  Adaptive experiments, including efficient average treatment effect estimation
and multi-armed bandit algorithms, have garnered attention in various
applications, such as social experiments, clinical trials, and online
advertisement optimization. This paper considers estimating the mean outcome of
an action from samples obtained in adaptive experiments. In causal inference,
the mean outcome of an action has a crucial role, and the estimation is an
essential task, where the average treatment effect estimation and off-policy
value estimation are its variants. In adaptive experiments, the probability of
choosing an action (logging policy) is allowed to be sequentially updated based
on past observations. Due to this logging policy depending on the past
observations, the samples are often not independent and identically distributed
(i.i.d.), making developing an asymptotically normal estimator difficult. A
typical approach for this problem is to assume that the logging policy
converges in a time-invariant function. However, this assumption is restrictive
in various applications, such as when the logging policy fluctuates or becomes
zero at some periods. To mitigate this limitation, we propose another
assumption that the average logging policy converges to a time-invariant
function and show the doubly robust (DR) estimator's asymptotic normality.
Under the assumption, the logging policy itself can fluctuate or be zero for
some actions. We also show the empirical properties by simulations.
