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In this thesis we present some interesting new results for simultaneous inference under dependence. One of the key idea behind establishing such asymptotic theory is an invariance principle where the partial sums of a mean-zero process are approximated by a corresponding Gaussian analogue. The first work discusses such a Gaussian approximation result and extends achieving the popular KMT-type optimal bound to non-linear, non-stationary and weakly dependent vector-valued processes. In the second work, we use the invariance principle to construct simultaneous confidence intervals for time-varying coefficient models. Using Bahadur representation, it was possible to construct such confidence bands for models as complicated as ARMA-GARCH or generalized regression under a single framework. As a summary, this thesis is a systematic presentation of several interesting problems in simultaneous inference for vector-valued processes that can be explored with the help of a powerful invariance principle.


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