Estimating stationary mass, frequency by frequency
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912416886947840 |
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| author | Nakul, Milind Muthukumar, Vidya Pananjady, Ashwin |
| author_facet | Nakul, Milind Muthukumar, Vidya Pananjady, Ashwin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_12808 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Estimating stationary mass, frequency by frequency Nakul, Milind Muthukumar, Vidya Pananjady, Ashwin Machine Learning Information Theory Probability Statistics Theory 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. |
| title | Estimating stationary mass, frequency by frequency |
| topic | Machine Learning Information Theory Probability Statistics Theory |
| url | https://arxiv.org/abs/2503.12808 |