Abstract
Let \((\Omega,\mathcal{A},P)\) be a probability space, S a metric space, μ a probability measure on the Borel σ-field of S, and \(X_n:\Omega\rightarrow S\) an arbitrary map, n = 1,2,.... If μ is tight and X n converges in distribution to μ (in Hoffmann–Jørgensen’s sense), then X∼μ for some S-valued random variable X on \((\Omega,\mathcal{A},P)\). If, in addition, the X n are measurable and tight, there are S-valued random variables \(\overset{\sim}{X}_n\) and X, defined on \((\Omega,\mathcal{A},P)\), such that \(\overset{\sim}{X}_n\sim X_n\), X∼μ, and \(\overset{\sim}{X}_{n_k}\rightarrow X\) a.s. for some subsequence (n k ). Further, \(\overset{\sim}{X}_n\rightarrow X\) a.s. (without need of taking subsequences) if μ{x} = 0 for all x, or if P(X n = x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to \(\sigma(\mathcal{A}\cup\{H\})\) for some H⊂Ω with P *(H) = 1. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken \(((0,1),\sigma(\mathcal{U}\cup\{H\}),m_H)\), for some H⊂ (0,1) with outer Lebesgue measure 1, where \(\mathcal{U}\) is the Borel σ-field on (0,1) and m H the only extension of Lebesgue measure such that m H (H) = 1. In order to prove the previous results, it is also shown that, if X n converges in distribution to a separable limit, then X n k converges stably for some subsequence (n k ).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Berti P., Pratelli L., Rigo P.(2004). Limit theorems for a class of identically distributed random variables. Ann. Probab. 32, 2029–2052
Berti P., Pratelli L., Rigo P.(2006). Asymptotic behaviour of the empirical process for exchangeable data. Stoch. Proc. Appl. 116, 337–344
Crimaldi, I., Letta, G., Pratelli, L.: A strong form of stable convergence. Sem Probab (to appear) (2005)
Dudley R.(1999). Uniform central limit theorems. Cambridge University Press, Cambridge
Hall P., Heyde C.C.(1980). Martingale limit theory and its applications. Academic, New York
Letta, G., Pratelli, L.: Le théorème de Skorohod pour des lois de Radon sur un espace métrisable. In: Rendiconti Accademia Nazionale delle Scienze detta dei XL, vol. 115, pp. 157–162 (1997)
Renyi A.(1963). On stable sequences of events. Sankhya A 25, 293–302
van der Vaart A., Wellner J.A.(1996). Weak convergence and empirical processes. Springer, Berlin Heidelberg New York
van der Vaart A., van Zanten H.(2005). Donsker theorems for diffusions: necessary and sufficient conditions. Ann. Probab. 33, 1422–1451
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Berti, P., Pratelli, L. & Rigo, P. Skorohod representation on a given probability space. Probab. Theory Relat. Fields 137, 277–288 (2007). https://doi.org/10.1007/s00440-006-0018-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-006-0018-1
Keywords
- Empirical process
- Non measurable random element
- Skorohod representation theorem
- Stable convergence
- Weak convergence of probability measures