Asymptotics for the Expected Maximum of Random Walks and L\'evy Flights with a Constant Drift

TL;DR

This paper derives the large-$n$ asymptotics of the expected maximum of $n$-step random walks and Lévy flights with a constant drift, revealing different growth behaviors depending on the Lévy index and drift sign.

Contribution

It provides the first comprehensive asymptotic analysis of the expected maximum for Lévy flights with drift, including explicit formulas and scaling forms.

Findings

Expected maximum diverges for $0<er 1$

For $1<er 2$, asymptotic expansion obtained

Expected maximum grows as $n^{2-er}$ for $1<er<2$

Abstract

In this paper, we study the large $n$ asymptotics of the expected maximum of an $n$-step random walk/L\'evy flight (characterized by a L\'evy index $1<\mu\leq 2$) on a line, in the presence of a constant drift $c$. For $0<\mu\leq 1$, the expected maximum is infinite, even for finite values of $n$. For $1<\mu\leq 2$, we obtain all the non-vanishing terms in the asymptotic expansion of the expected maximum for large $n$. For $c<0$ and $\mu =2$, the expected maximum approaches a non-trivial constant as $n$ gets large, while for $1<\mu < 2$, it grows as a power law $\sim n^{2-\mu}$. 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 $c\ne 0$. These results are borne out by numerical simulations in excellent agreement with our analytical predictions.