# Function-Specific Mixing Times and Concentration Away from Equilibrium

arXiv: https://arxiv.org/abs/1605.02077 · HTML: https://mlsanity.com/paper/1605.02077/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 May 2016.

## Built on by (most cited first, 4 of 4)

- [Fast Doubly-Adaptive MCMC to Estimate the Gibbs Partition Function with Weak Mixing Time Bounds](https://mlsanity.com/paper/2111.07372/) (2021)
- [Asymptotically Optimal Sequential Testing with Markovian Data](https://mlsanity.com/paper/2602.17587/) (2026)
- [Near-Optimal Clustering in Mixture of Markov Chains](https://mlsanity.com/paper/2506.01324/) (2025)
- [On the Pseudo-Mixing of Kac's Walk](https://mlsanity.com/paper/2608.17374/) (2026)

## Abstract (arXiv)

Slow mixing is the central hurdle when working with Markov chains, especially those used for Monte Carlo approximations (MCMC). In many applications, it is only of interest to estimate the stationary expectations of a small set of functions, and so the usual definition of mixing based on total variation convergence may be too conservative. Accordingly, we introduce function-specific analogs of mixing times and spectral gaps, and use them to prove Hoeffding-like function-specific concentration inequalities. These results show that it is possible for empirical expectations of functions to concentrate long before the underlying chain has mixed in the classical sense, and we show that the concentration rates we achieve are optimal up to constants. We use our techniques to derive confidence intervals that are sharper than those implied by both classical Markov chain Hoeffding bounds and Berry-Esseen-corrected CLT bounds. For applications that require testing, rather than point estimation, we show similar improvements over recent sequential testing results for MCMC. We conclude by applying our framework to real data examples of MCMC, providing evidence that our theory is both accurate and relevant to practice.