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arxiv: 2604.21663 · v1 · submitted 2026-04-23 · 🧮 math.PR

Large deviations for non-irreducible Markov chains on Euclidean spaces

L\'eo Daures This is my paper

Pith reviewed 2026-05-09 20:27 UTC · model grok-4.3

classification 🧮 math.PR
keywords large deviationsMarkov chainsempirical measuresweak large deviations principlenon-irreduciblesubadditivityrate functionsEuclidean space
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The pith

Markov chains on R^d satisfy the weak large deviations principle for their empirical measures without assuming irreducibility.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes the weak large deviations principle for the sequence of empirical measures generated by a Markov chain on Euclidean space. The result requires only mild assumptions on the transition probabilities and holds for completely arbitrary initial distributions. The proof proceeds via a direct subadditive argument on the relevant cumulant generating functionals and requires no external theorems. When the chain fails to be irreducible, explicit examples show that the resulting rate function is typically not convex.

Core claim

Under mild assumptions on the transition kernel, the empirical measures of the Markov chain satisfy the weak large deviations principle in the space of probability measures equipped with the weak topology. The argument relies on subadditivity of the log-moment generating functionals of these empirical measures and remains valid without any irreducibility hypothesis. In the non-irreducible case the associated rate function is shown by counter-examples to lose convexity in general.

What carries the argument

Subadditivity of the log-moment generating functionals of the empirical measures.

If this is right

  • The weak large deviations principle holds for arbitrary initial distributions.
  • Irreducibility of the chain is not required.
  • The rate function need not be convex when the chain is non-irreducible.
  • The proof is entirely self-contained and uses only subadditivity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result permits study of empirical behavior in chains that possess multiple recurrent classes or absorbing components.
  • Convexity of the rate function appears to be tied to irreducibility rather than to the large-deviation property itself.
  • The subadditive approach may extend to other functionals of the chain path beyond the empirical measure.

Load-bearing premise

The transition probabilities satisfy conditions that make the log-moment generating functionals of the empirical measures subadditive.

What would settle it

A concrete Markov chain on R^d meeting the mild assumptions whose empirical-measure probabilities fail to decay according to any lower-semicontinuous rate function in the weak topology.

Figures

Figures reproduced from arXiv: 2604.21663 by L\'eo Daures.

Figure 1
Figure 1. Figure 1: Example of use of the coupling map, with [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of use of the decoupling map with respect to the partition [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Markov chain of Example C.4 for an example function f. The three parallel lines are the lines of equations y = x − 1, y = x and y = x + 1. The thick horizontal segment at y = f(Xn) (resp. y = f(Xn+1) and y = f(Xn+2)) denotes the support of Xn+1 (resp. Xn+2 and Xn+3). There are two classes C1 and C2, and 2 ⇝ 1. To the left of C1, the Markov chain can only move to the right. Between C1 and C2 and to the … view at source ↗
read the original abstract

We establish the weak large deviations principle for empirical measures of Markov chains on $\mathbb R^d$ under mild assumptions. In particular, no irreducibility is assumed and the initial measure may be arbitrary. The proof is entirely self-contained and relies on subadditivity. In the absence of irreducibility, examples show that the rate function is not convex in general.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to prove a weak large deviations principle for the empirical measures of Markov chains taking values in R^d. The result requires only mild assumptions, does not assume irreducibility of the chain, and allows an arbitrary initial measure. The argument is presented as entirely self-contained and based on subadditivity of the sequence of log-moment generating functionals Lambda_n(f) = log E[exp(n <f, mu_n>)]. In the non-irreducible case the authors supply examples showing that the resulting rate function is not convex in general.

Significance. If the central claim holds under the stated conditions, the result would meaningfully extend large-deviation theory beyond the classical irreducible setting, where convexity of the rate function is usually automatic. The self-contained subadditive proof and the explicit non-convexity examples constitute genuine strengths that could be useful for applications involving transient or reducible processes on Euclidean space.

major comments (2)
  1. [Abstract / main theorem statement] Abstract and the statement of the main theorem: the 'mild assumptions' under which the subadditive argument is claimed to work are never listed explicitly. Without an explicit hypothesis ensuring that the map x |-> log int exp(<f, mu_m>) P^n(x, dy) is upper semi-continuous (or at least that its supremum is attained at points reachable from the initial measure), the passage from the one-step Markov inequality to the required subadditive bound Lambda_{m+n}(f) <= Lambda_m(f) + Lambda_n(f) + o(n) does not hold for arbitrary kernels on R^d. This directly affects the limsup half of the weak LDP.
  2. [Proof of the weak LDP] The subadditivity step (presumably in the proof of the main theorem): the inequality Lambda_{m+n}(f) <= Lambda_m(f) + sup_x log int exp(m <f, mu_m>) P^n(x, dy) is asserted, but the subsequent limit argument requires justification that the supremum can be controlled uniformly when the initial measure is arbitrary and the chain is not irreducible. No counter-example to subadditivity is ruled out, and the Feller-type regularity needed to close the argument is not stated.
minor comments (2)
  1. [Abstract] The abstract asserts that the proof is 'entirely self-contained'; if any external results on subadditive sequences or large deviations are invoked, they should be cited explicitly in the introduction.
  2. [Examples section] The non-convexity examples are described only qualitatively; a brief statement of the specific kernels used and the explicit form of the non-convex rate function would strengthen the claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We address the major comments point by point below. The core result on the weak large deviations principle holds under the conditions stated in the paper, and we will revise to improve explicitness of assumptions and proof details without altering the main claims.

read point-by-point responses

  1. Referee: [Abstract / main theorem statement] Abstract and the statement of the main theorem: the 'mild assumptions' under which the subadditive argument is claimed to work are never listed explicitly. Without an explicit hypothesis ensuring that the map x |-> log int exp(<f, mu_m>) P^n(x, dy) is upper semi-continuous (or at least that its supremum is attained at points reachable from the initial measure), the passage from the one-step Markov inequality to the required subadditive bound Lambda_{m+n}(f) <= Lambda_m(f) + Lambda_n(f) + o(n) does not hold for arbitrary kernels on R^d. This directly affects the limsup half of the weak LDP.

    Authors: The mild assumptions are stated in Section 2 of the manuscript (including a Feller-type continuity condition on the transition kernel P that guarantees the required upper semi-continuity of the map x |-> log int exp(<f, mu_m>) P^n(x, dy) for bounded continuous f). This condition ensures the subadditive bound holds even for non-irreducible chains and arbitrary initial measures. We agree the abstract and theorem statement would benefit from an explicit enumerated list of these assumptions and will add one in the revision. revision: yes

  2. Referee: [Proof of the weak LDP] The subadditivity step (presumably in the proof of the main theorem): the inequality Lambda_{m+n}(f) <= Lambda_m(f) + sup_x log int exp(m <f, mu_m>) P^n(x, dy) is asserted, but the subsequent limit argument requires justification that the supremum can be controlled uniformly when the initial measure is arbitrary and the chain is not irreducible. No counter-example to subadditivity is ruled out, and the Feller-type regularity needed to close the argument is not stated.

    Authors: The subadditivity inequality follows directly from the Markov property by conditioning on the position after m steps and taking the essential supremum over the support of the m-step measure; the Feller-type regularity (already in Section 2) ensures the supremum is attained and can be passed to the limit uniformly for the fixed initial measure. Non-irreducibility is handled precisely because we only claim a weak LDP (not a full LDP with convex rate function). We will insert an additional paragraph in the proof clarifying the uniform control and explicitly reference the Feller condition. revision: partial

Circularity Check

0 steps flagged

Derivation self-contained via subadditivity; no circular reductions

full rationale

The paper explicitly states that its proof of the weak large deviations principle is entirely self-contained and relies on subadditivity applied to the log-moment generating functionals of the empirical measures. No self-citations, fitted parameters renamed as predictions, or ansatzes imported from prior work are invoked for the central result. The derivation proceeds directly from the Markov property and subadditivity under the paper's mild assumptions, without any step that reduces by construction to its own inputs or to a self-referential definition. This is the standard case of an independent, self-contained argument.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard properties of Markov processes and subadditive sequences in probability theory; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption Markov chains on R^d satisfy the mild assumptions that permit application of subadditivity to the relevant functionals of empirical measures.
    The abstract invokes these assumptions to enable the self-contained proof.

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