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arxiv: 2507.09321 · v2 · submitted 2025-07-12 · 🧮 math.PR

Large Deviations for Iterated Sums and Integrals

Yuri Kifer , Ofer Zeitouni This is my paper

Pith reviewed 2026-05-19 04:21 UTC · model grok-4.3

classification 🧮 math.PR
keywords large deviationsiterated sumsiterated integralsstationary processestrajectorial large deviationsvector processeslarge deviation principles
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The pith

If the partial sums of a stationary process satisfy a trajectorial large deviations principle, then so do its normalized iterated sums and integrals.

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

This paper shows how large deviation principles extend from simple sums to their multiple iterated versions. It focuses on centered bounded stationary vector processes where the base sums or integrals already obey a pathwise large deviations principle. The normalized iterated objects, scaled by N to the power of minus the iteration order, then also satisfy such principles. A reader might care because this provides a way to understand rare events in systems that involve repeated integrations or summations of random fluctuations.

Core claim

The paper claims that for centered bounded stationary vector processes whose sums or integrals satisfy a trajectorial large deviations principle, the normalized multiple iterated sums and integrals of the form N^{-ν} times the sum or integral of the tensor products also satisfy large deviation principles.

What carries the argument

The normalized iterated sum or integral bbS_N^{(ν)}(t), defined as the N^{-ν} scaled multiple sum or integral of the tensor products of the process values over ordered indices or times up to Nt.

If this is right

  • If the base process satisfies the trajectorial LDP then the double iterated sum satisfies an LDP.
  • The result applies equally to the continuous-time integral version of the iterated object.
  • The same extension holds for any fixed iteration order ν.
  • The rate function for the iterated object is determined by the rate function of the base sums.

Where Pith is reading between the lines

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

  • The result could be applied to obtain large deviations for iterated stochastic integrals arising in solutions of differential equations.
  • It may connect to rough path theory for handling less regular driving signals.
  • Simulation of the base process could test whether observed frequencies of rare iterated events match the predicted rate.

Load-bearing premise

The sums or integrals of the underlying stationary processes must satisfy a trajectorial large deviations principle.

What would settle it

A concrete stationary process where the base sums satisfy a trajectorial LDP but the iterated sum of order two fails to satisfy the corresponding large deviation principle.

read the original abstract

We describe large deviations for normalized multiple iterated sums and integrals of the form $\bbS_N^{(\nu)}(t)=N^{-\nu}\sum_{0\leq k_1<...<k_\nu\leq Nt}\xi(k_1)\otimes\cdots\otimes\xi(k_\nu)$, $t\in[0,T]$ and $\bbS_N^{(\nu)}(t)=N^{-\nu}\int_{0\leq s_1\leq...\leq s_\nu\leq Nt}\xi(s_1)\otimes\cdots\otimes\xi(s_\nu)ds_1\cdots ds_\nu$, where $\{\xi(k)\}_{-\infty<k<\infty}$ and $\{\xi(s)\}_{-\infty<s<\infty}$ are centered bounded stationary vector processes whose sums or integrals satisfy a trajectorial large deviations principle.

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

2 major / 2 minor

Summary. The paper establishes large deviation principles (LDPs) for normalized multiple iterated sums S_N^{(ν)}(t) = N^{-ν} ∑_{0≤k1<...<kν≤Nt} ξ(k1)⊗⋯⊗ξ(kν) and the corresponding iterated integrals, for t∈[0,T], where {ξ(k)} and {ξ(s)} are centered bounded stationary vector processes. The central results are conditional on the assumption that the partial sums or integrals of the base processes ξ satisfy a trajectorial LDP; under this hypothesis the iterated objects are shown to satisfy an LDP via continuous mapping arguments on path space.

Significance. If the derivations hold, the work provides a systematic lifting of trajectorial LDPs to their ν-fold iterated versions. This is potentially useful in contexts such as rough-path theory, multiple stochastic integrals, or empirical-measure large deviations where higher-order iterated objects appear. The boundedness and stationarity assumptions ensure the maps are well-defined and Lipschitz in the uniform topology, and the conditional formulation avoids circularity by taking the base LDP as given.

major comments (2)
  1. [§3] §3 (Main Theorem): The statement of the LDP for the iterated objects relies on the contraction principle applied to the iterated-sum map. It is not immediately clear from the topology chosen on the path space whether the map is continuous at every point in the support of the base measure; a counter-example or explicit verification for the uniform topology would strengthen the argument.
  2. [Theorem 2.1] Theorem 2.1 (Base LDP hypothesis): The paper assumes the base trajectorial LDP without rate-function identification. While this is explicitly conditional, the iterated LDP rate function is then obtained only implicitly via the contraction principle; an explicit expression or variational formula for the rate function of the ν-fold objects would make the result more usable.
minor comments (2)
  1. Notation: The symbol bbS_N^{(ν)} is used for both the sum and integral versions; a subscript or superscript distinction would avoid confusion when both appear in the same statement.
  2. References: The introduction cites several works on trajectorial LDPs but omits recent papers on iterated integrals in the context of rough paths (e.g., works building on Friz-Victoir); adding one or two would improve context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive suggestions. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [§3] §3 (Main Theorem): The statement of the LDP for the iterated objects relies on the contraction principle applied to the iterated-sum map. It is not immediately clear from the topology chosen on the path space whether the map is continuous at every point in the support of the base measure; a counter-example or explicit verification for the uniform topology would strengthen the argument.

    Authors: We appreciate this observation. Upon review, the iterated-sum map is indeed continuous in the uniform topology for the class of bounded paths under consideration. In the revised version, we will insert a short lemma (Lemma 3.2) providing an explicit proof of this continuity: for any two base paths x, y with ||x - y||_∞ < δ, the difference in the iterated sums is bounded by C ν T^ν δ, where C depends on the bound of ξ. This ensures the map is Lipschitz continuous, hence continuous everywhere, including on the support of the base measure. revision: yes

  2. Referee: [Theorem 2.1] Theorem 2.1 (Base LDP hypothesis): The paper assumes the base trajectorial LDP without rate-function identification. While this is explicitly conditional, the iterated LDP rate function is then obtained only implicitly via the contraction principle; an explicit expression or variational formula for the rate function of the ν-fold objects would make the result more usable.

    Authors: The conditional formulation is deliberate to allow application to any base process satisfying a trajectorial LDP. However, we agree that stating the variational formula explicitly improves clarity. We will add a remark after Theorem 3.1 noting that the good rate function for the iterated object is I^{(ν)}(y) = inf{ I(x) : Φ(x) = y }, where Φ denotes the continuous map from the base path to the iterated sum/integral, and I is the base rate function. This is the standard explicit form obtained from the contraction principle. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation is explicitly conditional on the independent hypothesis that the base sums or integrals of the centered bounded stationary processes satisfy a trajectorial large deviations principle. The lifting to the iterated objects then proceeds via standard, topology-preserving operations (continuous maps on path space) to which the contraction principle applies directly. Boundedness ensures the iterated maps are well-defined and Lipschitz in the uniform topology; no internal equations reduce to self-definitions, no parameters are fitted and relabeled as predictions, and no load-bearing self-citations or uniqueness theorems imported from prior work by the same authors are required. The result is therefore self-contained against the stated external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the trajectorial LDP assumption for the base processes; no free parameters or new invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The sums or integrals of the stationary processes ξ satisfy a trajectorial large deviations principle.
    Explicitly stated as the hypothesis enabling the result for the iterated objects.

pith-pipeline@v0.9.0 · 5656 in / 1129 out tokens · 38834 ms · 2026-05-19T04:21:45.732999+00:00 · methodology

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Baldi, G

    P. Baldi, G. Ben Arous, G. Kerkyacharian, Large deviations and the Strassen theorem in H\" older norm , Stoch. Proc. Appl. 42 (1992), 171--180

    work page 1992

  2. [2]

    Bogolyubov and Yu.A

    N.N. Bogolyubov and Yu.A. Mitropol'skii, Asymptotic Methods in the

    work page

  3. [3]

    D. S. Clark, A short proof of a discrete Gronwall inequality ,

    work page

  4. [4]

    Borovskikh and N.C

    Y.V. Borovskikh and N.C. Weber, Large deviation results for a U -statistical sum with product kernel ,

    work page

  5. [5]

    Dembo and O

    A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications , Springer 2nd. ed., New York, 1998

    work page 1998

  6. [6]

    Diehl, K

    J. Diehl, K. Ebrahimi-Fard and Nicolas Tapia, Generalized iterated-sums signatores , arXiv:2012.04597

    work page arXiv 2012

  7. [7]

    Diehl and J

    J. Diehl and J. Reizenstein, Invariants of multidimensional time series based on their iterated-integral signature , Acta Appl. Math. 164 (2019), 83--122

    work page 2019

  8. [8]

    Eichelshacher and M

    P. Eichelshacher and M. L\" owe, A Large Deviation Principle for m -Variate von Mises-Statistics and U-Statistics , J. Theoretical Probab. 8 (1995), 807--824

    work page 1995

  9. [9]

    Freidlin, On the factorization of non-negative definite matrices ,

    M.I. Freidlin, On the factorization of non-negative definite matrices ,

    work page

  10. [10]

    Friz and M

    P.K. Friz and M. Hairer, A Course on Rough Paths , Springer, Switzerland, 2020

    work page 2020

  11. [11]

    Friz and Yu

    P.K. Friz and Yu. Kifer, Almost sure diffusion approximation in averaging via

    work page

  12. [12]

    Melbourne, Iterated invariance principle for slowly mixing

    M.Galton and I. Melbourne, Iterated invariance principle for slowly mixing

    work page

  13. [13]

    Gibbs and F.E

    A.L. Gibbs and F.E. Su, On choosing and bounding probability metrics , Internat

    work page

  14. [14]

    Hambly and T

    B. Hambly and T. Lyons, Uniqueness for the signature of a path of bounded variation and the reduced path group , Ann. Math. 171 (2010), 109--167

    work page 2010

  15. [15]

    Kelly and I

    D. Kelly and I. Melbourne, Smooth approximation of stochastic

    work page

  16. [16]

    Kifer, Large deviations in dynamical systems and stochastic processes ,

    Yu. Kifer, Large deviations in dynamical systems and stochastic processes ,

    work page

  17. [17]

    Kifer, Limit theorems for signatures , arXiv: 2306.13376

    Yu. Kifer, Limit theorems for signatures , arXiv: 2306.13376

    work page arXiv

  18. [18]

    Kifer, Almost sure approximations and laws of iterated logarithm for signatures , Stoch

    Yu. Kifer, Almost sure approximations and laws of iterated logarithm for signatures , Stoch. Proc. Appl. 182 (2025), 104576

    work page 2025

  19. [19]

    Kifer, Iterated ergodic theorems , arxiv: 2501.15633

    Yu. Kifer, Iterated ergodic theorems , arxiv: 2501.15633

    work page arXiv

  20. [20]

    Lyons, "Differential equations driven by rough signals , Revista Mat

    T. Lyons, "Differential equations driven by rough signals , Revista Mat. Iberoamericana 14.2 (1998):

    work page 1998

  21. [21]

    Nirenberg, Topics in Nonlinear Functional Analysis , Courant

    L. Nirenberg, Topics in Nonlinear Functional Analysis , Courant

    work page