  In this article, weak convergence of the general non-Markov state transition
probability estimator by Titman (2015) is established which, up to now, has not
been verified yet for other general non-Markov estimators. A similar theorem is
shown for the bootstrap, yielding resampling-based inference methods for
statistical functionals. Formulas of the involved covariance functions are
presented in detail. Particular applications include the conditional expected
length of stay in a specific state, given occupation of another state in the
past, as well as the construction of time-simultaneous confidence bands for the
transition probabilities. The expected lengths of stay in the two-sample liver
cirrhosis data-set by Andersen et al. (1993) are compared and confidence
intervals for their difference are constructed. With borderline significance
and in comparison to the placebo group, the treatment group has an elevated
expected length of stay in the healthy state given an earlier disease state
occupation. In contrast, the Aalen-Johansen estimator-based confidence
interval, which relies on a Markov assumption, leads to a drastically different
conclusion. Also, graphical illustrations of confidence bands for the
transition probabilities demonstrate the biasedness of the Aalen-Johansen
estimator in this data example. The reliability of these results is assessed in
a simulation study.
