Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2201.10256

Quantitative coarse-graining of Markov chains

Bastian Hilder·Upanshu Sharma
View original source ↗Read source document ↗Export metadata ↓

Abstract

Coarse-graining techniques play a central role in reducing the complexity of stochastic models, and are typically characterised by a mapping which projects the full state of the system onto a smaller set of variables which captures the essential features of the system. Starting with a continuous-time Markov chain, in this work we propose and analyse an effective dynamics, which approximates the dynamical information in the coarse-grained chain. Without assuming explicit scale-separation, we provide sufficient conditions under which this effective dynamics stays close to the original system and provide quantitative bounds on the approximation error. We also compare the effective dynamics and corresponding error bounds to the averaging literature on Markov chains which involve explicit scale-separation. We demonstrate our findings on an illustrative test example.

Explore related subjects

math.PRmath.CAmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bastian Hilder, Upanshu Sharma. 2023-09-27. Quantitative coarse-graining of Markov chains. https://arxiv.org/abs/2201.10256

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Hermite trace polynomials and chaos decompositions for the Hermitian Brownian motion

For a non-zero parameter $q$, we define Hermite trace polynomials, which are multivariate polynomials indexed by permutations. We prove several combinatorial properties for them, such as expansions and product formulas. The linear functional determined by these trace polynomials is a state for $q = \frac{1}{N}$ for $N$ a non-zero integer. For such $q$, Hermite trace polynomials of different degrees are orthogonal. The product formulas extend to the closure with respect to the state. The state can be identified with the expectation induced by the $N \times N$ Hermitian Brownian motion. Hermite trace polynomials are martingales for this Brownian motion, while the elements in the closure can be interpreted as stochastic integrals with respect to it. Using the grading on the algebra, we prove several chaos decompositions for such integrals, as well as analyze corresponding creation and annihilation operators. In the univariate, pure trace polynomial case, trace Hermite polynomials can be identified with the Hermite polynomials of matrix argument.

math.PR↗
arXiv2026

Coupling and Maximal Inequalities for Graph-Dependent Empirical Processes

We develop maximal inequalities for empirical processes indexed by graph-dependent observations. Our bounds separate the complexity of the indexing class from two features specific to graph dependence: the geometry of the underlying graph and the cost of coupling graph-separated blocks to independent copies. The coupling construction combines a novel graph-adapted dependence coefficient with a coloring of a block partition. As an application, we derive Glivenko--Cantelli results and characterize the associated effective sample size. A central implication is that graph-dependent empirical processes need not exhibit a generic root-$n$ rate: convergence is jointly determined by function-class complexity, graph geometry, and the decay of dependence with graph distance. We specialize the results to graphs with polynomial and exponential growth and to directed dyadic graphs. Finally, we apply the results to network autoregressive models, nonlinear local-propagation models, and treatment-interference settings.

math.PR↗