On the supremum of an infinitely divisible process

It was shown by Berman in a recent paper that, for any infinitely divisible process X = {Xt, t[greater-or-equal, slanted]0} with symmetric increments, P(sup0[less-than-or-equals, slant]s[less-than-or-equals, slant]t Xs[greater-or-equal, slanted]u) ~ P(Xt[greater-or-equal, slanted]u) (u --> [infinity]) if the right tail of the Lévy measure is regularly varying with index 0<[alpha]<2. In this note we use a simple argument to show that this result is true for a more general class of processes.