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

2010/01/09 by Ghosh, Subhankar
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1001.1396

Abstract

Let (W,W') be an exchangeable pair of vectors in ℝ^k. Suppose this pair satisfies \beas E(W'|W)=(I_k-Λ)W+R(W). \enas If ||W-W'||₂≤ K and R(W)=0, then concentration of measure results of following form is proved for all w\succeq 0 when the moment generating function of W is finite. \beas P(W\succeqw),P(W\preceq -w)≤ exp(-(||w||₂²)/(2K²ν₁)), \enas for an explicit constant ν₁, where \succeq 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 \citecnm involving bounded size bias couplings.

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