Combinatorial Stochastic Processes: Ecole d’Eté de by Jim Pitman (auth.), Jean Picard (eds.)

By Jim Pitman (auth.), Jean Picard (eds.)

Three sequence of lectures got on the thirty second likelihood summer season university in Saint-Flour (July 7–24, 2002), via the Professors Pitman, Tsirelson and Werner. ThecoursesofProfessorsTsirelson(“Scalinglimit,noise,stability”)andWerner (“Random planar curves and Schramm-Loewner evolutions”) were p- lished in a prior factor ofLectures Notes in arithmetic (volume 1840). This quantity comprises the path “Combinatorial stochastic techniques” of Professor Pitman. We cordially thank the writer for his functionality in Saint-Flour and for those notes. seventy six members have attended this college. 33 of them have given a quick lecture. The lists of members and of brief lectures are enclosed on the finish of the quantity. The Saint-Flour likelihood summer season institution used to be based in 1971. listed here are the references of Springer volumes which were released sooner than this one. All numbers consult with theLecture Notes in arithmetic series,except S-50 which refers to quantity 50 of the Lecture Notes in information sequence. 1971: vol 307 1980: vol 929 1990: vol 1527 1998: vol 1738 1973: vol 390 1981: vol 976 1991: vol 1541 1999: vol 1781 1974: vol 480 1982: vol 1097 1992: vol 1581 2000: vol 1816 1975: vol 539 1983: vol 1117 1993: vol 1608 2001: vol 1837 & 1851 1976: vol 598 1984: vol 1180 1994: vol 1648 2002: vol 1840 1977: vol 678 1985/86/87: vol 1362 & S-50 1995: vol 1690 2003: vol 1869 1978: vol 774 1988: vol 1427 1996: vol 1665 1979: vol 876 1989: vol 1464 1997: vol 1717

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Extra info for Combinatorial Stochastic Processes: Ecole d’Eté de Probabilités de Saint-Flour XXXII – 2002

Example text

This model turns up naturally in the theory of random graphs and has been studied and applied in several other contexts. The coalescent obtained by reversing the process of deletion of edges is the additive coalescent studied in [356]. The structure of large random trees is one of the main themes of this course, to be taken up in Chapter 6. This leads in Chapter 7 to the notion of continuum trees embedded in Brownian paths, then in Chapter 10 to a representation in terms of continuum trees of the large n asymptotics of the additive coalescent.

1. Finite partitions A random partition Πn of [n] is called exchangeable if its distribution is invariant under the natural action on partitions of [n] by the symmetric group of permutations of [n]. Equivalently, for each partition {A1 , . . , Ak } of [n], P(Πn = {A1 , . . , Ak }) = p(|A1 |, . . , |Ak |) for some symmetric function p of compositions (n1 , . . , nk ) of n. This function p is called the exchangeable partition probability function (EPPF) of Πn . For instance, given two positive sequences v• = (v1 , v2 , .

61), y = e, θ = 12 . See [27] for many more examples. Note that mixtures over θ of (0, θ) partitions (n) could arise by suitable choice of v• so that |Πn |/ log n had a non-degenerate limit distribution, but this phenomenon does not seem to arise naturally in combinatorial examples. Another case, treated by Pavlov [335, 336, 337], and Aldous-Pitman [17] covers a large number of examples involving random forests, where the limit involves the stable subordinator of index 12 . 5. Let w• = (wj ) be a sequence of weights with exponential gen∞ erating function w(ξ) := j=1 ξ j wj /j!

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