The principle of a single big jump: discrete and continuous time   modulated random walks with heavy-tailed increments
  We consider a modulated process S which, conditional on a background process X, has independent increments. Assuming that S drifts to -infinity and that its increments (jumps) are heavy-tailed (in a sense made precise in the paper), we exhibit natural conditions under which the asymptotics of the tail distribution of the overall maximum of S can be computed. We present results in discrete and in continuous time. In particular, in the absence of modulation, the process S in continuous time reduces to a Levy process with heavy-tailed Levy measure. A central point of the paper is that we make full use of the so-called ``principle of a single big jump'' in order to obtain both upper and lower bounds. Thus, the proofs are entirely probabilistic. The paper is motivated by queueing and Levy stochastic networks. 
Sergey Foss, Takis Konstantopoulos and Stan Zachary