Bernoulli

  • Bernoulli
  • Volume 13, Number 4 (2007), 952-965.

Monotonicity, asymptotic normality and vertex degrees in random graphs

Svante Janson

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Abstract

We exploit a result by Nerman which shows that conditional limit theorems hold when a certain monotonicity condition is satisfied. Our main result is an application to vertex degrees in random graphs, where we obtain asymptotic normality for the number of vertices with a given degree in the random graph G(n, m) with a fixed number of edges from the corresponding result for the random graph G(n, p) with independent edges. We also give some simple applications to random allocations and to spacings. Finally, inspired by these results, but logically independent of them, we investigate whether a one-sided version of the Cramér–Wold theorem holds. We show that such a version holds under a weak supplementary condition, but not without it.

Article information

Source
Bernoulli, Volume 13, Number 4 (2007), 952-965.

Dates
First available in Project Euclid: 9 November 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1194625597

Digital Object Identifier
doi:10.3150/07-BEJ6103

Mathematical Reviews number (MathSciNet)
MR2364221

Zentralblatt MATH identifier
1132.60024

Keywords
asymptotic normality conditional limit theorem Cramér–Wold theorem random allocations random graphs vertex degrees

Citation

Janson, Svante. Monotonicity, asymptotic normality and vertex degrees in random graphs. Bernoulli 13 (2007), no. 4, 952--965. doi:10.3150/07-BEJ6103. https://projecteuclid.org/euclid.bj/1194625597


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References

  • Arfwedson, G. (1951). A probability distribution connected with Stirling's second class numbers., Skand. Aktuarietidskr. 34 121--132.
  • Barbour, A.D., Karoński, M. and Ruciński, A. (1989). A central limit theorem for decomposable random variables with applications to random graphs., J. Combin. Theory Ser. B 47 125--145.
  • Békéssy, A. (1963). On classical occupancy problems. I., Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 59--71.
  • Billingsley, P. (1968)., Convergence of Probability Measures. New York: Wiley.
  • Bollobás, B. (2001)., Random Graphs, 2nd ed. Cambridge: Cambridge Univ. Press.
  • Cramér, H. and Wold, H. (1936). Some theorems on distribution functions., J. London Math. Soc. 11 290--295.
  • Feller, W. (1957)., An Introduction to Probability Theory and Its Applications. I, 2nd ed. New York: Wiley.
  • Feller, W. (1971)., An Introduction to Probability Theory and its Applications. II, 2nd ed. New York: Wiley.
  • Holst, L. (1979). Two conditional limit theorems with applications., Ann. Statist. 7 551--557.
  • Holst, L. (1979). Asymptotic normality of sum-functions of spacings., Ann. Probab. 7 1066--1072.
  • Holst, L. (1981). Some conditional limit theorems in exponential families., Ann. Probab. 9 818--830.
  • Holst, L. (1981). On sequential occupancy problems., J. Appl. Probab. 18 435--442.
  • Holst, L. (1986). On birthday, collectors', occupancy and other classical urn problems., Internat. Statist. Rev. 54 15--27.
  • Hwang, H.-K. and Janson, S. (2007). Local limit theorems for finite and infinite urn models., Ann. Probab. To appear.
  • Janson, S. (1990). A functional limit theorem for random graphs with applications to subgraph count statistics., Random Struct. Alg. 1 15--37.
  • Janson, S. (1995). The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph., Random Struct. Alg. 7 337--355.
  • Janson, S. (2001). Moment convergence in conditional limit theorems., J. Appl. Probab. 38 421--437.
  • Janson, S. and Luczak, M. (2006). Asymptotic normality of the $k$-core in random graphs. Preprint., arXiv:math.CO/0612827
  • Janson, S., Łuczak, T. and Ruciński, A. (2000)., Random Graphs. New York: Wiley.
  • Kolchin, V.F., Sevast'yanov, B.A. and Chistyakov, V.P. (1976)., Random Allocations. (In Russian.) Moscow: Nauka. English transl.: Washington, D.C.: V.H. Winston & Sons (1978).
  • Le Cam, L. (1958). Un théorème sur la division d'un intervalle par des points pris au hasard., Publ. Inst. Statist. Univ. Paris 7 7--16.
  • von Mises, R. (1939). Über Aufteilungs- und Besetzungs-Wahrscheinlichkeiten., Acta [Trudy] Univ. Asiae Mediae. Ser. V-a. 27 1--21.
  • Nerman, O. (1998). Stochastic monotonicity and conditioning in the limit., Scand. J. Statist. 25 569--572.
  • Pittel, B. (1990). On tree census and the giant component in sparse random graphs., Random Struct. Alg. 1 311--342.
  • Rényi, A. (1962). Three new proofs and a generalization of a theorem of Irving Weiss., Magyar Tud. Akad. Mat. Kutató Int. Közl. 7 203--214.
  • Rosén, B. (1967). On the central limit theorem for sums of dependent random variables., Z. Wahrsch. Verw. Gebiete 7 48--82.
  • Rosén, B. (1969). Asymptotic normality in a coupon collector's problem., Z. Wahrsch. Verw. Gebiete 13 256--279.
  • Steck, G.P. (1957). Limit theorems for conditional distributions., Univ. California Publ. Statist. 2 237--284.
  • Weiss, I. (1958). Limiting distributions in some occupancy problems., Ann. Math. Statist. 29 878--884.