# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The coalescent in population models with time-inhomogeneous environment. (English) Zbl 1064.92034
Summary: The coalescent theory, well developed for the class of exchangeable population models with time-homogeneous reproduction law, is extended to a class of population models with time-inhomogeneous environment, where the population size is allowed to vary deterministically with time and where the distribution of the family sizes is allowed to change from generation to generation. A new class of time-inhomogeneous coalescent limit processes with simultaneous multiple mergers arises. Its distribution can be characterized in terms of product integrals.

##### MSC:
 92D15 Problems related to evolution 60J70 Applications of Brownian motions and diffusion theory 60F17 Functional limit theorems; invariance principles
Full Text:
##### References:
 [1] Billingsley, P.: Convergence of probability measures. (1968) · Zbl 0172.21201 [2] Cannings, C.: The latent roots of certain Markov chains arising in genetics: a new approach, I. Haploid models. Adv. in appl. Probab. 6, 260-290 (1974) · Zbl 0284.60064 [3] Cannings, C.: The latent roots of certain Markov chains arising in genetics: a new approach, II. Further haploid models. Adv. in appl. Probab. 7, 264-282 (1975) · Zbl 0339.60067 [4] Donnelly, P.: A genealogical approach to variable-population-size models in population genetics. J. appl. Probab. 23, 283-296 (1986) · Zbl 0635.92008 [5] Donnelly, P.; Tavaré, S.: Coalescents and genealogical structure under neutrality.. Annu. rev. Genet. 29, 401-421 (1995) [6] Gill, R. D.; Johansen, S.: A survey of product-integration with a view toward application in survival analysis. Ann. statist. 18, 1501-1555 (1990) · Zbl 0718.60087 [7] Griffiths, R. C.; Tavaré, S.: Sampling theory for neutral alleles in a varying environment. Philos. trans. Roy. soc. London B 344, 403-410 (1994) [8] Kingman, J. F. C.: On the genealogy of large populations. J. appl. Probab. 19A, 27-43 (1982) · Zbl 0516.92011 [9] Kingman, J. F. C.: Exchangeability and the evolution of large populations. Exchangeability in probability and statistics, 97-112 (1982) · Zbl 0494.92011 [10] Kingman, J. F. C.: The coalescent. Stochastic process appl. 13, 235-248 (1982) · Zbl 0491.60076 [11] Möhle, M.: Robustness results for the coalescent. J. appl. Probab. 35, 438-447 (1998) · Zbl 0913.60022 [12] Möhle, M.: Weak convergence to the coalescent in neutral population models. J. appl. Probab. 36, 446-460 (1999) · Zbl 0938.92024 [13] Möhle, M., Sagitov, S., 1999. A classification of coalescent processes for haploid exchangeable population models. Preprint 10, Department of Mathematical Statistics, Göteborg University, Sweden, Ann. Probab., to appear. · Zbl 1013.92029 [14] Péano, G.: Intégration par séries des équations différentielles linéaires. Math. ann. 32, 450-456 (1888) [15] Pitman, J.: Coalescents with multiple collisions. Ann. probab. 27, 1870-1902 (1999) · Zbl 0963.60079 [16] Sagitov, S.: The general coalescent with asynchronous mergers of ancestral lines. J. appl. Probab. 36, 1116-1125 (1999) · Zbl 0962.92026 [17] Schweinsberg, J.: A necessary and sufficient condition for the ${\Lambda}$-coalescent to come down from infinity. Electron. comm. Probab. 5, 1-11 (2000) · Zbl 0953.60072 [18] Schweinsberg, J.: Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5, 1-50 (2000) · Zbl 0959.60065