  Consider the case that one observes a single time-series, where at each time
t one observes a data record O(t) involving treatment nodes A(t), possible
covariates L(t) and an outcome node Y(t). The data record at time t carries
information for an (potentially causal) effect of the treatment A(t) on the
outcome Y(t), in the context defined by a fixed dimensional summary measure
Co(t). We are concerned with defining causal effects that can be consistently
estimated, with valid inference, for sequentially randomized experiments
without further assumptions. More generally, we consider the case when the
(possibly causal) effects can be estimated in a double robust manner, analogue
to double robust estimation of effects in the i.i.d. causal inference
literature. We propose a general class of averages of conditional
(context-specific) causal parameters that can be estimated in a double robust
manner, therefore fully utilizing the sequential randomization. We propose a
targeted maximum likelihood estimator (TMLE) of these causal parameters, and
present a general theorem establishing the asymptotic consistency and normality
of the TMLE. We extend our general framework to a number of typically studied
causal target parameters, including a sequentially adaptive design within a
single unit that learns the optimal treatment rule for the unit over time. Our
work opens up robust statistical inference for causal questions based on
observing a single time-series on a particular unit.
