Concentration inequalities via zero bias couplings

Larry Goldstein, Ümit Işlak

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


The tails of the distribution of a mean zero, variance σ2 random variable Y satisfy concentration of measure inequalities of the form P(Y≥t) ≤ exp(-B(t)) for B(t)=t2/2(σ2 + ct) for t ≥ 0, and B(t)=t/c(log t -log log t-σ2/c)for t>e whenever there exists a zero biased coupling of Y bounded by c, under suitable conditions on the existence of the moment generating function of Y. These inequalities apply in cases where Y is not a function of independent variables, such as for the Hoeffding statistic Y=∑i=1naiπ(i) where A=(aij)1≤i,j≤n ∈Rn×n and the permutation π has the uniform distribution over the symmetric group, and when its distribution is constant on cycle type.

Original languageEnglish (US)
Pages (from-to)17-23
Number of pages7
JournalStatistics and Probability Letters
Issue number1
StatePublished - Mar 2014
Externally publishedYes

Bibliographical note

Funding Information:
The first author’s work is partially supported by NSA grant H98230-11-1-0162 .


  • Primary
  • Stein's method
  • Tail probabilities
  • Zero bias coupling


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