Asymptotic behavior of continuous set-indexed martingales (Q5939304)

The author proves the following version of the asymptotic Knight's theorem for set-indexed martingales. Let \(M=(M_1^n,\dots ,M_k^n)\) be a sequence of continuous vector-valued set-indexed square integrable strong martingales such that for each component \(i\) there exists a sequence of stopping set processes \({\xi}_i^n\) and \(M_i^n\) \(({\xi}_i^n\) is a set-indexed continuous Brownian motion denoted by \(B_i^n)\). Suppose that for each \(i \neq j\) mutual brackets \(\langle M_i^n, M_i^n \rangle\) tend to zero. Then under some additional technical restrictions, \((B_1^n,\dots ,B_k^n)\) converges in distribution to a \(k\)-dimensional continuous Brownian motion. To create the proof, the author defines a tightness criterion for set-indexed continuous processes. The core of this characterization is connected with a weaker definition of continuity and hence the use of the corresponding topology, and with the fact that indices take values in a semilattice of closed subsets. An effective tightness criterion by means of an estimate for a majorizing measure defined on the space is presented. The author also emphasizes that in the set-indexed case it is impossible to define the inverse of the stopping sets introduced for the set-indexed version of Dubins-Schwarz theorem and to get the same characterization of Brownian sheet as given by Yor.