This paper discusses the convergence of stochastic processes, focusing on improving the "invariance principles" in probability theory. The invariance principles typically involve showing that a sequence of stochastic processes induces a sequence of measures that converge weakly to a limiting process. The paper shows that under certain conditions, the class of functions for which this convergence holds can be significantly enlarged by considering function spaces other than the usual ones, such as the space of continuous functions (C) or the space of Hölder continuous functions (Lip_α). The paper introduces the concept of Hölder continuity and shows that if a stochastic process satisfies a certain condition involving the expectation of the α-th power of the difference between two points, then its sample functions are Hölder continuous. This result is used to improve a theorem by Prokhorov, showing that the space of Hölder continuous functions can be used instead of the space of continuous functions, leading to a larger class of functions for which the convergence holds. The paper also discusses the implications of these results for the invariance principle of Donsker, showing that if additional moment conditions are satisfied, the convergence can be strengthened. The paper provides an application to the convergence of Gaussian processes, showing that under certain conditions, the paths of the limiting process are Hölder continuous of any order less than a certain value. The paper concludes with a corollary that shows that if the random variables have bounded 2pth moments, then the convergence holds for any functional that is continuous in the topology of Lip_γ for some γ < (p-1)/2p. The paper also notes that the moment assumption is essential for the convergence in the Lip_α topology.This paper discusses the convergence of stochastic processes, focusing on improving the "invariance principles" in probability theory. The invariance principles typically involve showing that a sequence of stochastic processes induces a sequence of measures that converge weakly to a limiting process. The paper shows that under certain conditions, the class of functions for which this convergence holds can be significantly enlarged by considering function spaces other than the usual ones, such as the space of continuous functions (C) or the space of Hölder continuous functions (Lip_α). The paper introduces the concept of Hölder continuity and shows that if a stochastic process satisfies a certain condition involving the expectation of the α-th power of the difference between two points, then its sample functions are Hölder continuous. This result is used to improve a theorem by Prokhorov, showing that the space of Hölder continuous functions can be used instead of the space of continuous functions, leading to a larger class of functions for which the convergence holds. The paper also discusses the implications of these results for the invariance principle of Donsker, showing that if additional moment conditions are satisfied, the convergence can be strengthened. The paper provides an application to the convergence of Gaussian processes, showing that under certain conditions, the paths of the limiting process are Hölder continuous of any order less than a certain value. The paper concludes with a corollary that shows that if the random variables have bounded 2pth moments, then the convergence holds for any functional that is continuous in the topology of Lip_γ for some γ < (p-1)/2p. The paper also notes that the moment assumption is essential for the convergence in the Lip_α topology.