Abstract
According to the Dudley-Wichura extension of the Skorohod representation theorem, convergence in distribution to a limit in a separable set is equivalent to the existence of a coupling with elements converging a.s.in the metric. A density analogue of this theorem says that a sequence of probability densities on a general measurable space has a probability density as a pointwise lower limit if and only if there exists a coupling with elements converging a.s.in the discrete metric. In this paper the discrete-metric theorem is extended to stochastic processes considered in a widening time window. The extension is then used to prove the separability version of the Skorohod representation theorem. The paper concludes with an application to Markov chains.
Highlights
Let X1, X2, . . . , X be random elements in a general space (E, E) with distributions P1, P2, . . . , P
In a 1995 paper [10], Section 5.4, this author showed that convergence in density is equivalent to the existence of a coupling converging in the discrete metric, that is, (1) holds if and only if there exists a coupling (X1, X2, . . . , X ) of X1, X2, . . . , X such that for some random variable N taking values in N = {1, 2, . . . }, Xn = X, n N. This density result is analogous to the Skorohod representation theorem which says that convergence in distribution on a complete separable metric E with E the Borel sets is equivalent to the existence of a coupling converging a.s. in the metric
In the present paper we extend the above mentioned density result at (2) to stochastic processes considered in a widening time window
Summary
Let X1, X2, . . . , X be random elements in a general space (E, E) with distributions P1, P2, . . . , P. This density result is analogous to the Skorohod representation theorem which says that convergence in distribution on a complete separable metric E with E the Borel sets (a Polish space) is equivalent to the existence of a coupling converging a.s. in the metric. Skorohod proved this theorem in the 1956 paper [9], Dudley removed the completeness assumption in the 1968 paper [6], and Wichura showed in the 1970 paper [12] that it is enough that the limit probability measure P is concentrated on a separable Borel set; for historical notes, see [7].
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