Foundations of Modern Probability pp 202-223 | Cite as

# Special Notions of Symmetry and Invariance

Chapter

## Abstract

Palm distributions and inversion formulas; stationarity and cycle stationarity; local hitting and conditioning; ergodic properties of Palm measures; exchangeable sequences and processes; strong stationarity and predictable sampling; ballot theorems; entropy and information

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## References

- Palm distributions are named after the Swedish engineer Palm (1943), who in a pioneering study of intensity fluctuations in telephone traffic considered some basic Palm probabilities associated with simple, stationary point processes on ℝ, using an elementary conditioning approach. Palm also derived some primitive inversion formulas. An extended and more rigorous account of Palm’s ideas was given by Khinchin (1955), in a monograph on queuing theory.Google Scholar
- Independently of Palm’s work, Kaplan (1955) first obtained Theorem 11.4 as an extension of some results for renewal processes by Doob (1948). A partial discrete-time result in this direction had already been noted by Kac (1947). Kaplan’s result was rediscovered in the setting of Palm distributions, independently by Ryll-Nardzewski (1961) and Slivnyak (1962). In the special case of intervals on the real line, Theorem 11.5 (i) was first noted by Korolyuk (as cited by Khinchin (1955)), and part (iii) of the same theorem was obtained by Ryll-Nardzewski (1961). The general versions are due to König and Matthes (1963) and Matthes (1963) for
*d*= 1 and to Matthes et al. (1978) for*d >*1. A more primitive setwise version of Theorem 11.8 (i), due to Slivnyak (1962), was strengthened by Zähle (1980) to convergence in total variation.Google Scholar - de Finetti (1930,1937) proved that an infinite sequence of exchangeable random variables is mixed i.i.d. The result became a cornerstone in his theory of subjective probability and Bayesian statistics. Ryll-Nardzewski (1957) noted that the theorem remains valid under the weaker hypothesis of spreadability, and Bühlmann (1960) extended the result to continuous time. The predictable sampling property in Theorem 11.13 was first noted by Doob (1936) for i.i.d. random variables and increasing sequences of predictable times. The general result and its continuous-time counterpart appear in Kallenberg (1988). Sparre-Andersen’s (1953–54) announcement of his Corollary 11.14 was (according to Feller) “a sensation greeted with incredulity, and the original proof was of an extraordinary intricacy and complexity.” A simplified argument (different from ours) appears in Feller (1971). Lemma 11.9 is quoted from Kallenberg (1999b).Google Scholar
- Bertrand (1887) noted that if two candidates A and B in an election get the proportions
*p*and 1 —*p*of the votes, then the probability that A will lead throughout the counting of ballots equals (2*p*- 1) V 0. More general “ballot theorems” and alternative proofs have been discovered by many authors, beginning with André (1887) and Barbier (1887). Takács (1967) obtained the version for cyclically stationary processes on a finite interval and gave numerous applications to queuing theory. The present statement is cited from Kallenberg (1999a).Google Scholar - The first version of Theorem 11.18 was obtained by Shannon (1948), who proved the convergence in probability for stationary and ergodic Markov chains in a finite state space. The Markovian restriction was lifted by McMillan (1953), who also strengthened the result to convergence in
*L*^{1}. Carleson (1958) extended McMillan’s result to countable state spaces. The a.s. convergence is due to Breiman (1957–60) and A. Ionescu Tul-Cea (1960) for finite state spaces and to Chung (1961) for the countable case.Google Scholar - More information about Palm measures is available in Matthes et al. (1978), Daley and Vere-Jones (1988), and Thorisson (2000). Applications to queuing theory and other areas are discussed by many authors, including Franken et al. (1981) and Baccelli and Bremaud (1994). Aldous (1985) gives a comprehensive survey of exchangeability theory. A nice introduction to information theory is given by Billingsley (1965).Google Scholar

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© Springer Science+Business Media New York 2002