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 language||English (US)|
|Number of pages||7|
|Journal||Statistics and Probability Letters|
|State||Published - Mar 2014|
Bibliographical noteFunding Information:
The first author’s work is partially supported by NSA grant H98230-11-1-0162 .
- Stein's method
- Tail probabilities
- Zero bias coupling