A multivariate invariance principle is given for dependent processes exhibiting trending variances and other types of global nonstationarity. The limit processes obtained in these results are not Brownian motion, but members of a related class of Gaussian diffusion processes. Also derived is the limit in distribution of the mean product of a vector partial sum process with its increments, a standard result for asymptotic analysis of regressions in integrated variables. These statistics converge to members of a class of stochastic integrals under the broad assumptions yielding the invariance principle. An important application of these results is the analysis of regressions with I(2) variables. The distributions of the regression coefficient and t-value in a simple model are derived and tabulated by simulation.