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arxiv: 2606.31455 · v1 · pith:VXZFR4LJnew · submitted 2026-06-30 · 🧮 math.PR

Discrete time-multidimensional renewal theory and applications

Leonidas Kordalis , Samis Trevezas This is my paper

Pith reviewed 2026-07-01 04:16 UTC · model grok-4.3

classification 🧮 math.PR
keywords renewal theorymulti-dimensional processesdiscrete timemulti-index convolutionformal power seriescentral limit theoremsnonparametric estimationwarranty models
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The pith

A multi-dimensional discrete renewal theory is constructed on the multi-index lattice using convolution and formal power series.

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

The paper builds a renewal framework where events advance simultaneously along several time coordinates, with the waiting mechanism given by an arbitrary distribution over multi-indices. Renewal equations are solved explicitly through the algebra of multivariate power series, which supplies coefficient formulas and inversion methods. Computation relies on FFT convolution paired with Newton iteration, while asymptotics deliver strong laws and central limit theorems when all time coordinates grow proportionally. Fixed-horizon data produce a multivariate censoring scheme that admits an exact nonparametric maximum-likelihood estimator with asymptotic normality.

Core claim

We develop a discrete-time renewal framework in which renewal events evolve along multiple time coordinates and the sojourn mechanism is described by a general distribution on the multi-index lattice. The resulting processes, called multi-time renewal chains, are studied through multi-index convolution and the associated algebra of multivariate formal power series. This algebraic formulation gives explicit representations for multi-time renewal equations, constructive coefficient formulas, and practical inversion schemes.

What carries the argument

multi-index convolution together with the algebra of multivariate formal power series, which converts renewal equations into explicit algebraic identities on the lattice

If this is right

  • Explicit algebraic solutions replace recursive computation for multi-time renewal equations.
  • FFT-based multidimensional convolution combined with Newton reciprocal iteration yields practical numerical evaluation on large grids.
  • Strong laws and central limit theorems hold for additive functionals and the renewal counting process when all coordinates grow proportionally.
  • The terminal age vector under fixed-horizon observation induces a multivariate right-censoring scheme that admits an exact nonparametric maximum-likelihood estimator with asymptotic normality.
  • The framework directly supplies closed-form identities for two-attribute warranty costs and alternating-renewal availability.

Where Pith is reading between the lines

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

  • The same power-series machinery could be applied to discretizations of continuous-time multivariate renewal models beyond the bivariate case treated in the applications.
  • The proportional-growth central limit theorem suggests analogous limit results for other lattice-indexed processes such as multi-type branching processes.
  • Fixed-horizon multivariate censoring may generalize to other incomplete-observation schemes in reliability or queueing when several clocks run in parallel.

Load-bearing premise

A general probability distribution on the multi-index lattice is assumed to exist and to support a well-defined convolution algebra without additional measurability or existence conditions being verified.

What would settle it

A lattice distribution for which the associated multi-index convolution fails to produce a consistent algebra of formal power series, or a proportional-growth regime in which the stated central limit theorem for the renewal counting process does not hold.

Figures

Figures reproduced from arXiv: 2606.31455 by Leonidas Kordalis, Samis Trevezas.

Figure 1
Figure 1. Figure 1: Logarithm of the diagonal ratio M(t, t)/t for the FGM bivariate exponential model: (λ1, λ2) = (1, 1) [PITH_FULL_IMAGE:figures/full_fig_p029_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Logarithm of the diagonal ratio M(t, t)/t for the FGM bivariate exponential model: (λ1, λ2) = (5, 4). 29 [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Logarithm of the diagonal ratio M(t, t)/t for the FGM bivariate exponential model: (λ1, λ2) = (6, 7). Continuous-time bivariate availability and its discrete approximation. A second quantity of interest is the availability of a continuous-time alternating renewal system on R 2 +. Let (Un)n≥1 and (Dn)n≥1 be independent i.i.d. sequences in R 2 + with cdfs FU , FD, and define Cn := Un + Dn and Sn := Pn j=1 Cj… view at source ↗
read the original abstract

We develop a discrete-time renewal framework in which renewal events evolve along multiple time coordinates and the sojourn mechanism is described by a general distribution on the multi-index lattice. The resulting processes, called multi-time renewal chains, are studied through multi-index convolution and the associated algebra of multivariate formal power series. This algebraic formulation gives explicit representations for multi-time renewal equations, constructive coefficient formulas, and practical inversion schemes. For computation, we combine FFT-based multidimensional convolution with Newton-type reciprocal iteration to evaluate renewal quantities on large grids. For asymptotics, we prove strong laws and central limit theorems under proportional growth of the observation horizon, including a general central limit theorem for additive functionals and a Gaussian limit for the renewal counting process in directions with a unique rate-determining coordinate. We also study fixed-horizon observations: the terminal age vector induces a genuinely multivariate right-censoring mechanism, leading to an exact nonparametric maximum likelihood estimator and its asymptotic normality. Applications include a binomial--multiset identity, two-attribute warranty evaluation, alternating-renewal availability computation, and discretization-based approximations of continuous-time bivariate renewal and availability models.

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

1 major / 1 minor

Summary. The manuscript develops a discrete-time multi-dimensional renewal theory for processes called multi-time renewal chains, defined using a general distribution on the multi-index lattice. It employs multi-index convolution and the algebra of multivariate formal power series to obtain explicit representations for renewal equations and coefficient formulas. Computational methods combine multidimensional FFT with Newton-type reciprocal iteration. Asymptotic results include strong laws and central limit theorems under proportional growth of the observation horizon, with a general CLT for additive functionals and a Gaussian limit for the renewal counting process. For fixed-horizon observations, a nonparametric MLE for the terminal age vector is constructed under multivariate right-censoring, with asymptotic normality established. Applications to a binomial-multiset identity, warranty evaluation, alternating-renewal availability, and approximations of continuous-time models are included.

Significance. If the multi-dimensional framework is properly constructed, the paper provides a valuable extension of classical renewal theory, offering algebraic tools for solving multi-dimensional renewal equations and practical computational schemes. The asymptotic theory under proportional growth and the exact nonparametric MLE for the terminal age vector represent significant contributions, particularly for applications in reliability and warranty analysis. The work is grounded in standard one-dimensional theory but extends it systematically.

major comments (1)
  1. [Abstract] The framework is built directly on an arbitrary probability distribution on the multi-index lattice N_0^d as the sojourn mechanism (abstract, first sentence). No construction of the underlying probability space, sigma-algebra, or verification that the multi-dimensional convolution remains a probability measure is supplied; this is load-bearing for the renewal equations, power series algebra, FFT inversion, strong laws, CLTs, and MLE, and the one-dimensional case does not automatically extend.
minor comments (1)
  1. The abstract refers to 'constructive coefficient formulas' without citing specific equations or examples from the main text, making it hard to evaluate their explicitness or novelty.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of major revision. The single major comment is addressed below; we agree that an explicit probabilistic construction strengthens the paper and will revise accordingly.

read point-by-point responses

  1. Referee: [Abstract] The framework is built directly on an arbitrary probability distribution on the multi-index lattice N_0^d as the sojourn mechanism (abstract, first sentence). No construction of the underlying probability space, sigma-algebra, or verification that the multi-dimensional convolution remains a probability measure is supplied; this is load-bearing for the renewal equations, power series algebra, FFT inversion, strong laws, CLTs, and MLE, and the one-dimensional case does not automatically extend.

    Authors: We agree that the manuscript would benefit from an explicit construction of the underlying probability space. In the revised version we will insert a short preliminary subsection (new Section 2.1) that proceeds as follows: let μ be an arbitrary probability measure on the countable set N_0^d; the underlying space is the infinite product Ω = (N_0^d)^ℕ equipped with the product σ-algebra and the product probability P = μ^ℕ. The coordinate maps X_i : Ω o N_0^d are i.i.d. with law μ. The partial-sum process S_n = X_1 + … + X_n is then a random walk on the abelian group Z^d, and the multi-index convolution μ^{*n} is simply the law of S_n, which is again a probability measure because convolution of probability measures on a countable group preserves total mass (by Fubini or direct summation over the countable support). All subsequent objects—renewal measures, generating functions, FFT inversions, strong laws, CLTs, and the nonparametric MLE—are defined pathwise on this space and inherit the required measurability and normalization from the product construction. This is the direct multi-dimensional analogue of the classical one-dimensional construction and requires no additional assumptions. The revision will be purely expository and will not alter any theorems or computational procedures. revision: yes

Circularity Check

0 steps flagged

No circularity; definitional extension of renewal theory with independent algebraic and asymptotic content

full rationale

The paper defines multi-time renewal chains from an arbitrary distribution on the multi-index lattice and develops multi-index convolutions, formal power series, FFT inversion, strong laws, CLTs, and nonparametric MLE directly from that setup. No step reduces a claimed result to a fitted parameter renamed as prediction, a self-citation chain, or an ansatz smuggled via prior work by the same authors. The derivation chain consists of constructive definitions and standard extensions of one-dimensional renewal theory; the initial distribution is taken as primitive input rather than derived, which is normal for a framework paper and does not constitute circularity under the enumerated patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on treating the multi-dimensional waiting times as a general probability distribution on the lattice and on standard results from multivariate formal power series; no free parameters are fitted in the abstract description.

axioms (2)
  • domain assumption The sojourn mechanism is described by a general distribution on the multi-index lattice.
    Invoked in the opening sentence as the foundation for defining the processes.
  • standard math Standard results on multi-index convolution and multivariate formal power series hold in this setting.
    Used to obtain explicit representations and inversion schemes.
invented entities (1)
  • multi-time renewal chains no independent evidence
    purpose: To model renewal events evolving along multiple time coordinates simultaneously.
    Newly introduced process class defined via the multi-index lattice distribution.

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