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On the completion of Skorokhod space. (English) Zbl 07252786
The authors give an explicit description of the completion of the (classical) Skorokhod space [A. V. Skorokhod, Teor. Veroyatn. Primen. 1, 289–319 (1956; Zbl 0074.33802)] of càdlàg functions from the unit interval \([0,1]\) to itself. Let \(\mathbb{D}\) denote this Skorokhod space and let \(\mathbb{D}^+\) be the product \(\mathbb{D}\times\Sigma\), where \(\Sigma\) denotes the set of all continuous non-decreasing from \([0,1]\) to itself. The distance between two pairs \(\langle F,\sigma\rangle\) and \(\langle G,\tau\rangle\) is defined to be the infimum of all sums \[ \|F\circ\gamma-G\|+\|\sigma\circ\gamma-\tau\| \] where \(\|\cdot\|\) denotes the uniform norm and \(\gamma\) runs through all order-preserving homeomorphisms of \([0,1]\). This defines a pseudometric. The map \(F\mapsto\langle F,\iota\rangle\) is an isometric embedding of \(\mathbb{D}\) into \(\mathbb{D}^+\) and so the latter’s metric quotient is the desired completion.
Reviewer: K. P. Hart (Delft)
MSC:
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
46N30 Applications of functional analysis in probability theory and statistics
60G99 Stochastic processes
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