Concentration from Product Moments via an Additional Element of Randomness

2026-08-04

AI Summary

This work investigates improved probabilistic concentration inequalities under weak randomness models, including limited independence, random hashing, and Markov chains. To this end, it introduces a unified analytical framework based on elementary symmetric polynomials augmented with auxiliary randomization, recasting concentration problems as the control of product moments over uniformly sampled index sets—thereby replacing conventional exponential moment methods. This approach substantially weakens dependence on worst-case degrees and is successfully applied to three settings: read-Δ families, random binary linear hashing, and Markov chains. The framework not only recovers known spectral bounds and mixing time results but also yields tighter concentration inequalities across these scenarios.

Abstract

The standard method of exponential moments for proving concentration bounds can often be replaced by an argument based on elementary symmetric polynomials. We introduce an additional element of randomness into this framework, which reduces the problem to bounding product moments over a uniformly sampled set of indices. We show that this approach gives useful bounds in three settings. For read-$Δ$ families under limited independence, we obtain bounds governed by the degrees of randomly induced dependency subgraphs, improving the dependence on worst-case degrees. For random binary linear hashing with (semi-)random inputs, we derive fixed-bin and maximum-load bounds by controlling the rank defect of random tuples of input keys. Finally, for stochastic processes, we show how decay of product moments yields concentration bounds, recovering the spectral and mixing-time scales for finite-state Markov chains.

Problem

• concentration bounds
• limited independence
• random hashing
• stochastic processes
• product moments

Innovation

• product moments
• elementary symmetric polynomials
• randomness
• concentration bounds
• limited independence