Purpose: To present testing procedures for equality of means in two functional samples having temporal dependence using normal approximation of the functional sample mean. Summary: Functional time series data form a type of data structure but many particulars regarding estimation and testing of this data are yet to be examined. This paper is on the estimation and inference of the mean function of the functional time series data. In particular the paper develops a methodology and an asymptotic theory for the estimation of the variance of the sample mean, and also proposes procedures for testing equality of two mean functions in functional samples. The temporal dependence is the major issue in the analysis of functional time series. Here it is assumed that the functional time series is stationary and are dependent. In a time series the variance of the sample mean is asymptotically approximated by long-run variance and this variance is used for drawing inferences regarding sample mean functions. Refs. 1 & 2 had proposed the use of the use of linear process with L~p m-notation for functional time series models. This paper develops a general framework in for estimation of covariance in this setting. (16 refs.)
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