  We consider a discrete time random walk in one dimension. At each time step
the walker jumps by a random distance, independent from step to step, drawn
from an arbitrary symmetric density function. We show that the expected
positive maximum E[M_n] of the walk up to n steps behaves asymptotically for
large n as, E[M_n]/\sigma=\sqrt{2n/\pi}+ \gamma +O(n^{-1/2}), where \sigma^2 is
the variance of the step lengths. While the leading \sqrt{n} behavior is
universal and easy to derive, the leading correction term turns out to be a
nontrivial constant \gamma. For the special case of uniform distribution over
[-1,1], Coffmann et. al. recently computed \gamma=-0.516068...by exactly
enumerating a lengthy double series. Here we present a closed exact formula for
\gamma valid for arbitrary symmetric distributions. We also demonstrate how
\gamma appears in the thermodynamic limit as the leading behavior of the
difference variable E[M_n]-E[|x_n|] where x_n is the position of the walker
after n steps. An application of these results to the equilibrium
thermodynamics of a Rouse polymer chain is pointed out. We also generalize our
results to L\'evy walks.
