Multivariate concentration of measure type results using exchangeable pairs and size biasing

الأصل · EN

Let (W,W') be an exchangeable pair of vectors in Rᵏ. Suppose this pair satisfies E(W'|W)=(Iₖ-Λ)W+R(W). If ||W-W'||₂≤ K and R(W)=0, then concentration of measure results of following form is proved for all w 0 when the moment generating function of W is finite. P(W),P(W -w)≤ (-||w||₂²/2K²ν₁), for an explicit constant ν₁, where stands for coordinate wise ≥ ordering. This result is applied to examples like complete non degenerate U-statistics. Also, we deal with the example of doubly indexed permutation statistics where R(W)≠ 0 and obtain similar concentration of measure inequalities. Practical examples from doubly indexed permutation statistics include Mann-Whitney-Wilcoxon statistic and random intersection of two graphs. Both these two examples are used in nonparametric statistical testing. We conclude the paper with a multivariate generalization of a recent concentration result due to Ghosh and Goldstein cnm involving bounded size bias couplings.