Let μ n be a probability measure on the Borel σ-field on D[0, 1] with respect to Skorohod distance, n ≥ 0. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are D[0, 1]-valued random variables X n such that X n ~ μ n for all n ≥ 0 and ||X n − X 0|| → 0 in probability, where ||·|| is the sup-norm. Such conditions do not require μ 0 separable under ||·||. Applications to exchangeable empirical processes and to pure jump processes are given as well.
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Berti, P., Pratelli, L. & Rigo, P. A Skorohod representation theorem for uniform distance. Probab. Theory Relat. Fields 150, 321–335 (2011). https://doi.org/10.1007/s00440-010-0279-6
- Cadlag function
- Exchangeable empirical process
- Separable probability measure
- Skorohod representation theorem
- Uniform distance
- Weak convergence of probability measures
Mathematics Subject Classification (2000)