# Rosenthal-type inequalities for linear statistics of Markov chains

arXiv: https://arxiv.org/abs/2303.05838 · HTML: https://mlsanity.com/paper/2303.05838/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 66 references that are in the map.
- First posted on arXiv 10 Mar 2023.

## Builds on (most cited first, 9 of 9)

- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration](https://mlsanity.com/paper/2004.04719/) (2020)
- [Finite-time High-probability Bounds for Polyak-Ruppert Averaged Iterates of Linear Stochastic Approximation](https://mlsanity.com/paper/2207.04475/) (2022)
- [Adapting to Mixing Time in Stochastic Optimization with Markovian Data](https://mlsanity.com/paper/2202.04428/) (2022)
- [Optimal and instance-dependent guarantees for Markovian linear stochastic approximation](https://mlsanity.com/paper/2112.12770/) (2021)
- [Bias and Extrapolation in Markovian Linear Stochastic Approximation with Constant Stepsizes](https://mlsanity.com/paper/2210.00953/) (2022)
- [Tight High Probability Bounds for Linear Stochastic Approximation with Fixed Stepsize](https://mlsanity.com/paper/2106.01257/) (2021)
- [Asymptotic bias of inexact Markov Chain Monte Carlo methods in high dimension](https://mlsanity.com/paper/2108.00682/) (2021)

## Built on by (most cited first, 6 of 6)

- [First Order Methods with Markovian Noise: from Acceleration to Variational Inequalities](https://mlsanity.com/paper/2305.15938/) (2023)
- [Nonasymptotic Analysis of Stochastic Gradient Descent with the Richardson-Romberg Extrapolation](https://mlsanity.com/paper/2410.05106/) (2024)
- [High-Order Error Bounds for Markovian LSA with Richardson-Romberg Extrapolation](https://mlsanity.com/paper/2508.05570/) (2025)
- [Differential privacy guarantees of Markov chain Monte Carlo algorithms](https://mlsanity.com/paper/2502.17150/) (2025)
- [Statistical analysis of Inverse Entropy-regularized Reinforcement Learning](https://mlsanity.com/paper/2512.06956/) (2025)
- [Minimax estimation of the pseudo-spectral gap of continuous Markov chains, with applications to empirical concentration bounds](https://mlsanity.com/paper/2610.20007/) (2026)

## Abstract (arXiv)

In this paper, we establish novel concentration inequalities for additive functionals of geometrically ergodic Markov chains similar to Rosenthal inequalities for sums of independent random variables. We pay special attention to the dependence of our bounds on the mixing time of the corresponding chain. Precisely, we establish explicit bounds that are linked to the constants from the martingale version of the Rosenthal inequality, as well as the constants that characterize the mixing properties of the underlying Markov kernel. Finally, our proof technique is, up to our knowledge, new and is based on a recurrent application of the Poisson decomposition.