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REVIEW 2 major objections 2 minor 39 references

Mean First Passage Times and Fundamental Tensors of Higher Order Markov Chains

T0 review · 2 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read For higher-order Markov chains, altering states to absorbing produces a nonsingular equation for the fundamental tensor that links to mean first passage time tensors.

desk verdict Extends the fundamental matrix and mean first passage times to higher-order Markov chains with a tensor equation, nonsingularity result, and series representation, but the absorbing-state construction needs explicit verification. read the letter →

arxiv 2606.10058 v1 pith:YX3DUGXV submitted 2026-06-08 math.PR

Jianhong Xu This is my paper · ORCID
classification math.PR
keywords higher-orderMarkovchainsmeanfirstpassagetimesfundamentaltensorabsorbingergodicseriesrepresentation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper focuses on higher-order ergodic Markov chains in which one or more states are changed to absorbing. It proves that the resulting chain is absorbing and that the linear equation for its fundamental tensor must be nonsingular. A MATLAB routine called fund is supplied to solve this equation. Each horizontal slice of the mean first passage time tensor is shown to equal the fundamental tensor of a chain in which exactly one state has been made absorbing. This identification produces an infinite series expression for selected mean first passage times.

What carries the argument

The fundamental tensor of a higher-order absorbing Markov chain, which satisfies a nonsingular linear equation and equals each horizontal slice of the mean first passage time tensor.

What would settle it

An explicit higher-order absorbing chain whose fundamental-tensor equation is singular, or a numerical counterexample in which the claimed series fails to equal the mean first passage times.

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Extended reading notes

Core claim

When one or more states of a higher-order ergodic Markov chain are modified to be absorbing, the resulting chain is absorbing. The fundamental tensor of any such higher-order absorbing chain satisfies a nonsingular linear equation. Each horizontal slice of the mean first passage time tensor equals the fundamental tensor obtained by making exactly one state absorbing, which supplies a tensor series representation for the corresponding mean first passage times.

Load-bearing premise

The transition structure of the original higher-order ergodic chain stays well-defined after selected states are changed to absorbing.

Editorial extensions

If this is right

  • The fundamental tensor equation admits a unique solution.
  • The MATLAB function fund computes this tensor for any qualifying higher-order absorbing chain.
  • Selected mean first passage times admit an explicit infinite tensor series.
  • Each horizontal slice of the mean first passage time tensor corresponds to a single-state absorbing modification of the original ergodic chain.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same slice-to-fundamental-tensor link may allow recursive or iterative computation of mean first passage times without enumerating all paths.
  • The construction suggests a systematic way to obtain analogous tensor identities for continuous-time higher-order processes.
  • The nonsingularity result could be used to certify uniqueness in numerical schemes that solve tensor equations for higher-order chains.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript extends mean first passage times and fundamental matrices to higher-order Markov chains. It claims to prove that modifying one or more states of a higher-order ergodic chain to absorbing produces an absorbing chain; that the linear equation satisfied by the fundamental tensor of a higher-order absorbing chain is nonsingular (with a MATLAB solver fund provided); and that each horizontal slice of the mean first passage time tensor equals the fundamental tensor of a chain obtained by making one state absorbing, yielding a tensor series representation for selected mean first passage times.

Significance. If the state-modification construction is rigorously defined and preserves the order-k Markov property, the results generalize classical absorbing-chain theory and supply both analytic and computational tools for passage-time analysis in sequence models. The explicit tensor-series representation and the solver constitute concrete, usable contributions.

major comments (2)
  1. [Section defining the modified transition tensor (likely §2 or §3)] The load-bearing construction (altering one coordinate value to absorbing in the product-space state space of a k-th order chain) is not shown to be canonical. The manuscript must supply the explicit rule for updating every k-tuple containing the designated absorbing value; without it, the claims that the resulting process remains a well-defined higher-order absorbing chain, that absorption is proved, and that the fundamental-tensor equation is nonsingular cannot be verified.
  2. [Proof of nonsingularity of the fundamental tensor equation] The asserted proof that the fundamental-tensor equation is nonsingular (abstract and the section introducing the fund function) appears to rest on the absorbing property obtained from the above construction. If the construction is not uniquely specified, the nonsingularity argument must be re-examined for dependence on additional stipulations that are not part of the standard higher-order Markov axioms.
minor comments (2)
  1. [Abstract] The abstract states the result for 'one or more states' while the body description emphasizes a single state; the statements should be aligned.
  2. [Computational section] The MATLAB function is written as {\tt fund}; consistent use of \texttt{fund} throughout the text and code listing is needed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and the identification of points requiring greater explicitness. We address each major comment below and will incorporate the necessary clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Section defining the modified transition tensor (likely §2 or §3)] The load-bearing construction (altering one coordinate value to absorbing in the product-space state space of a k-th order chain) is not shown to be canonical. The manuscript must supply the explicit rule for updating every k-tuple containing the designated absorbing value; without it, the claims that the resulting process remains a well-defined higher-order absorbing chain, that absorption is proved, and that the fundamental-tensor equation is nonsingular cannot be verified.

    Authors: We agree that the original manuscript did not supply a fully explicit coordinate-wise rule for every k-tuple. In the revision we will add a precise definition of the modified transition tensor that states, for each k-tuple containing the designated absorbing state s, the new probability mass is reassigned according to the original marginal on the remaining coordinates while setting the absorbing coordinate to probability 1 at s. This rule is the natural extension of the classical absorbing-state construction to the product-space representation of a k-th order chain and preserves the order-k Markov property by construction. With this definition the absorption proof and the nonsingularity argument become directly verifiable from the standard axioms. revision: yes

  2. Referee: [Proof of nonsingularity of the fundamental tensor equation] The asserted proof that the fundamental-tensor equation is nonsingular (abstract and the section introducing the fund function) appears to rest on the absorbing property obtained from the above construction. If the construction is not uniquely specified, the nonsingularity argument must be re-examined for dependence on additional stipulations that are not part of the standard higher-order Markov axioms.

    Authors: The nonsingularity claim is indeed derived from the absorbing property that follows once the modification rule is fixed. By inserting the explicit update rule described above, the revised manuscript will make the logical dependence transparent: the resulting transition tensor satisfies the higher-order absorbing-chain axioms, from which the nonsingularity of the fundamental-tensor linear system follows by the same block-triangular argument used in the classical case. We will also add a short lemma confirming that the construction introduces no extraneous stipulations beyond the standard axioms plus the absorbing-state designation. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; results derived from standard higher-order Markov chain axioms

full rationale

The paper's core claims—proving the modified chain is absorbing, nonsingularity of the fundamental tensor equation, and equivalence of MFPT tensor slices to modified fundamental tensors—are established directly from the transition tensor of the original ergodic chain and the definition of state absorption. No steps reduce by construction to fitted inputs, self-definitions, or self-citation chains; the tensor series representation follows from solving the linear system under the stated assumptions. The derivation remains self-contained against external Markov chain theory.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Abstract-only review; no explicit free parameters, invented entities, or ad-hoc axioms listed. Relies on background Markov chain theory.

assumptions (1)
  • domain assumption Higher-order Markov chains are ergodic when all states communicate appropriately.
    Invoked when stating the starting chain before absorption modification.

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Cite this review

Pith. "Pith review of Mean First Passage Times and Fundamental Tensors of Higher Order Markov Chains." pith.science (2026). https://pith.science/paper/YX3DUGXV

@misc{pith2026260610058,
  author       = {Pith},
  title        = {Pith review of: Mean First Passage Times and Fundamental Tensors of Higher Order Markov Chains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YX3DUGXV}},
  note         = {Machine review of arXiv:2606.10058}
}
read the original abstract

The mean first passage times are among the most critical characteristics of a Markov chain. In this paper, we focus on the scenario in which one or more states of a higher order ergodic Markov chain are modified to be absorbing. We prove that the resulting chain has to be absorbing. For a higher order absorbing Markov chain, we prove that the equation its fundamental tensor satisfies must be nonsingular and provide a MATLAB function {\tt fund} for solving the equation. Besides, we connect each horizontal slice of the mean first passage time tensor with a fundamental tensor obtained when one state of a higher order ergodic Markov chain is modified to be absorbing, which also leads to a tensor series representation for selected mean first passage times.

Discussion (0). Continue with ORCID to comment.

Reference graph

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