  In this paper we study a substantial generalization of the model of excited
random walk introduced in [Electron. Commun. Probab. 8 (2003) 86-92] by
Benjamini and Wilson. We consider a discrete-time stochastic process
$(X_n,n=0,1,2,...)$ taking values on ${\mathbb{Z}}^d$, $d\geq2$, described as
follows: when the particle visits a site for the first time, it has a
uniformly-positive drift in a given direction $\ell$; when the particle is at a
site which was already visited before, it has zero drift. Assuming uniform
ellipticity and that the jumps of the process are uniformly bounded, we prove
that the process is ballistic in the direction $\ell$ so that
$\liminf_{n\to\infty}\frac{X_n\cdot \ell}{n}>0$. A key ingredient in the proof
of this result is an estimate on the probability that the process visits less
than $n^{{1/2}+\alpha}$ distinct sites by time n, where $\alpha$ is some
positive number depending on the parameters of the model. This approach
completely avoids the use of tan points and coupling methods specific to the
excited random walk. Furthermore, we apply this technique to prove that the
excited random walk in an i.i.d. random environment satisfies a ballistic law
of large numbers and a central limit theorem.
