Autocorrelation Function Characterization of Continuous Time Markov Chains

Douglas G. Down; G. Rama Murthy

REVIEW 5 major objections 3 minor 16 references

Autocorrelation Function Characterization of Continuous Time Markov Chains

T0 review · 5 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For a finite-state Markov chain, any nonzero constant term in the autocorrelation function forces every finite Lp norm to be infinite.

desk verdict The general conclusion is true but nearly tautological, and the two-state calculation that is supposed to motivate it is wrong, so the paper is not publishable as it stands. read the letter →

arxiv 1908.09284 v1 pith:VPO2PO5D submitted 2019-08-25 math.PR

G. Rama Murthy , Douglas G. Down This is my paper · ORCID

classification math.PR MSC 60J2760G10

keywords autocorrelationfunctioncontinuous-timeMarkovchainLpnormintegrabilityunitstochasticprocessequilibriumdistributionpointsuperpositionspectraldecomposition

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

This paper establishes a structural fact about continuous-time Markov chains with finitely many states: their autocorrelation function splits into a decaying exponential part and a constant offset, and whenever that offset is nonzero the whole function is not in $L_p$ for any $p \geq 1$. The two-state $\{-1,+1\}$ case is worked out first, showing that the offset is the product of the mean initial state and the mean equilibrium state, and that it vanishes only for special symmetric choices such as equal transition rates with a $q=1/2$ start. The same decomposition is then derived for arbitrary finite state spaces from the spectral decomposition of the generator matrix. If the claim is right, standard tools that assume integrable autocorrelations, such as power spectral densities and some simulation variance estimates, need to be applied to such chains with extra care or replaced by generalized formulations.

What carries the argument

The load-bearing object is the spectral expansion of the transition semigroup, $e^{Q\tau}=\sum_{k=1}^{N} e^{\gamma_k \tau} E_k$ with $E_k=\bar{f}_k \bar{g}_k$, where the zero eigenvalue contributes a time-independent residue matrix $\bar{f}_N \bar{g}_N$. Feeding this expansion into $R(\tau)=\sum_{i,j} i\,j\,q_i\,(e^{Q\tau})_{ij}$ isolates the constant term $c=(\sum_i i q_i)(\sum_j j\pi_j)=E[X(0)]E[Z]$. Because the nonzero eigenvalues of a generator have negative real parts, their contribution $f(\tau)$ is integrable; the zero-eigenvalue contribution is what decides integrability of the whole autocorrelation.

What would settle it

For $\alpha=1$, $\beta=3$, $q=1/4$, compute the exact two-state autocorrelation without dropping any terms: the correct expression is $R(\tau)=1.25e^{-4\tau}-0.25$. Since the constant $-0.25$ makes $\int |R(\tau)|^p d\tau$ infinite for every $p\geq 1$, this confirms the claim; if a full calculation for some finite-state chain with nonzero $c$ instead produced an integrable autocorrelation, the claim would be falsified.

Watch

Extended reading notes

Core claim

The central claim is that for a homogeneous finite-state continuous-time Markov chain, $R(\tau)=E[X(0)X(\tau)]$ has the form $f(\tau)+c$, where $f(\tau)$ is a sum of decaying exponentials coming from the nonzero eigenvalues of the generator and $c=E[X(0)]E[Z]$ is the product of the initial mean and the equilibrium mean. When $c\neq 0$, the constant term alone makes $R(\tau)$ fail to belong to $L_p(\mathbb{R})$ for every $p\geq 1$, so the autocorrelation is not integrable in any Lebesgue sense; the $L_p$ norm instead approaches a finite constant as $p\to\infty$. The paper derives this explicitly for a unit two-state chain with rates $\alpha,\beta$: with a general initial distribution $q$, $c=2[(\beta-\alpha)/(\alpha+\beta)]q+(\alpha-\beta)/(\alpha+\beta)$, and only the choice $q=1/2$ removes the constant, leaving a purely exponential autocorrelation. It also observes the asymptotic identity $\lim_{\tau\to\infty}R(\tau)=E[X(0)]E[Z]$, which it reads as asymptotic independence between the initial and equilibrium variables.

Load-bearing premise

The central claim rests on assuming the autocorrelation depends only on the time lag even when the chain is started away from equilibrium, and on treating the zero-eigenvalue projection as the only long-time part; if wide-sense stationarity or that projection step fails, the constant-offset formula for those cases needs re-examination.

Editorial extensions

If this is right

Any finite-state chain started in equilibrium with a nonzero mean state yields an autocorrelation with infinite $L_p$ norm for every finite $p$, so its power spectral density is not defined in the usual Fourier sense.
For a two-state unit chain, the only integrable-autocorrelation cases are those with $q=1/2$; in the symmetric-rate case $\alpha=\beta$ this gives $R(\tau)=e^{-2\alpha|\tau|}$, the classical telegraph-signal form.
The asymptotic limit $\lim_{\tau\to\infty}R(\tau)=E[X(0)]E[Z]$ gives a practical way to detect the constant offset: a nonzero long-lag autocorrelation estimate indicates non-integrability.
When a CTMC is viewed as a superposition of point processes, the equilibrium probability of the $j$-th constituent process is $\pi_j$ and its transient probability at time $\tau$ is $[\pi(0)e^{Q\tau}]_j$, so the chain's transient and equilibrium computations transfer directly to that setting.

Reading between the lines

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

The non-integrability conclusion survives even if the paper's simplification that drops the exponential term when the chain starts in equilibrium is corrected: the exact equilibrium-start autocorrelation contains the same constant $c$ plus a decaying exponential, and the constant alone already forces infinite $L_p$ norm.
A natural sharpening, suggested by the paper's conjecture on symmetric state spaces, is that integrable autocorrelations occur exactly when the equilibrium mean is zero; a testable extension would search for a finite-state chain with zero equilibrium mean but nonzero initial mean whose exact $R(\tau)$ is not integrable, which would disprove the sufficiency part of that conjecture.
If the constant offset is present, standard spectral density estimators should show a growing spike near zero frequency as the observation window lengthens; detecting that empirically would confirm non-integrability without computing $L_p$ norms.
For countably infinite CTMCs the paper states the same decomposition should generalize; a testable extension is to check that a nonzero limit $E[X(0)]E[Z]$ with finite second moments still yields non-integrability when the remaining spectrum has infinitely many negative eigenvalues.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Here is my honest read of 1908.09284. The paper's main message is that for a finite-state CTMC, when the autocorrelation R(τ) has a nonzero limit as τ→∞, R cannot be in Lp for any p≥1. That statement is true. The explicit computation c=E[X(0)]E[Z] in Section 3 is a clean way to identify the limit, and the spectral decomposition used there is standard. I also think the two-state calculation for a symmetric initial distribution (q=1/2) gives the familiar random telegraph autocorrelation correctly.

The soft spot is not in the conclusion but in the key demonstration. In Section 2, when the initial distribution is the equilibrium distribution, the paper claims P{X(τ)=X(0)} = (α²+β²)/(α+β)² and hence R(τ) is a constant. That drops the exponential term in the transition matrix. The correct joint probability has an extra 2αβ e^{-(α+β)τ}/(α+β)², so R(τ) = (α-β)²/(α+β)² + 4αβ e^{-(α+β)τ}/(α+β)². Lemma 1's proof relies on R being identically a nonzero constant; as written, the proof is invalid. The lemma's conclusion survives because the limit is still nonzero when α≠β, but the derivation is wrong. The paper also contradicts itself: for α=β the constant formula gives R=0, while later the same paper uses R=e^{-2α|τ|}.

Section 4's point-process "inferences" are paraphrases of Markov chain transition probabilities and carry no new content. There is also a notational slip in Section 3 where left and right eigenvectors are swapped, though the final c formula is correct.

Who is this for? A reader looking for a one-line reason why some Markov chain autocorrelations are not integrable would get that from Section 3. But the paper is not a rigorous mathematical contribution in its current form. The authors could fix the two-state computation and reframe the paper as a short note on nonzero asymptotic autocorrelation. As is, I would not encourage the editor to spend referee time on it. Desk reject seems right; the flaw sits in the motivating example and the remaining content is too thin to justify a full review.

Referee Report

5 major / 3 minor

Summary. The paper studies autocorrelation functions of finite-state continuous-time Markov chains (CTMCs), focusing on unit ({+1,−1}) and symmetric state spaces. For the two-state chain it computes R(τ) and claims that, under a stationary initial distribution, the autocorrelation is the nonzero constant (α−β)²/(α+β)². It then uses this claim to prove that the autocorrelation is not in L_p for any p ≥ 1. A general finite-state argument is given via the spectral decomposition of the generator, in which the zero-eigenvalue term contributes a constant c = E[X(0)]E[Z], and the paper concludes non-integrability whenever c ≠ 0. The final section draws inferences about point processes associated with CTMCs and DTMCs.

Significance. The core structural observation—that a finite-state Markov chain autocorrelation with a nonzero limit at infinity cannot be integrable—is valid and worth stating. The identification c = E[X(0)]E[Z] in the zero-eigenvalue term is a useful and essentially correct decomposition in the diagonalizable case. However, the two-state calculation that motivates the paper is incorrect, the L_p-limit statements are mathematically confused, and the point-process claims contain a false assertion. The central theorem is salvageable, but the present manuscript would require substantial correction before it could be considered for publication.

major comments (5)

[§2, Note after Eq. (3) and Eq. (4)] The claim that P{X(τ)=X(0)} = (α²+β²)/(α+β)² under a stationary initial distribution does not follow from equality of the transient and equilibrium marginal distributions, because the joint event requires the transition probabilities. Using the paper's own e^{Qτ}, P{X(τ)=X(0)} = (α²+β²)/(α+β)² + 2αβ/(α+β)² e^{-(α+β)τ}, and hence R(τ) = (α−β)²/(α+β)² + 4αβ/(α+β)² e^{-(α+β)τ}, not the constant in Eq. (4). This also creates an internal contradiction: for α=β, Eq. (4) gives R(τ)=0, whereas the arbitrary-initial-distribution calculation immediately below gives R(τ)=e^{-(α+β)τ}, and the correct stationary value is e^{-2ατ}.
[§2, first paragraph and arbitrary-initial-distribution paragraph] The paper begins by assuming the process is wide-sense stationary, but then computes R(τ)=E[X(0)X(τ)] for an arbitrary initial distribution q. For q different from the stationary distribution, E[X(t)] varies with t, so the process is not wide-sense stationary and E[X(0)X(τ)] is not a stationary lag autocorrelation. Moreover, R(τ) need not be symmetric in τ, although the paper later symmetrizes to expressions such as e^{-2α|τ|}. The non-integrability analysis for arbitrary q is therefore applied to a different object—a covariance computed from a fixed nonstationary initial law—and this distinction must be addressed explicitly.
[§3.1, first paragraph and Lemma 2] The statement 'We now prove that for any finite state space CTMC, the autocorrelation function is not integrable' is false, as the paper itself shows in Section 2: for a unit CTMC with q=1/2 and α=β, R(τ)=e^{-2α|τ|}, which is in L_p for every p ≥ 1. The claimed 'without loss of generality' restriction to a positive state space changes the value of c; for state spaces containing negative values, c=E[X(0)]E[Z] can vanish (e.g., symmetric uniform equilibrium), so Lemma 2's conditional formulation with c ≠ 0 cannot be promoted to an unconditional statement about arbitrary finite-state CTMCs.
[§3.1, paragraph after Eq. (5); Lemmas 1 and 2] The L_p-limit statements are internally inconsistent. The paper states that for c ≠ 0, ∫(R(τ))^p dτ is infinite for every p ≥ 1, and then adds 'Further if |c| < 1, then ∫(R(τ))^p dτ approaches zero as p → ∞.' An integral that is infinite for every finite p cannot approach zero, and the L_p norm with the 1/p power is likewise infinite for every finite p. The repeated claim that the L_p-norm approaches a finite constant as p → ∞ is therefore unsupported and should be removed or redefined.
[§4, first paragraph] The assertion that 'when successive visits to a state of a CTMC are stitched together, a Poisson process naturally results' is not generally true. The point process of visits to a fixed state of an irreducible CTMC is a renewal process whose interarrival times include both the sojourn time in that state and the random time needed to return, and this return time is not exponentially distributed in general. The numbered inferences in Section 4 are merely transient and equilibrium probabilities of the CTMC; they do not establish the claimed point-process characterization.

minor comments (3)

[Introduction, first paragraph] The phrase 'has not ben investigated' contains a typo; it should read 'has not been investigated.'
[§3.1, spectral decomposition] The notation for left and right eigenvectors is inconsistent: Eq. (5) calls f_k a right eigenvector and g_k a left eigenvector, while in Section 2 the roles of f and g are the opposite. This should be reconciled throughout.
[Lemma 1 statement and proof] Lemma 1 states α ≠ β, but the proof also treats the case q=1/2 and α=β; the statement should be aligned with the cases actually considered, and it should explicitly exclude or handle the integrable case q=1/2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Lp conclusion follows from a direct spectral decomposition and a nonzero constant term, not from fitting or self-referential inputs.

full rationale

The paper's derivation chain is a direct computation: it forms the generator matrix, computes e^{Qt} by spectral decomposition, decomposes the autocorrelation as R(tau) = f(tau) + c, and observes that a nonzero constant c forces R to have infinite Lp norm for every finite p. No parameter is fitted to a subset of data and then renamed a prediction; no load-bearing result is imported from the authors' prior work; no uniqueness theorem is invoked; and the conclusion is not assumed in the definition of c. The constant c is explicitly computed from the zero-eigenvalue eigenprojectors as E[X(0)]E[Z], and the remaining f(tau) is a sum of decaying exponentials, so the non-integrability statement is a mathematical consequence rather than a circular reduction. The apparent defect in the Note after Eq. (3), where the transient joint probability is treated as if the exponential terms vanish, is a mathematical correctness concern about the proof as written, not evidence that the paper's central claim reduces to its inputs by construction. The manuscript is therefore self-contained for circularity purposes.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The spectral representation of e^{Qt} and Perron-Frobenius are standard. The questionable domain assumption is the mixing of stationarity with arbitrary initial distributions. Alpha and beta are model inputs, not fitted parameters.

assumptions (3)

standard math Spectral representation of the matrix exponential for a diagonalizable generator Q, plus Perron-Frobenius for the zero eigenvalue.
Used in Sections 2 and 3 to compute e^{Qt} and to isolate the constant term c = E[X(0)]E[Z].
domain assumption The process is wide-sense stationary, so the autocorrelation depends only on the lag τ.
Invoked at the start of Section 2, but then the paper computes with arbitrary initial distributions q, which make the process non-stationary and invalidate the lag-only formulation.
ad hoc to paper The state space is positive, {1,...,N}, so E[X(0)] and E[Z] are both positive when the relevant probabilities are positive.
Used in Section 3 to assert that c is nonzero; for symmetric state spaces the paper only conjectures a condition for c=0.

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Cite this review

Pith. "Pith review of Autocorrelation Function Characterization of Continuous Time Markov Chains." pith.science (2026). https://pith.science/paper/VPO2PO5D

@misc{pith2026190809284,
  author       = {Pith},
  title        = {Pith review of: Autocorrelation Function Characterization of Continuous Time Markov Chains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VPO2PO5D}},
  note         = {Machine review of arXiv:1908.09284}
}

abstract

We study certain properties of the function space of autocorrelation functions of Unit Continuous Time Markov Chains (CTMCs). It is shown that under particular conditions, the $L^p$ norm of the autocorrelation function of arbitrary finite state space CTMCs is infinite. Several interesting inferences are made for point processes associated with CTMCs/ Discrete Time Markov Chains (DTMCs).

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

McMillan, History of a problem, SIAM Journal of Applie d Mathematics 3 (1955) 114–128

B. McMillan, History of a problem, SIAM Journal of Applie d Mathematics 3 (1955) 114–128

work page 1955
[2]

Shepp, Covariance of unit processes, in: Working Conf erence on Stochastic Processes, Santa Barbara, CA, 1967, pp

L. Shepp, Covariance of unit processes, in: Working Conf erence on Stochastic Processes, Santa Barbara, CA, 1967, pp . 205–218

work page 1967
[3]

Masry, On covariance functions of unit processes, SIA M Journal of Applied Mathematics 23 (1972) 28–33

E. Masry, On covariance functions of unit processes, SIA M Journal of Applied Mathematics 23 (1972) 28–33

work page 1972
[4]

D¸ ebowski, On processes with hyperbolically decaying autocorrelations, Journal of Time Series Analysis 32 (20 11) 580–584

/suppress L. D¸ ebowski, On processes with hyperbolically decaying autocorrelations, Journal of Time Series Analysis 32 (20 11) 580–584

work page
[5]

D´ egerine, S

S. D´ egerine, S. Lambert-Lacroix, Characterization of the partial autocorrelation function of nonstationary tim e series, Journal of Multivariate Analysis 87 (2003) 46–59

work page 2003
[6]

Inoue, AR and MA representation of partial autocorrel ation functions, with applications, Probability Theory an d Related Fields 140 (2008) 523–551

A. Inoue, AR and MA representation of partial autocorrel ation functions, with applications, Probability Theory an d Related Fields 140 (2008) 523–551

work page 2008
[7]

Y. Eun, S. Jin, Y. Hong, H. Song, Frequency hopping sequen ces with optimal partial autocorrelation properties, IEEE Transactions on Information Theory 50 (2004) 2438–2442

work page 2004
[8]

H¨ aggstr¨ om, M

O. H¨ aggstr¨ om, M. Van Lieshout, J. Møller, Characterization results and Markov chain Monte Carlo algorithms inclu ding exact simulation for some spatial point processes, Bernoul li 5 (1999) 641–658

work page 1999

Show all 16 references

[9]

Huang, On the characterization of point processes wit h the exchangeable and Markov properties, Sankyh¯ a: The Indian Journal of Statistics, Series A (1990) 16–27

W. Huang, On the characterization of point processes wit h the exchangeable and Markov properties, Sankyh¯ a: The Indian Journal of Statistics, Series A (1990) 16–27

1990
[10]

Ivanoﬀ, E

B. Ivanoﬀ, E. Merzbach, M. Plante, A compensator charac terization of point processes on topological lattices, Ele ctronic Journal of Probability 12 (2007) 47–74

2007
[11]

Whitt, The eﬃciency of one long run versus independen t replications in steady-state simulation, Management Sci ence 37 (1991) 645–666

W. Whitt, The eﬃciency of one long run versus independen t replications in steady-state simulation, Management Sci ence 37 (1991) 645–666

1991
[12]

Klemm, C

A. Klemm, C. Lindemann, M. Lohmann, Modeling IP traﬃc us ing the batch Markovian arrival process, Performance Evaluation 54 (2003) 149–173

2003
[13]

C ¸ inlar, Introduction to Stochastic Processes, Prentice-Hall, Inc., Englewood Cliﬀs, New Jersey, 1975

E. C ¸ inlar, Introduction to Stochastic Processes, Prentice-Hall, Inc., Englewood Cliﬀs, New Jersey, 1975

1975
[14]

Sriram, W

B. Sriram, W. Whitt, Characterizing superposition arr ival processes in packet multiplexers for voice and data, IE EE Journal on Selected Areas in Communications 4 (1986) 833–84 6

1986
[15]

Heﬀes, D

H. Heﬀes, D. Lucantoni, A Markov modulated characteriz ation of packetized voice and data traﬃc and related statist ical multiplexer performance, IEEE Journal on Selected Areas in Communications 4 (1986) 856–868

1986
[16]

Neuts, A versatile Markovian point process, Journal of Applied Probability 16 (1979) 764–779

M. Neuts, A versatile Markovian point process, Journal of Applied Probability 16 (1979) 764–779. 8

1979

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

McMillan, History of a problem, SIAM Journal of Applie d Mathematics 3 (1955) 114–128

B. McMillan, History of a problem, SIAM Journal of Applie d Mathematics 3 (1955) 114–128

work page 1955

[2] [2]

Shepp, Covariance of unit processes, in: Working Conf erence on Stochastic Processes, Santa Barbara, CA, 1967, pp

L. Shepp, Covariance of unit processes, in: Working Conf erence on Stochastic Processes, Santa Barbara, CA, 1967, pp . 205–218

work page 1967

[3] [3]

Masry, On covariance functions of unit processes, SIA M Journal of Applied Mathematics 23 (1972) 28–33

E. Masry, On covariance functions of unit processes, SIA M Journal of Applied Mathematics 23 (1972) 28–33

work page 1972

[4] [4]

D¸ ebowski, On processes with hyperbolically decaying autocorrelations, Journal of Time Series Analysis 32 (20 11) 580–584

/suppress L. D¸ ebowski, On processes with hyperbolically decaying autocorrelations, Journal of Time Series Analysis 32 (20 11) 580–584

work page

[5] [5]

D´ egerine, S

S. D´ egerine, S. Lambert-Lacroix, Characterization of the partial autocorrelation function of nonstationary tim e series, Journal of Multivariate Analysis 87 (2003) 46–59

work page 2003

[6] [6]

Inoue, AR and MA representation of partial autocorrel ation functions, with applications, Probability Theory an d Related Fields 140 (2008) 523–551

A. Inoue, AR and MA representation of partial autocorrel ation functions, with applications, Probability Theory an d Related Fields 140 (2008) 523–551

work page 2008

[7] [7]

Y. Eun, S. Jin, Y. Hong, H. Song, Frequency hopping sequen ces with optimal partial autocorrelation properties, IEEE Transactions on Information Theory 50 (2004) 2438–2442

work page 2004

[8] [8]

H¨ aggstr¨ om, M

O. H¨ aggstr¨ om, M. Van Lieshout, J. Møller, Characterization results and Markov chain Monte Carlo algorithms inclu ding exact simulation for some spatial point processes, Bernoul li 5 (1999) 641–658

work page 1999