Asymptotics for the expected maximum of random walks and Lévy flights with a constant drift. (6th August 2018) - Mounaix, Philippe; Majumdar, Satya N; Schehr, Grégory
Abstract: 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.