Asymptotic properties of permanental sequences

Jay Rosen; Michael B. Marcus

REVIEW 2 major objections 3 minor 15 references

Asymptotic properties of permanental sequences

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Adding an excessive function to a Markov potential transfers the Gaussian law of the iterated logarithm to permanental sequences.

desk verdict A real transfer principle with a load-bearing lim/limsup gap in the proof; repairable, and worth sending to a referee. read the letter →

arxiv 1908.04155 v1 pith:ASUP6U66 submitted 2019-08-12 math.PR

Michael B. Marcus , Jay Rosen This is my paper · ORCID

classification math.PR MSC 60J2760F2060G17

keywords permanentalsequencesalpha-permanentalprocesseslawoftheiteratedlogarithmGaussianMarkovchainpotentialsexcessivefunctionsinverseM-matricesbirthanddeath

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 general transfer principle for permanental sequences, the nonnegative stochastic processes whose Laplace transform is $\det(I+KS)^{-\alpha}$. If the kernel is written as $\widetilde U = U + f$, where $U$ is the symmetric potential of a transient symmetric Markov process (in the Borel right-process sense) and $f$ is an excessive function, then under three explicit conditions the almost-sure limsup of $\widetilde X_{\alpha,j}/\phi_j$ equals the limsup of the Gaussian sequence with covariance $U$, namely $\limsup \eta_j/(2\phi_j)^{1/2}=1$. The conditions are that the augmented matrix is an inverse M-matrix, that the inverse-potential row sums applied to $f$ vanish uniformly as the window moves to infinity, and that $f_j=o(\phi_j)$. The payoff is a collection of sharp, explicit log-log and log asymptotics for permanental sequences based on birth-death processes, birth-death processes with emigration, first- and higher-order Gaussian autoregressions, and L\'evy processes on the integers.

What carries the argument

The load-bearing object is the symmetrize-then-invert operation: for an inverse M-matrix $K$ (a positive kernel whose inverse has non-positive off-diagonal entries), set $A=K^{-1}$, form $A^{\mathrm{sym}}$ by keeping the diagonal and replacing off-diagonal entries by $-(A_{i,j}A_{j,i})^{1/2}$, and define $K^{\mathrm{isymi}}=(A^{\mathrm{sym}})^{-1}$. For the extended kernels $K(l,n+1)$ built from $\widetilde U$, the paper proves $1\le \nu_{l,n}=|A(l,n+1)^{\mathrm{sym}}|/|A(l,n+1)|\le 1+\rho_{l,n}$, where $\rho_{l,n}=\sum_{j,k}(U(l,n))^{-1}_{j,k}f_{k+l}$. Condition (1.7) makes $\rho_{l,n}=o_l(1)$, so the symmetrized permanental sequence is close in probability to the original one; its Gaussian representation then transfers the Gaussian law of the iterated logarithm. For the concrete examples, Koval's Gaussian LIL supplies the correct denominator $\phi_j$, and inverses of the triangular potentials $s_j\wedge s_k$ are computed explicitly.

What would settle it

For the birth-death potential $V_{j,k}=s_j\wedge s_k$, take $f_j=\delta s_j$, which is excessive but not a potential; the explicit inverse in Lemma 2.6 gives $\rho_{l,n}=\delta$, independent of $l$, so hypothesis (1.7) fails. If simulation of the permanental sequence with kernel $V_{j,k}+\delta s_k$ shows that $\limsup \widetilde X_{\alpha,j}/(s_j K_s(j))$ is not $1$, this confirms that the uniformity estimate is doing the load-bearing work.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is Theorem 1.2: let $X$ be a transient symmetric Borel right process with potential $U$, let $f$ be a finite excessive function, and form $\widetilde U_{j,k}=U_{j,k}+f_k$. Then $\widetilde U$ is the kernel of an $\alpha$-permanental sequence $\widetilde X_\alpha$ for every $\alpha>0$. If the nonnegativity condition (1.6) holds, if the quantitative estimate $\sum_{j,k=1}^n (U(l,n))^{-1}_{j,k} f_{k+l}=o_l(1)$ holds uniformly in $n$, and if the Gaussian sequence $\eta$ with covariance $U$ satisfies $\limsup_{j\to\infty}\eta_j/(2\phi_j)^{1/2}=1$ a.s. with $f_j=o(\phi_j)$, then $\limsup_{j\to\infty}\widetilde X_{\alpha,j}/\phi_j=1$ a.s. for every $\alpha\ge 1/2$. The upper bound is noted to hold for all $\alpha>0$, and in the applications the lower bound is extended to all $\alpha>0$ using earlier permanental-process technology. The proof compares the target permanental sequence with a symmetrized permanental sequence whose Gaussian representation is explicit, and controls the error by the ratio of the inverse M-matrix and its symmetrization.

Load-bearing premise

The transfer rests on the quantitative estimate (1.7): the row sums of the inverse of every finite block of the potential, applied to the excessive function $f$, must tend to zero uniformly as the block is pushed out to infinity; if that estimate fails, the probability comparison in (6.35) has an uncontrollable error and the limsup transfer does not follow.

Editorial extensions

If this is right

For every symmetric transient Borel right process satisfying (1.6)-(1.9), the Gaussian LIL transfers to the permanental sequence with kernel $U+f$: the upper bound holds for all $\alpha>0$ and the lower bound for $\alpha\ge 1/2$, with the lower bound extended to all $\alpha>0$ in the paper's applications.
For birth-death chains without emigration, $\limsup_{j\to\infty}\widetilde Y_{\alpha,j}/(s_j K_s(j))=1$ for every $\alpha>0$, and this simplifies to $s_j\log\log s_j$ or $s_j\log j$ according to the growth rate of $s_j$.
For birth-death chains with emigration and for first-order Gaussian autoregressive potentials, the same principle gives $\limsup \widetilde Z_{\alpha,j}/(W_{j,j}K_s(j))=1$ and the explicit $U_{j,j}\log\log$ or $U_{j,j}\log j$ corollaries, including the constant $1-\beta$ when $U_{j,j}$ is regularly varying with index $0<\beta<1$.
For higher-order autoregressive potentials, if $\sum_l p_l=1$ then $\limsup \widetilde Y_{\alpha,j}/(j\log\log j)=1/(\sum_l l p_l)^2$; if $\sum_l p_l<1$ then $\limsup \widetilde Y_{\alpha,j}/\log j=c^*$, where $c^*$ is expressed through the roots of $P(x)=1-\sum_l p_l x^l$.
For L\'evy processes on $\mathbb Z$ killed at an independent exponential time, even without symmetry of the kernel, $\limsup \widetilde X_{\alpha,n}/\log n=U_{0,0}$ a.s. whenever the added excessive function vanishes at infinity.

Reading between the lines

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

The same transfer should work for other symmetric Markov-chain potentials whose finite-block inverses are explicit enough to verify (1.7), for instance potentials of random walks on trees or on finitely generated groups.
The ratio $\nu_{l,n}$ can be read as a quantitative measure of how non-symmetric the kernel is: keeping it close to $1$ is what makes the permanental sequence mimic its Gaussian counterpart, so it could serve as a diagnostic for Gaussian-like extremes in other permanental processes.
The paper itself notes that the restriction $f_j=o(j^{1/2})$ in Theorem 1.10 is probably unnecessary; a natural next step is to replace it by $f_j=o(j)$ and check whether the all-$\alpha>0$ lower bound persists.
Because the upper bound is trivial for all $\alpha>0$ by infinite divisibility, the delicate point is the lower bound; the Gaussian decomposition in (6.39) suggests a route toward proving the lower bound for all $\alpha>0$ under (1.7) alone, without the additional hypotheses used in the examples.

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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

Two things to know. First, this is a serious paper with a genuinely new transfer principle: a Gaussian LIL for a sequence with covariance U is transferred to a permanental LIL for the nonsymmetric kernel U + f, where U is a symmetric transient Markov potential and f is excessive. That is a real organizer for the class of non-Gaussian infinitely divisible sequences the authors have been developing, and the applications — birth-death processes with and without emigration, first- and higher-order AR covariances, Lévy potentials — are actually worked, not just asserted. The example-specific verification of the inverse-matrix estimates (Lemmas 2.6, 3.3, 5.14) is the real work and it holds up on my read.

Second, the proof of the main theorem has a genuine lim/limsup mismatch at its load-bearing joint. Equation (6.44) is justified by Lemma 6.5, but Lemma 6.5 requires a full limit in (6.47) while Theorem 1.2 only assumes limsup in (1.8). The step is false as written: for iid N(0,1) with φ_j = log j the limsup of the normalized absolute value is 1 a.s., but the squares have liminf 0, so the stated limit of the average squares fails. The stress-test note is right that this is an internal gap, not a too-weak hypothesis in an isolated lemma.

That said, the gap is repairable. A limsup version of Lemma 6.5 is true with the same proof: the upper bound comes from the finite net, the lower bound from projecting onto any fixed unit vector. Then (6.44) becomes a limsup statement, which is all the surrounding argument needs to get (1.10). My assessment is that Theorem 1.2 is true as stated and the applications are not in danger; the written proof just doesn't yet establish what it claims.

Minor issues: the abstract advertises (0.2) for all α > 0, while Theorem 1.2 states α ≥ 1/2 and recovers all α through extra work in each example; that should be aligned. There is a small typo in Theorem 6.1 (δ_l = o(l)). The citation profile leans hard on the authors' own [9] and [10], but those are used as black-box lemmas, the normalizer φ is fixed by U and Koval's Gaussian LIL rather than fitted, and I see no circularity.

Who it's for: people working on permanental processes, Markov-chain potentials, or Gaussian LIL transfer; the isymi comparison technique is worth studying on its own. I'd send it to a serious referee, with the instruction to require a corrected Lemma 6.5 or an explicit subsequence argument before passing the proof of Theorem 1.2.

Referee Report

2 major / 3 minor

Summary. The paper studies alpha-permanental sequences whose kernels are obtained by adding a finite excessive function f to the symmetric potential U of a transient symmetric Borel right process. The main result, Theorem 1.2, gives conditions under which a Gaussian law of the iterated logarithm for the sequence with covariance U, limsup_j eta_j/(2 phi_j)^{1/2}=1, transfers to a sharp limsup for the permanental sequence, limsup_j tilde X_{alpha,j}/phi_j=1. The proof is based on a comparison with a symmetrized inverse-M-matrix kernel, developed in Section 6, and is then applied to birth-and-death processes with and without emigration, first- and higher-order Gaussian autoregressive sequences, and Levy processes on Z.

Significance. If the proof is made fully rigorous, the paper gives a substantial and useful transfer principle: sharp asymptotic behavior of nonsymmetric permanental sequences is derived from the classical Gaussian LIL, with explicit verification for several families of Markov chains. The applications are non-trivial and the verification of the inverse-matrix hypotheses (1.6)-(1.7) in Sections 2-5 is detailed. The paper is also honest in separating the general theorem from the application-specific lower bounds for all alpha>0. The main proof, however, contains a mismatch between the limsup hypothesis of Theorem 1.2 and the limit hypothesis used in Lemma 6.5, which currently leaves the central transfer argument incomplete.

major comments (2)

[Section 6, Eq. (6.44) and Lemma 6.5] The proof of Theorem 1.2 asserts that 'It follows from (1.8) and Lemma 6.5 below that lim_{j to infinity} sum_{i=1}^k eta_{i,j}^2/(2 phi_j) = 1 a.s.' This implication is not justified. Lemma 6.5 has the hypothesis lim_{j to infinity} |eta_j|/(2 phi_j)^{1/2} = 1 (Eq. (6.47)), while Theorem 1.2 assumes only the limsup condition (1.8). The stronger hypothesis is essential for the lemma as stated: for iid N(0,1) variables and phi_j = log j, (1.8) holds but eta_j^2/(2 log j) has liminf 0 and limsup 1, so it does not converge; hence Lemma 6.5 cannot be invoked, and the limit asserted in (6.44) is not established. Since (6.44) is the bridge between the probability comparison (6.38) and the claimed permanental limit in (6.45)-(6.46), the proof of (1.10) for alpha = k/2 is incomplete as written. The applications likely satisfy the stronger limit through Koval-type theorems, e.g., (2.52), (3.28), and (5.85), so the gap may be repairable by strengthening the hypothesis of Theorem 1.2 or by a subsequence argument for the lower bound, but the general theorem as stated is not proved.
[Section 6, Theorem 6.1, Eq. (6.34)] The hypothesis in Theorem 6.1 that rho_{l,n} <= delta_l with delta_l = o(l) is insufficient for its use in the proof of Theorem 1.2: the error term 2 alpha delta_l must tend to zero as l tends to infinity in the inequalities (6.35) and subsequently in (6.42)-(6.43). The proof of Theorem 1.2 actually requires delta_l = o_l(1), which does follow from (1.7). The statement of Theorem 6.1 should therefore be corrected to delta_l = o_l(1), or delta_l -> 0, to match the argument that follows.

minor comments (3)

[Abstract and Theorem 1.2] The abstract states the transfer for all alpha > 0, but Theorem 1.2 proves the lower bound only for alpha >= 1/2, with additional arguments needed for all alpha > 0 in the applications. This discrepancy should be clarified in the statement of the abstract or the theorem.
[Section 2, around Eq. (2.15)] The notation Q(s) = 2 Q(s) is confusing because the same symbol is used for the rescaled generator; a different symbol, such as Q'(s) or a sentence explaining the rescaling, would make the argument easier to follow.
[Theorem 1.9, Eq. (1.55)] The displayed formula in (1.55) appears to have a missing fraction: the denominator should be (sum_{l=1}^k l p_l)^2, as used later in the proof.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Gaussian-to-permanental transfer is derived from an external Gaussian LIL plus explicit estimates; self-citations are background lemmas, not the target conclusion.

full rationale

The paper's main transfer principle, Theorem 1.2, does not reduce to its inputs by construction. The Gaussian normalization φ is not fitted to permanental data; it is determined by the covariance U through Koval's external Gaussian LIL theorem [5], which is cited as an independent benchmark. The permanental sequence with kernel U~ = U + f is obtained from the prior existence result [10, Theorem 1.11], and the probability comparison with the symmetrized kernel comes from [9, Corollary 3.1]; both are used as black-box lemmas from earlier published work, and neither states or assumes the target conclusion (1.10). The hypotheses (1.6)-(1.9) are then verified case by case with explicit inverse-matrix computations (Lemmas 2.6, 3.3, 5.14) and Riesz decomposition arguments, not by assuming the permanental limit. The lower bounds for all α > 0 are proved separately through subsequence extensions and [10, Lemma 7.1], again as an external input rather than a re-statement of the conclusion. There is a genuine internal proof gap at equation (6.44): Lemma 6.5 requires a true limit in (6.47), while Theorem 1.2 assumes only the limsup in (1.8), so the inference to the limit of normalized sums of squares is not justified as written. This is a correctness issue, not circularity; it does not make the theorem's conclusion identical to an assumption or to a fitted parameter. Overall, the derivation chain is self-contained against external benchmarks and the self-citations are not load-bearing in a circular way.

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

No fitted free parameters and no invented entities. The central theorem relies on explicit hypotheses (1.6)-(1.7) and on the import of prior results: the existence theorem from [10], the comparison lemma from [9], and Koval's external Gaussian LIL. Koval's theorem is parameter-free and external; the self-cited [9] and [10] are used as lemmas, not as the target result.

assumptions (3)

domain assumption The kernel ~U = U+f is the kernel of an α-permanental sequence for all α>0 when f is a finite excessive function (Theorem 1.1, imported from [10, Theorem 1.11]).
This existence step is central to the paper and is taken from the authors' prior work without reproof.
standard math Comparison lemma for permanental variables (Lemma 6.2, from [9, Corollary 3.1]) and the fact that K^{isymi} is an inverse M-matrix (Lemma 6.1).
Used throughout Section 6 to compare the non-symmetric permanental sequence with a symmetric Gaussian-related sequence.
domain assumption Koval's law of the iterated logarithm for Gaussian sequences, giving lim_j ξ_j/(2 s_j K_s(j))^{1/2}=1 a.s. for covariance s_i∧s_j (Eq. (2.52)).
External theorem from reference [5], used to verify the Gaussian LIL hypothesis (1.8) in the birth-death and AR examples.

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Cite this review

Pith. "Pith review of Asymptotic properties of permanental sequences." pith.science (2026). https://pith.science/paper/ASUP6U66

@misc{pith2026190804155,
  author       = {Pith},
  title        = {Pith review of: Asymptotic properties of permanental sequences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ASUP6U66}},
  note         = {Machine review of arXiv:1908.04155}
}

abstract

Let $U=\{U_{j,k},j,k\in \overline {\mathbb N}\}$ be the potential of a transient symmetric Borel right process $X$ with state space $\overline {\mathbb N}$. For any excessive function $f=\{f_{k,k\in \overline {\mathbb N}}\}$ for $X$ , $\widetilde U=\{\widetilde U_{j,k},j,k\in\overline {\mathbb N}\}$, where \begin{equation} \widetilde U_{j,k}= U_{j,k} +f_{ k},\qquad j,k\in\overline {\mathbb N},\label{a.1} \end{equation} is the kernel of an $\alpha$-permanental sequence $\widetilde X_{\alpha}=(\widetilde X_{\alpha, 1} ,\ldots)$ for all $\alpha>0$. The symmetric potential $U$ is also the covariance of a mean zero Gaussian sequence $\eta=\{\eta_{j},j\in \overline {\mathbb N}\}$. Conditions are given on the potentials $U$ and excessive functions $f$ under which, \begin{equation} \limsup_{j\to \infty}\frac{ \eta_{j}}{( 2\,\phi_{j})^{1/2} }=1 \quad a.s. \quad \implies \quad \limsup_{n\to \infty}\frac{\widetilde X_{\alpha, j}}{\phi_{j} }=1\quad a.s.,\label{a.2} \end{equation} for all $\alpha>0$, and sequences $\phi=\{\phi_{j}\}$ such that $f_{j}=o(\phi_{j})$. The function $\phi$ is determined by $U$. Many examples are given in which $U$ is the potential of symmetric birth and death processes with and without emigration, first and higher order Gaussian autoregressive sequences and L\'evy processes on $\mathbf Z$.

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[9]

M. B. Marcus and J. Rosen, Permanental random variables, M -matrices and α-permanents, High Dimensional Probability VII, The Carg` e se Vol- ume, Progress in Probability, 71 (2016), 363–379, Birkhaus er, Boston. Pages 82, 85

work page 2016
[10]

M. B. Marcus and J. Rosen, Sample path properties of perm anental pro- cesses, EJP, (2018), No. 58, 1-49. Pages 2, 4, 5, 5, 14, 15, 27, 77, 84, 84, 95, 95, 96

work page 2018
[1]

Blumenthal, and R

R. Blumenthal, and R. Getoor, Markov Processes and Potential Theory . New York: Academic Press, (1968). Page 95

work page 1968
[2]

Dellacherie, S

C. Dellacherie, S. Martinez and J. San Martin, Inverse M-Matrices and Ultrametric Matrices, Lecture Notes in Mathematics 2118, Springer, NY, (2014). Pages 85, 85

work page 2014
[3]

Eisenbaum and H

N. Eisenbaum and H. Kaspi, On permanental processes, Stochastic Pro- cesses and their Applications , 119, (2009), 1401-1415. Pages 2, 27, 82

work page 2009
[4]

Freedman, Markov chains , Second Edition, Springer, NY, (1983)

D. Freedman, Markov chains , Second Edition, Springer, NY, (1983). Pages 6, 99

work page 1983
[5]

Koval, The law of the iterated logarithm for Gaussia n sequences and its applications, Theor

V, A. Koval, The law of the iterated logarithm for Gaussia n sequences and its applications, Theor. Probability and Math. Stat. , 54, (1997), 69-75. Pages 8, 24

work page 1997
[6]

M. B. Marcus and J. Rosen, Markov Processes, Gaussian Processes and Local Times, Cambridge University Press, New York, (2006). Pages 3, 4

work page 2006

Show all 15 references

[7]

M. B. Marcus and J. Rosen, A suﬃcient condition for the con tinuity of permanental processes with applications to local times of M arkov pro- cesses, Annals of Probability, (2013), Vol. 41, No. 2, 671-6 98. Page 15. 104

2013
[8]

M. B. Marcus and J. Rosen, Conditions for permanental pro cesses to be unbounded, Annals of Probability, 45, (2017), 2059–2086. P ages 15, 73, 73, 82, 84

2017
[11]

M. B. Marcus and J. Rosen, Permanental sequences with ke rnels that are not equivalent to a symmetric matrix. Proceedings of the Eig hth Confer- ence on High Dimensional Probability, Oaxaca, Mexico. Page 2
[12]

J. R. Norris, Markov Chains , Cambridge University Press, New York, (1999). Pages 17, 17, 73

1999
[13]

Continuous martingales and Brownian mo- tion, volume 293

Daniel Revuz and Marc Yor. Continuous martingales and Brownian mo- tion, volume 293. Springer-Verlag, Berlin, third edition, 1999 . Pages 17, 17

1999
[14]

Revesz, Random walk in random and non-random environments , Cam- bridge University Press, New York, (1990)

P. Revesz, Random walk in random and non-random environments , Cam- bridge University Press, New York, (1990). Page 90

1990
[15]

Taylor and S

H. Taylor and S. Karlin, An Introduction to Stochastic Modeling , Third Edition, Academic Press, NY, (1998). Pages 8, 16. 105

1998

Pith tools

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

[1] [9]

M. B. Marcus and J. Rosen, Permanental random variables, M -matrices and α-permanents, High Dimensional Probability VII, The Carg` e se Vol- ume, Progress in Probability, 71 (2016), 363–379, Birkhaus er, Boston. Pages 82, 85

work page 2016

[2] [10]

M. B. Marcus and J. Rosen, Sample path properties of perm anental pro- cesses, EJP, (2018), No. 58, 1-49. Pages 2, 4, 5, 5, 14, 15, 27, 77, 84, 84, 95, 95, 96

work page 2018

[3] [1]

Blumenthal, and R

R. Blumenthal, and R. Getoor, Markov Processes and Potential Theory . New York: Academic Press, (1968). Page 95

work page 1968

[4] [2]

Dellacherie, S

C. Dellacherie, S. Martinez and J. San Martin, Inverse M-Matrices and Ultrametric Matrices, Lecture Notes in Mathematics 2118, Springer, NY, (2014). Pages 85, 85

work page 2014

[5] [3]

Eisenbaum and H

N. Eisenbaum and H. Kaspi, On permanental processes, Stochastic Pro- cesses and their Applications , 119, (2009), 1401-1415. Pages 2, 27, 82

work page 2009

[6] [4]

Freedman, Markov chains , Second Edition, Springer, NY, (1983)

D. Freedman, Markov chains , Second Edition, Springer, NY, (1983). Pages 6, 99

work page 1983

[7] [5]

Koval, The law of the iterated logarithm for Gaussia n sequences and its applications, Theor

V, A. Koval, The law of the iterated logarithm for Gaussia n sequences and its applications, Theor. Probability and Math. Stat. , 54, (1997), 69-75. Pages 8, 24

work page 1997

[8] [6]

M. B. Marcus and J. Rosen, Markov Processes, Gaussian Processes and Local Times, Cambridge University Press, New York, (2006). Pages 3, 4

work page 2006