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 ) d f ( t )  and∫ 0 1 g ( a ( t ) + M t ) d f ( t ), and thus of∫ 0 ∞ g ( a ( t ) + M t ) d f ( t ) , where M t = sup{| X s |: s ⩽ t } is the two‐sided maximum process corresponding to a Lévy process ( X t ) t ⩾ 0 , 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, Π(·), of the process X t . 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 M t with the largest jump of X t in (0, t ]. Some results concerning the convergence at zero and infinity of integrals like ∈ t g ( a ( t ) + | X t |) dt , ∈ t g ( S t ) dt , and ∈ t 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 ∞ E g ( a ( t ) + M t ) d f ( t ) , etc.