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The central limit theorem can be generalized to independent but not iid random variables, provided they satisfy the Lyapunov condition (which looks something like a variance bound), see https://en.wikipedia.org/wiki/Central_limit_theorem#Lyapunov_CLT. The Lindeberg central limit theorem provides another such condition. Meanwhile, Donsker's theorem states that a random walk with iid mean 0 variance 1 increments converges to Brownian motion. I am looking for a version of Donsker's theorem for independent but not iid random variables, subject to something like the Lyapunov or Lindeberg condition, but I've been unable to find it written down so far. Any reference to something like this would be appreciated!

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Such results were obtained by A. A. Borovkov and his students. See e.g.

Borovkov, A. A. Estimates in the invariance principle. (Russian) Dokl. Akad. Nauk SSSR 206 (1972), 1037–1039.

Borovkov, A. A. The rate of convergence in the invariance principle. (Russian. English summary) Teor. Verojatnost. i Primenen. 18 (1973), 217–234.

Borovkov, A. A. Rate of convergence and large deviations in invariance principle. Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 725–731, Acad. Sci. Fennica, Helsinki, 1980.

Borovkov, A. A.; Sahanenko, A. I. On the rate of convergence in invariance principle. Probability theory and mathematical statistics (Tbilisi, 1982), 59–66, Lecture Notes in Math., 1021, Springer, Berlin, 1983. https://link.springer.com/chapter/10.1007/BFb0072903

Borovkov, K. A. The rate of convergence in the invariance principle for a Hilbert space. (Russian) Teor. Veroyatnost. i Primenen. 29 (1984), no. 3, 532–535.

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