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Unformatted text preview: Chapter 8 More on Continuity Section 8.1 constructs separable modifications of reasonable but non-separable random functions, and explains how separability re- lates to non-denumerable properties like continuity. Section 8.2 constructs versions of our favorite one-parameter pro- cesses where the sample paths are measurable functions of the pa- rameter. Section 8.3 gives conditions for the existence of cadlag versions. Section 8.4 gives some criteria for continuity, and for the existence of “continuous modifications” of discontinuous processes. Recall the story so far: last time we saw that the existence of processes with given finite-dimensional distributions does not guarantee that they have desir- able and natural properties, like continuity, and in fact that one can construct discontinuous versions of processes which ought to be continuous. We therefore need extra theorems to guarantee the existence of continuous versions of pro- cesses with specified FDDs. To get there, we will first prove the existence of separable versions. This will require various topological conditions on both the index set T and the value space Ξ. In the interest of space (or is it time?), Section 8.1 will provide complete and detailed proofs. The other sections will simply state results, and refer proofs to standard sources, mostly Gikhman and Skorokhod (1965/1969). (They in turn follow Doob (1953), but are explicit about what he regarded as obvious generalizations and extensions, and they cost about $20, whereas Doob costs $120 in paperback.) 8.1 Separable Versions We can show that separable versions of our favorite stochastic processes exist under quite general conditions, but first we will need some preliminary results, living at the border between topology and measure theory. This starts by re- calling some facts about compact spaces. 40 CHAPTER 8. MORE ON CONTINUITY 41 Definition 82 (Compactness, Compactification) A set A in a topological space Ξ is compact if every covering of A by open sets contains a finite sub-cover. Ξ is a compact space if it is itself a compact set. Every non-compact topological space Ξ is a sub-space of some compact topological space ˜ Ξ . The super-space ˜ Ξ is a compactification of Ξ . Every compact metric space is separable. 1 Example 83 The real numbers R are not compact: they have no finite covering by open intervals (or other open sets). The extended reals, R ≡ R ∪ + ∞ ∪-∞ , are compact, since intervals of the form ( a, ∞ ] and [-∞ , a ) are open. This is a two-point compactification of the reals. There is also a one-point compactification, with a single point at ±∞ , but this has the undesirable property of making big negative and positive numbers close to each other....
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- Spring '06
- Tn, Metric space, sK, Doob, Borel