  An excited random walk is a non-Markovian extension of the simple random
walk, in which the walk's behavior at time $n$ is impacted by the path it has
taken up to time $n$. The properties of an excited random walk are more
difficult to investigate than those of a simple random walk. For example, the
limiting speed of an excited random walk is either zero or unknown depending on
its initial conditions. While its limiting speed is unknown in most cases, the
qualitative behavior of an excited random walk is largely determined by a
parameter $\delta$ which can be computed explicitly. Despite this, it is known
that the limiting speed cannot be written as a function of $\delta$. We offer a
new proof of this fact, and use techniques from this proof to further
investigate the relationship between $\delta$ and speed. We also generalize the
standard excited random walk by introducing a "bias" to the right, and call
this generalization an excited asymmetric random walk. Under certain initial
conditions we are able to compute an explicit formula for the limiting speed of
an excited asymmetric random walk.
