  Consider a sequence {X(i,0) : i = 1, ..., n} of i.i.d. random variables.
Associate to each X(i,0) an independent mean-one Poisson clock. Every time a
clock rings replace that X-variable by an independent copy. In this way, we
obtain i.i.d. stationary processes {X(i,t) : t >= 0} (i=1,2, ...) whose
invariant distribution is the law of X(1,0). Benjamini, Haggstrom, Peres, and
Steif (2003) introduced the dynamical walk S(n,t) = X(1,t) + ... + X(n,t), and
proved among other things that the LIL holds for {S(n,t) : n =1,2, ...}
simultaneously for all t. In other words, the LIL is dynamically stable.
Subsequently, we showed that in the case that the X(i,0)'s are standard normal,
the classical integral test is not dynamically stable. Presently, we study the
set of times t when {S(n,t) : n=1,2, ...} exceeds a given envelope infinitely
often. Our analysis is made possible thanks to a connection to the Kolmogorov
epsilon-entropy. When used in conjunction with the invariance principle of this
paper, this connection has other interesting by-products some of which we
relate. We prove also that viewed as an infinite-dimensional process, the
rescaled dynamical random walk converges weakly in D(D([0,1])) to the
Ornstein-Uhlenbeck process in C([0,1]). For this we assume only that the
increments have mean zero and variance one. In addition, we extend a result of
Benjamini, Haggstrom, Peres and Steif (2003) by proving that if the X(i,0)'s
are lattice, mean-zero variance-one, and possess 2 + epsilon finite absolute
moments for some positive epsilon, then the recurrence of the origin is
dynamically stable. To prove this we derive a gambler's ruin estimate that is
valid for all lattice random walks that have mean zero and finite variance. We
believe the latter may be of independent interest.
