Skorohod representation on a given probability space |
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Authors: | Patrizia Berti Luca Pratelli Pietro Rigo |
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Affiliation: | (1) Dipartimento di Matematica Pura ed Applicata “G. Vitali”,, Universita’ di Modena e Reggio-Emilia, via Campi 213/B, 41100 Modena, Italy;(2) Accademia Navale, viale Italia 72, 57100 Livorno, Italy;(3) Dipartimento di Economia Politica e Metodi Quantitativi, Universita’ di Pavia, via S. Felice 5, 27100 Pavia, Italy |
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Abstract: | Let $(Omega,mathcal{A},P)Let be a probability space, S a metric space, μ a probability measure on the Borel σ-field of S, and an arbitrary map, n = 1,2,.... If μ is tight and X n converges in distribution to μ (in Hoffmann–J?rgensen’s sense), then X∼μ for some S-valued random variable X on . If, in addition, the X n are measurable and tight, there are S-valued random variables and X, defined on , such that , X∼μ, and a.s. for some subsequence (n k ). Further, 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 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 , for some H⊂ (0,1) with outer Lebesgue measure 1, where 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 ). |
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Keywords: | Empirical process Non measurable random element Skorohod representation theorem Stable convergence Weak convergence of probability measures |
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