Consider a sequence of i.i.d.\ random variables, where each variable is
refreshed (i.e., replaced by an independent variable
with the same law) independently, according to a Poisson clock.
At any fixed time t, the resulting sequence has the same
law as at time 0, but there can be exceptional random
times at
which certain almost sure properties of the time 0 sequence are violated.
We prove that there are no such exceptional times for the law of large
numbers and the law of the iterated logarithm,
so these laws are dynamically stable. However, there are
times at which run lengths are exceptionally long, that is, run tests are
dynamically sensitive. We obtain a
multifractal analysis of exceptional times for run lengths
and for prediction. In particular, starting from an i.i.d.
sequence of unbiased random bits, the random set of times t
where $\alpha \log_2(n)$ bits among the first n bits can be predicted
from their predecessors, has Hausdorff dimension $1-\alpha$ a.s.
Finally, we consider simple random walk in the lattice $\Z^d$, and prove that
transience is dynamically stable for $d \ge 5$,
and dynamically sensitive for $d=3,4$.
Moreover, for $d=3,4$, the nonempty
random set of exceptional times t where the walk
is recurrent has Hausdorff dimension $(4-d)/2$ a.s.