## Abstract

The tails of the distribution of a mean zero, variance σ^{2} random variable Y satisfy concentration of measure inequalities of the form P(Y≥t) ≤ exp(-B(t)) for B(t)=t^{2}/2(σ^{2} + ct) for t ≥ 0, and B(t)=t/c(log t -log log t-σ^{2}/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=1}^{n}aiπ(i) where A=(a_{ij})_{1≤i,j≤n} ∈R^{n×n} and the permutation π has the uniform distribution over the symmetric group, and when its distribution is constant on cycle type.

Original language | English (US) |
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Pages (from-to) | 17-23 |

Number of pages | 7 |

Journal | Statistics and Probability Letters |

Volume | 86 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2014 |

Externally published | Yes |

### Bibliographical note

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

## Keywords

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