Foundations of Modern Probability pp 307-328 | Cite as

# Convergence of Random Processes, Measures, and Sets

Chapter

## Abstract

Relative compactness and tightness; uniform topology on C(K, S); Skorohod’s J_{1}-topology; equicontinuity and tightness; convergence of random measures; superposition and thinning; exchangeable sequences and processes; simple point processes and random closed sets

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## References

- After Donsker (1951–52) had proved his functional limit theorems for random walks and empirical distribution functions, a general theory of weak convergence in function spaces was developed by the Russian school, in seminal papers by Prohorov (1956), Skorohod (1956, 1957), and Kolmogorov (1956). Thus, Prohorov (1956) proved his fundamental compactness Theorem 16.3, in a setting for separable and complete metric spaces. The abstract theory was later extended in various directions by Le Cam (1957), Varadarajan (1958), and Dudley (1966, 1967). The elementary inequality of Ottaviani is from (1939).Google Scholar
- Originally Skorohod (1956) considered the space
*D*([0,1]) endowed with four different topologies, of which the*J*_{1}-topology considered here is by far the most important for applications. The theory was later extended to*D*(ℝ_{+}) by C.J. Stone (1963) and Lindvall (1973). Tightness was originally verified by means of various product moment conditions, developed by Chentsov (1956) and Billingsley (1968), before the powerful criterion of Aldous (1978) became available. Kurtz (1975) and Mitoma (1983) noted that criteria for tightness in*D*(ℝ_{+},*S*) can often be expressed in terms of one-dimensional projections, as in Theorem 16.27.Google Scholar - The weak convergence theory for random measures and point processes originated with Prohorov (1961), who noted the equivalence of (i) and (ii) in Theorem 16.16 when
*S*is compact. The development continued with seminal papers by Debes et al. (1970–71), Harris (1971), and Jagers (1974). The one-dimensional criteria in Proposition 16.17 and Theorems 16.16 and 16.29 are based on results in Kallenberg (1973a, 1986, 1996b) and a subsequent remark by Kurtz. Random sets had already been studied extensively by many authors, including Choquet (1953–54), Kendall (1974), and Matheron (1975), when an associated weak convergence theory was developed by Norberg (1984).Google Scholar - The applications considered in this chapter have a long history. Thus, primitive versions of Theorem 16.18 were obtained by Palm (1943), Khinchin (1955), and Ososkov (1956). The present version is due for
*S =*ℝ to Grigelionis (1963) and for more general spaces to Goldman (1967) and Jagers (1972). Limit theorems under simultaneous thinning and rescal-ing of a given point process were obtained by Rényi (1956), Nawrotzki (1962), Belyaev (1963), and Goldman (1967). The general version in Theorem 16.19 was proved by Kallenberg (1986) after Mecke (1968) had obtained his related characterization of Cox processes. Limit theorems for sampling from a finite population and for general exchangeable sequences have been proved in varying generality by many authors, including Chernov and Teicher (1958), Hájek (1960), Rosen (1964), Billingsley (1968), and Hagberg (1973). The results of Theorems 16.23 and 16.21 first appeared in Kallenberg (1973b).Google Scholar - Detailed accounts of weak convergence theory and its applications may be found in several excellent textbooks and monographs, including Billingsley (1968), Pollard (1984), Ethier and Kurtz (1986), and Jacod and Shiryaev (1987). More information on limit theorems for random measures and point processes is available in Matthes et al. (1978) and Kallenberg (1986). A good general reference for random sets is Matheron (1975).Google Scholar

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