Which properties of a random sequence are dynamically sensitive?

Abstract

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.