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

Monotonicity, asymptotic normality and vertex degrees in random graphs

Svante Janson

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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.

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

First available in Project Euclid: 9 November 2007

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


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

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