Two classes of processes are considered. One is a class of spectrally positive infinitely divisible processes which includes all such stable processes. The other is a class of processes constructed from the sequence of partial sums of independent identically distributed positive random variables. A condition analogous to regular variation of the tails is imposed. Then a large deviation principle and a Strassen-type law of the iterated logarithm are presented. These theorems focus on unusually large values of the processes. They are expressed in terms of Skorokhod’s $M_1$ topology.
"Unusually Large Values for Spectrally Positive Stable and Related Processes." Ann. Probab. 27 (2) 990 - 1008, April 1999. https://doi.org/10.1214/aop/1022677393