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)+ M_t) df (t)and ∫₀₁ g(a(t) + Mt) df (t), and thus of ∫ _0∞ g(a(t) + Mt)df (t), where M_t = sup{|X _s| : s ≤ t} is the two-sided maximum process corresponding to a Lévy process (X_t)t≥o, a (̇) is a non-decreasing function on [0, ∞)with a(0) =0, g(-)is a positive non-increasing function on (0, ∞), possibly with g(0+) = ∞,and f (̇) is a positive non-decreasing function on [0, ∞)with f (0) = 0. The conditions are expressed in terms of the canonical measure, Il(̇), of the process X _t. The special case when a(x) = 0, f (x) = x and g(̇) isequivalenttothe tail of Il(atzeroorinfinity) leads to an interesting comparison of Mt with the largest jump of Xt in (0,t]. ome results concerning the convergence at zero and infinity of integrals like ∫g(a(t) + |X _t|)dt, ∫ g(S_t) dt, and ∫g(R_t) dt,where S_t is the supremum process and R_t = S_t - X _t is the process reflected in its supremum, are also given. We also consider the convergence of integrals such as ∫₀∞ Eg(a(t) + Mt) df (t), etc.