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Unformatted text preview: Chapter 7 Continuity of Stochastic Processes Section 7.1 describes the leading kinds of continuity for stochastic processes, which derive from the modes of convergence of random variables. It also defines the idea of versions of a stochastic process. Section 7.2 explains why continuity of sample paths is often prob- lematic, and why we need the whole “paths in U ” song-and-dance. As an illustration, we consider a Gausssian process which is close to the Wiener process, except that it’s got a nasty non-measurability. Section 7.3 introduces separable random functions. 7.1 Kinds of Continuity for Processes Continuity is a convergence property: a continuous function is one where con- vergence of the inputs implies convergence of the outputs. But we have several kinds of convergence for random variables, so we may expect to encounter several kinds of continuity for random processes. Note that the following definitions are stated broadly enough that the index set T does not have to be one-dimensional. Definition 70 (Continuity in Mean) A stochastic process X is continuous in the mean at t if t → t implies E | X ( t )- X ( t ) | 2 → . X is continuous in the mean if this holds for all t . It would, of course, be more natural to refer to this as “continuity in mean square ”, or even “continuity in L 2 ”, and one can define continuity in L p for arbitrary p . Definition 71 (Continuity in Probability) X is continuous in probability at t if t → t implies X ( t ) P → X ( t ) . X is continuous in probability or stochas- tically continuous if this holds for all t . 34 CHAPTER 7. CONTINUITY 35 Note that neither L p-continuity nor stochastic continuity says that the indi- vidual sample paths, themselves, are continuous. Definition 72 (Continuous Sample Paths) A process X is continuous at t if, for almost all ω , t → t implies X ( t, ω ) → X ( t , ω ) . A process is continu- ous if, for almost all ω , X ( · , ω ) is a continuous function. Obviously, continuity of sample paths implies stochastic continuity and L p- continuity. A weaker pathwise property than strict continuity, frequently used in prac- tice, is the combination of continuity from the right with limits from the left. This is usually known by the term “cadlag”, abbreviating the French phrase “continues ` a droite, limites ` a gauche”; “rcll” is an unpronounceable synonym. Definition 73 (Cadlag) A sample function x on a well-ordered set T is cadlag if it is continuous from the right and limited from the left at every point. That is, for every t ∈ T , t ↓ t implies x ( t ) → x ( t ) , and for t ↑ t , lim t ↑ t x ( t ) exists, but need not be x ( t ) . A stochastic process X is cadlag if almost all its sample paths are cadlag....
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This note was uploaded on 12/20/2011 for the course STAT 36-754 taught by Professor Schalizi during the Spring '06 term at University of Michigan.
- Spring '06