On the weak invariance principle for ortho-martingale in Banach spaces. Application to stationary random fields

In this paper, we study the weak invariance principle for stationary ortho-martingales with values in 2-smooth or cotype 2 Banach spaces. Then, with the help of a suitable maximal ortho-martingale approximation, we derive the weak invariance principle for stationary random fields in L 1 under a condition in the spirit of Hannan. As an application, we get an asymptotic result for the L 1-Wasserstein distance between the common distribution function and the corresponding empirical distribution function of stationary random fields.