  In recent years, there has been an interest in deriving certain important
probabilistic results as consequences of deterministic ones; see for instance
\cite{beig} and \cite{acc}. In this work, we continue on this path by deducing
a well known equivalence between the speed of random walks on the integers and
the growth of the size of their ranges. This result is an immediate consequence
of the Kesten-Spitzer-Whitman theorem, and by appearances is probabilistic in
nature, but we will show that it follows easily from an elementary
deterministic result. We also investigate the common property of recurrent
random walks of having speed zero, and show by example that this property need
not be shared by deterministic sequences. However, if we consider the
inter-arrival times (times at which the sequence is equal to 0) then we find a
sufficient deterministic condition for a sequence to have zero speed, and show
that this can be used to derive several probabilistic results.
