  Motivated by novel results in the theory of correlated sequences, we analyze
the dynamics of random walks with long-term memory (binary chains with
long-range correlations). In our model, the probability for a unit bit in a
binary string depends on the fraction of unities preceding it. We show that the
system undergoes a dynamical phase-transition from normal diffusion, in which
the variance D_L scales as the string's length L, into a super-diffusion phase
(D_L ~ L^{1+|alpha|}), when the correlation strength exceeds a critical value.
We demonstrate the generality of our results with respect to alternative
models, and discuss their applicability to various data, such as coarse-grained
DNA sequences, written texts, and financial data.
