An application of Skorokhod's \(M_1\)-topology (Q2726256)

The main topic of this paper is a variation of the \(M_{1}\)-topology for \(D[0,\infty]\) defined by \textit{A. V. Skorokhod} [Teor. Veroyatn. Primen. 1, 289-319 (1956; Zbl 0074.33802)]. This topology has not been as useful as his finer \(J_{1}\)-topology defined in the same famous paper. However, one case where it is often useful and natural is the study of processes with nondecreasing paths. A large deviation principle (LDP) for the class of spectrally positive stable processes is presented (without a full proof). The results can be extended to a larger class of infinitely divisible processes and to some partial sums processes. The processes do not have nondecreasing paths but they almost have this property in the sense that the rate function of the LDP is infinite for the set of nondecreasing paths. The corresponding vague or weak LDP holds, but the rate function does not have \(J_{1}\)-compact level sets. A coarser topology must be used to obtain the full LDP. The paper also considers an alternative formulation of tightness and its use in the context of weak convergence and large deviations.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00022].