# Estimating stationary mass, frequency by frequency

arXiv: https://arxiv.org/abs/2503.12808 · HTML: https://mlsanity.com/paper/2503.12808/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 43 references that are in the map.
- First posted on arXiv 17 Mar 2025.

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

- [Calibrated Language Models Must Hallucinate](https://mlsanity.com/paper/2311.14648/) (2023)
- [Conformal prediction for time series](https://mlsanity.com/paper/2010.09107/) (2020)
- [Optimal and instance-dependent guarantees for Markovian linear stochastic approximation](https://mlsanity.com/paper/2112.12770/) (2021)
- [Accelerated and instance-optimal policy evaluation with linear function approximation](https://mlsanity.com/paper/2112.13109/) (2021)
- [On Learning Markov Chains](https://mlsanity.com/paper/1810.11754/) (2018)
- [Statistical Estimation of Ergodic Markov Chain Kernel over Discrete State Space](https://mlsanity.com/paper/1809.05014/) (2018)
- [Just Wing It: Near-Optimal Estimation of Missing Mass in a Markovian Sequence](https://mlsanity.com/paper/2404.05819/) (2024)
- [How good is Good-Turing for Markov samples?](https://mlsanity.com/paper/2102.01938/) (2021)
- [Markov Chain Variance Estimation: A Stochastic Approximation Approach](https://mlsanity.com/paper/2409.05733/) (2024)

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

- [Predictive inference for time series: why is split conformal effective despite temporal dependence?](https://mlsanity.com/paper/2510.02471/) (2025)
- [Leave a Window Out: Modifying the Jackknife for Predictive Inference in Time Series](https://mlsanity.com/paper/2605.30292/) (2026)
- [Multiscale replay: A robust algorithm for stochastic variational inequalities with a Markovian buffer](https://mlsanity.com/paper/2601.01502/) (2026)
- [Next-token functional estimation](https://mlsanity.com/paper/2609.19529/) (2026)

## Abstract (arXiv)

Suppose we observe a trajectory of length n from an exponentially α-mixing stochastic process over a finite but potentially large state space. We consider the problem of estimating the probability mass placed by the stationary distribution of any such process on elements that occur with a certain frequency in the observed sequence. We estimate this vector of probabilities in total variation distance, showing universal consistency in n and recovering known results for i.i.d. sequences as special cases. Our proposed methodology -- implementable in linear time -- carefully combines the plug-in (or empirical) estimator with a recently-proposed modification of the Good--Turing estimator called WingIt, which was originally developed for Markovian sequences. En route to controlling the error of our estimator, we develop new performance bounds on WingIt and the plug-in estimator for exponentially α-mixing stochastic processes. Importantly, the extensively used method of Poissonization can no longer be applied in our non i.i.d. setting, and so we develop complementary tools -- including concentration inequalities for a natural self-normalized statistic of mixing sequences -- that may prove independently useful in the design and analysis of estimators for related problems. Simulation studies corroborate our theoretical findings.