On the Fourier coefficients of linear fractional stable motion

Abstract

Inspired by a theorem of Marcinkiewicz [J. Marcinkiewicz, On a class of functions and their Fourier series, C. R. Soc. Sci. Varsovie, 26:71–77, 1934. Reprinted in: J. Marcinkiewicz, Collected Papers (A. Zygmund (Ed.)), PanstwoweWydawnictwo Naukowe,Warsaw, 1964] stating that the maximum of the absolute values of real Fourier coefficients an and bn of a function of bounded p-variation \( \left( {p \geqslant 1} \right) \) on an interval [0, 1] is of order O(n−1/p) as n → ∞, we compute the Fourier coefficients of the linear fractional stable motion (LFSM) and of the closely related Riemann–Liouville (RL) process and investigate the rate of their decay.