  A representation of Gaussian distributed sparsely sampled longitudinal data
in terms of predictive distributions for their functional principal component
scores (FPCs) maps available data for each subject to a multivariate Gaussian
predictive distribution. Of special interest is the case where the number of
observations per subject increases in the transition from sparse (longitudinal)
to dense (functional) sampling of underlying stochastic processes. We study the
convergence of the predicted scores given noisy longitudinal observations
towards the true but unobservable FPCs, and under Gaussianity demonstrate the
shrinkage of the entire predictive distribution towards a point mass located at
the true FPCs and also extensions to the shrinkage of functional $K$-truncated
predictive distributions when the truncation point $K=K(n)$ diverges with
sample size $n$. To address the problem of non-consistency of point
predictions, we construct predictive distributions aimed at predicting outcomes
for the case of sparsely sampled longitudinal predictors in functional linear
models and derive asymptotic rates of convergence for the $2$-Wasserstein
metric between true and estimated predictive distributions. Predictive
distributions are illustrated for longitudinal data from the Baltimore
Longitudinal Study of Aging.
