Summary
Let X be a stochastic process with sample paths in the usual Skorohod space D[0, 1]. For a sequence {X n} of independent copies of X, let S n=X1+⋯+Xn. Conditions which are either necessary or sufficient for the weak convergence of n −1/2(S n−ESn) to a Gaussian process with sample paths in D[0, 1] are discussed. Stochastically continuous processe are considered separately from those with fixed discontinuities. A bridge between the two is made by a Decomposition central limit theorem.
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Hahn, M.G. Central limit theorems in D[0, 1]. Z. Wahrscheinlichkeitstheorie verw Gebiete 44, 89–101 (1978). https://doi.org/10.1007/BF00533047
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DOI: https://doi.org/10.1007/BF00533047