Finiteness of integrals of functions of Lévy processes

We prove necessary and sufficient conditions for the almost sure convergence of the integrals ∫∞ 1 g(a(t)+Mt)df(t), ∫ 1 0 g(a(t)+Mt)df(t), and thus of ∫∞ 0 g(a(t)+Mt)df(t), where Mt = sup{|Xs| : s ≤ t} is the two-sided maximum process corresponding to a Lévy process (Xt)t≥0, a(·) is a nondecreasing function on [0,∞) with a(0) = 0, g(·) is a positive nonincreasing function on (0,∞), possibly with g(0+) = ∞, and f(·) is a positive nondecreasing function on [0,∞) with f(0) = 0. The conditions are expressed in terms of the canonical measure, Π(·), of the process Xt. The special case when a(x) = 0, f(x) = x and g(·) is equivalent to the tail of Π (at zero or infinity) leads to an interesting comparison of Mt with the largest jump of Xt in (0, t]. Some results concerning the convergence at zero and infinity of integrals like ∫ g(a(t) + |Xt|)dt, ∫ g(St)dt, and ∫ g(Rt)dt, where St is the supremum process and Rt = St −Xt is the process reflected in its supremum, are also given. We also consider the convergence of some integrals such as ∫∞ 0 Eg(a(t) + Mt)df(t), etc. 2000 MSC Subject Classifications: primary: 60H05, 60G51 secondary: 60J15, 60F15, 60G52