  We study the minimal random walk introduced by Kumar, Harbola and Lindenberg.
It is a random process on $\{0, 1, \ldots \}$ with unbounded memory which
exhibits subdiffusive, diffusive and superdiffusive regimes. We prove the law
of large numbers for the whole parameter set. Then we prove the central limit
theorem and the law of the iterated logarithm for the minimal random walk under
diffusive and marginally superdiffusive behaviors. More interestingly, we
establish a result for the minimal random walk when it possesses the three
regimes; we show the convergence of its rescaled version to a non-normal random
variable.
