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Concentration of measure for the number of isolated vertices in the Erdos-Rényi random graph by size bias couplings

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  • Ghosh, Subhankar
  • Goldstein, Larry
  • Raic, Martin

Abstract

A concentration of measure result is proved for the number of isolated vertices Y in the Erdos-Rényi random graph model on n edges with edge probability p. When [mu] and [sigma]2 denote the mean and variance of Y respectively, P((Y-[mu])/[sigma]>=t) admits a bound of the form e-kt2 for some constant positive k under the assumption p[set membership, variant](0,1) and np-->c[set membership, variant](0,[infinity]) as n-->[infinity]. The left tail inequality holds for all n[set membership, variant]{2,3,...},p[set membership, variant](0,1) and t>=0. The results are shown by coupling Y to a random variable Ys having the Y-size biased distribution, that is, the distribution characterized by E[Yf(Y)]=[mu]E[f(Ys)] for all functions f for which these expectations exist.

Suggested Citation

  • Ghosh, Subhankar & Goldstein, Larry & Raic, Martin, 2011. "Concentration of measure for the number of isolated vertices in the Erdos-Rényi random graph by size bias couplings," Statistics & Probability Letters, Elsevier, vol. 81(11), pages 1565-1570, November.
    • Handle: RePEc:eee:stapro:v:81:y:2011:i:11:p:1565-1570
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    References listed on IDEAS

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    1. Yosef Rinott & Vladimir Rotar, 2000. "Normal approximations by Stein's method," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 23(1), pages 15-29.
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