this two sample procedure enjoys good size and power in finite samples. We provide the details of its numerical implementation. 
Lajos Horvath, Piotr Kokoszka and Ron Reeder
math.ST stat.TH