Skorohod representation on a given probability space |
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Authors: | Patrizia Berti Luca Pratelli Pietro Rigo |
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Institution: | (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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