Consistent nonparametric tests for detecting gradual changes in the marginals and the copula of multivariate time series

From a series of observations $${\mathbf Y}_1, \ldots , {\mathbf Y}_n$$ Y 1 , … , Y n in $$\mathbb {R}^d$$ R d taken sequentially, an interesting question is to know whether or not a significant change occurred in their stochastic behavior. The problem has been largely investigated both for univariate and multivariate observations, where the null hypothesis states that $$F_1 = \cdots = F_n$$ F 1 = ⋯ = F n , where $$F_j({\mathbf y}) = \mathrm{P}({\mathbf Y}_j \le {\mathbf y})$$ F j ( y ) = P ( Y j ≤ y ) . In most of the works done so far, the alternative hypothesis is generally that of an abrupt change at some unknown time K, i.e. $$F_j = D_1$$ F j = D 1 for $$j \le K$$ j ≤ K and $$F_j = D_2$$ F j = D 2 when $$j > K$$ j > K . This assumption is unrealistic in applications where changes tend to occur gradually. In this paper, a more general gradual-change model is proposed in which one admits the existence of times $$K_1