Asymptotics for the expected maximum of random walks and Lévy flights with a constant drift

In this paper, we study the large n asymptotics of the expected maximum of an n-step random walk/Lévy flight (characterized by a Lévy index ) on a line, in the presence of a constant drift c. For , the expected maximum is infinite, even for finite values of n. For , we obtain all the non-vanishing terms in the asymptotic expansion of the expected maximum for large n. For c < 0 and , the expected maximum approaches a non-trivial constant as n gets large, while for , it grows as a power law ∼ . For c > 0, the asymptotic expansion of the expected maximum is simply related to the one for c < 0 by adding to the latter the linear drift term cn, making the leading term grow linearly for large n, as expected. Finally, we derive a scaling form interpolating smoothly between the cases c = 0 and . These results are borne out by numerical simulations in excellent agreement with our analytical predictions.