Algorithms of Phase Space Reduction and Asymptotics of Hitting Times for Perturbed Semi-Markov Processes

Dmitrii Silvestrov

arxiv: 1907.03529 · v1 · pith:WZOTQDYXnew · submitted 2019-07-08 · 🧮 math.PR

Algorithms of Phase Space Reduction and Asymptotics of Hitting Times for Perturbed Semi-Markov Processes

Dmitrii Silvestrov This is my paper

Pith reviewed 2026-05-25 01:13 UTC · model grok-4.3

classification 🧮 math.PR

keywords semi-Markov processesphase space reductionhitting timesasymptoticsregular perturbationssingular perturbationsweak convergencerecurrent algorithms

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The pith

Recurrent algorithms of phase space reduction for regularly and singularly perturbed semi-Markov processes yield conditions for weak convergence of hitting time distributions together with explicit formulas for normalisation functions and a

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

This paper introduces asymptotic recurrent algorithms for reducing the phase space of regularly and singularly perturbed semi-Markov processes. The algorithms establish conditions under which the distributions of hitting times converge weakly and their expectations converge. They also supply recurrent formulas for normalisation functions, Laplace transforms of the limiting distributions, and the limiting expectations themselves. These results matter for models in which hitting times represent first-passage or absorption behaviour, because the methods avoid solving the full system of equations at each step. If the algorithms apply, they turn the asymptotic analysis into a sequence of lower-dimensional calculations that can be performed recursively.

Core claim

The paper claims that new asymptotic recurrent algorithms of phase space reduction for regularly and singularly perturbed semi-Markov processes give effective conditions of weak convergence for distributions and convergence of expectations for hitting times as well as recurrent formulas for computing the corresponding normalisation functions, Laplace transforms for limiting distributions and limits for expectations.

What carries the argument

Asymptotic recurrent algorithms of phase space reduction that operate on the embedded Markov chain and the holding-time distributions of the semi-Markov process.

If this is right

Effective conditions for weak convergence of the distributions of hitting times are obtained directly from the reduction procedure.
Convergence of the expectations of hitting times follows from the same conditions.
Normalisation functions are computed via recurrent relations that avoid inverting large matrices.
Laplace transforms of the limiting distributions are given by explicit recursive expressions.
Limits of the expectations are obtained as the output of the final step of the reduction.

Where Pith is reading between the lines

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

The recursive structure suggests that the algorithms can be coded once and applied to families of models that differ only in the perturbation parameter.
Similar reduction steps might be tested on piecewise-deterministic Markov processes or on semi-Markov processes with infinite state spaces.
The explicit Laplace-transform formulas open the possibility of inverting them numerically to obtain density approximations for moderate time horizons.
The method could be combined with existing numerical schemes for solving the unperturbed embedded chain when the state space remains large after reduction.

Load-bearing premise

The semi-Markov processes under study must admit regular or singular perturbations for which the required asymptotic conditions on the hitting times hold.

What would settle it

For a concrete regularly perturbed finite-state semi-Markov process, direct Monte Carlo simulation of hitting times that produces a limiting distribution different from the one predicted by the reduction algorithm would falsify the claim.

read the original abstract

The paper presents new asymptotic recurrent algorithms of phase space reduction for regularly and singularly perturbed semi-Markov processes. These algorithms give effective conditions of weak convergence for distributions and convergence of expectations for hitting times as well as recurrent formulas for computing the corresponding normalisation functions, Laplace transforms for limiting distributions and limits for expectations.

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.

Desk Editor's Note private letter to a colleague

Silvestrov supplies recurrent algorithms for phase-space reduction in perturbed semi-Markov processes plus explicit formulas for hitting-time asymptotics.

read the letter

The main thing here is a set of recurrent algorithms for reducing the phase space of regularly and singularly perturbed semi-Markov processes, together with the attendant conditions and formulas for hitting-time asymptotics. These algorithms deliver effective conditions for weak convergence of the distributions and convergence of the expectations, plus recurrent expressions for the normalization functions, Laplace transforms of the limits, and the limiting expectations themselves. The work frames the results as following directly from the perturbation assumptions rather than presupposing the convergence, which keeps the logic straightforward. For someone already working inside this corner of applied probability, the constructive, step-by-step character of the algorithms is the useful part; it turns what might otherwise be ad-hoc analysis into a repeatable procedure. The paper stays within its stated domain and does not claim broader impact. The clearest limitation is that the abstract gives no derivations, worked examples, or checks on how the conditions behave in practice. If the full text supplies those details and verifies the formulas on at least one non-trivial case, the contribution is solid; if it remains at the level of statements without supporting calculations, the results stay harder to assess. This is a technical paper aimed at researchers who already handle semi-Markov models under perturbation, for instance in queueing or reliability settings. A reader looking for new computational tools in asymptotic analysis of hitting times would find the recurrent formulas relevant. It deserves peer review because the central claims are consistent with the problem setup and the algorithmic approach is new on its own terms.

Referee Report

1 major / 0 minor

Summary. The manuscript presents new asymptotic recurrent algorithms of phase space reduction for regularly and singularly perturbed semi-Markov processes. These algorithms are claimed to deliver effective conditions of weak convergence for distributions and convergence of expectations for hitting times, together with recurrent formulas for the corresponding normalisation functions, Laplace transforms for limiting distributions, and limits for expectations.

Significance. If the algorithms and attendant convergence statements are rigorously derived, the work would supply a systematic, recurrent method for reducing phase space in perturbed semi-Markov models while furnishing explicit computational formulae; such tools could be valuable in applications requiring asymptotic analysis of hitting times. The emphasis on recurrent formulae is a potential strength for implementability, but the absence of any derivations, explicit conditions, or illustrative examples in the abstract leaves the soundness of the central claims unverified from the available information.

major comments (1)

Abstract: the abstract asserts the existence of the algorithms and the convergence results but supplies neither derivations nor the precise perturbation assumptions under which the weak-convergence statements hold; without these, the load-bearing claims cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report. We address the single major comment below.

read point-by-point responses

Referee: Abstract: the abstract asserts the existence of the algorithms and the convergence results but supplies neither derivations nor the precise perturbation assumptions under which the weak-convergence statements hold; without these, the load-bearing claims cannot be assessed.

Authors: Abstracts are concise summaries by design and do not contain derivations or full technical assumptions. The manuscript itself develops the recurrent algorithms for phase-space reduction of regularly and singularly perturbed semi-Markov processes, states the precise perturbation conditions, proves the weak-convergence results for hitting-time distributions and expectations, and supplies the recurrent formulae for normalising functions, Laplace transforms of the limits, and the limiting expectations. revision: no

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper introduces recurrent asymptotic algorithms for phase-space reduction of regularly and singularly perturbed semi-Markov processes. These algorithms are stated to produce effective conditions for weak convergence of hitting-time distributions, convergence of expectations, and explicit recurrent formulas for normalization functions, Laplace transforms, and limiting expectations. The provided abstract and description contain no quoted equations or steps that reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations. The domain of applicability is explicitly the class of processes admitting the stated perturbations, making the derivation self-contained against external assumptions rather than circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the processes are semi-Markov and admit regular or singular perturbations allowing the stated convergences; no free parameters or invented entities are indicated in the abstract.

axioms (1)

domain assumption The processes are semi-Markov and subject to regular or singular perturbations for which weak convergence and expectation convergence hold under the reduction algorithms.
Directly invoked in the abstract as the setting in which the new algorithms operate.

pith-pipeline@v0.9.0 · 5564 in / 1237 out tokens · 27580 ms · 2026-05-25T01:13:33.325950+00:00 · methodology

discussion (0)

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Works this paper leans on

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