Karapiperakis, Nikolaos and Korshunov, Dmitry (2026) On Tail Equivalence to the Supremum : From Lévy to Compound Renewal Processes. PhD thesis, Lancaster University.
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Abstract
The supremum of a stochastic process is an important object in applied probability, with applications in insurance, finance, queueing theory, and risk analysis. Although its distribution is typically intractable, in many settings the tail of the supremum is asymptotically equivalent to that of the underlying process, a phenomenon we call tail equivalence to the supremum. In this thesis we study certain cases where such equivalence holds. By highlighting a certain property, which we term synergistic at extremes, we show that processes attaining extreme values through the combined effect of all increments admit simpler and more intuitive arguments. This perspective allows us to give alternative proofs of existing results on tail equivalence for Lévy processes and to treat special cases such as the perturbed ruin process. The main contribution of the thesis is to extend the theory beyond the Lévy setting to compound renewal processes, removing the assumption of exponential waiting times. The key step is to establish extreme lower-tail asymptotics for sums of i.i.d. random variables, using exponential tilting and Edgeworth-type expansions. This leads to tail equivalence results for a broad class of compound renewal processes with light-tailed jumps.