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A functional time series is a time series whose observation at each time is a whole function, often modeled as a random element of the Hilbert space $L^2[0,1]$. For example, $X_j(s)$ might be the temperature profile within day $j$, with $s$ marking time of day. Dependence across days is different from dependence between nearby positions within one curve.

For a simple model, let $X_j(s)=A_j+B_{j}s$, where $(A_j,B_j)$ is a stationary bivariate time series. Its mean curve is $\mu (s)=\mathbb{E}{A}_j+s\mathbb{E}{B}_j$. The covariance between curves at lag $h$ is a kernel $C_h(s,t)=\mathrm{Cov}(X_0(s),X_h(t))$, retaining both within-curve coordinates.

Estimating $\mu$ by $n^{-1}{\sum }_{j=1}^n{X}_j$ resembles scalar averaging, but its uncertainty depends on serial covariance operators rather than only the variance of one curve. Treating every densely measured point as an independent observation can severely overstate precision.^[ref]

Lajos Horváth works on mathematical statistics, time series and econometrics. His University of Utah profile identifies these research interests.^[ref]

His joint paper with Piotr Kokoszka and Ron Reeder studies comparisons of mean curves in a functional time series. Its Hilbert space formulation yields Gaussian limits and a test based on the distance between two estimated mean functions, retaining dependence within each sample.^[ref]

Ron Reeder coauthored Estimation of the Mean of Functional Time Series and a Two-Sample Problem by Lajos Horváth and others with Lajos Horváth and Piotr Kokoszka. Their problem treats successive daily curves as observations of a functional time series. The paper develops a normal approximation for their mean and a comparison of two mean curves that allows dependence across days.^[ref]