Limit theorems of Hilbert valued semimartingales and Hilbert valued martingale measures

In this paper, we study tight criteria of càdlàg Hilbert valued processes and prove the tightness of Hilbert valued square integrable martingales and Hilbert valued semimartingales by using their characteristics. These extend appropriate results of Jacod and Shiryaev (1987). We also discuss the property of Hilbert valued martingale measure and introduce the concept of convergence of martingale measures in distribution. The sufficient and necessary conditions are provided for strongly orthogonal martingale measures with independent increments. The conditions are given for convergence of martingale measures.