The paper "ON CONVERGENCE OF STOCHASTIC PROCESSES" by John Lamperti explores the "invariance principles" in probability theory, which involve the weak convergence of measures induced by sequences of stochastic processes. The main focus is on extending the class of functions for which this weak convergence holds, particularly by using spaces other than the standard continuous function spaces. The author aims to improve known invariance principles by considering spaces like $\text{Lip}_n$ (spaces of functions with Hölder continuous derivatives) instead of $\mathcal{C}$ (spaces of continuous functions). This approach allows for a broader class of functionals to be considered, enhancing the applicability of the results. Key contributions include: 1. ** Criterion for Hölder Continuity**: The paper uses a criterion from Loève to show that if a separable process satisfies a certain condition, its sample functions are almost surely Hölder continuous. 2. ** Precompactness in $\text{Lip}_\alpha$**: Lemmas are provided to show that bounded sets in $\text{Lip}_\alpha$ are precompact in $\mathcal{C}_\beta$ and $\text{Lip}_\beta$ for $\beta < \alpha$. 3. ** Precompactness of Measures**: A theorem by Prokhorov is extended to show that a family of measures on $\text{Lip}_\alpha$ is precompact if they satisfy a specific condition. 4. ** Strengthened Invariance Principle**: The main result is an improved version of a theorem by Prokhorov, showing that under certain conditions, a sequence of stochastic processes can be approximated by a process with Hölder continuous paths, and the weak convergence of functionals holds for a larger class of functionals. The paper also provides corollaries and applications, including a strengthened version of Donsker's theorem and a result on the convergence of Gaussian processes, demonstrating the practical significance of the theoretical advancements.The paper "ON CONVERGENCE OF STOCHASTIC PROCESSES" by John Lamperti explores the "invariance principles" in probability theory, which involve the weak convergence of measures induced by sequences of stochastic processes. The main focus is on extending the class of functions for which this weak convergence holds, particularly by using spaces other than the standard continuous function spaces. The author aims to improve known invariance principles by considering spaces like $\text{Lip}_n$ (spaces of functions with Hölder continuous derivatives) instead of $\mathcal{C}$ (spaces of continuous functions). This approach allows for a broader class of functionals to be considered, enhancing the applicability of the results. Key contributions include: 1. ** Criterion for Hölder Continuity**: The paper uses a criterion from Loève to show that if a separable process satisfies a certain condition, its sample functions are almost surely Hölder continuous. 2. ** Precompactness in $\text{Lip}_\alpha$**: Lemmas are provided to show that bounded sets in $\text{Lip}_\alpha$ are precompact in $\mathcal{C}_\beta$ and $\text{Lip}_\beta$ for $\beta < \alpha$. 3. ** Precompactness of Measures**: A theorem by Prokhorov is extended to show that a family of measures on $\text{Lip}_\alpha$ is precompact if they satisfy a specific condition. 4. ** Strengthened Invariance Principle**: The main result is an improved version of a theorem by Prokhorov, showing that under certain conditions, a sequence of stochastic processes can be approximated by a process with Hölder continuous paths, and the weak convergence of functionals holds for a larger class of functionals. The paper also provides corollaries and applications, including a strengthened version of Donsker's theorem and a result on the convergence of Gaussian processes, demonstrating the practical significance of the theoretical advancements.