On large deviation principles for general random processes

A. A. Borovkov; K. A. Borovkov

arxiv: 2604.27485 · v1 · submitted 2026-04-30 · 🧮 math.PR

On large deviation principles for general random processes

A. A. Borovkov , K. A. Borovkov This is my paper

Pith reviewed 2026-05-07 10:12 UTC · model grok-4.3

classification 🧮 math.PR

keywords large deviationslarge deviation principlestochastic processesconditional probabilityLegendre transformmoment generating functionSkorokhod space

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The pith

If the conditional log-moment generating function of a general stochastic process converges uniformly to an essentially smooth function A, then the normalized position Z(T)/T obeys a uniform conditional local large deviation principle with

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

The paper establishes a conditional large deviation principle for general stochastic processes Z with paths in the Skorokhod space. It assumes that the scaled conditional log-moment generating function converges uniformly on events where Z(0)/T is close to a fixed alpha, to an essentially smooth function A. Under this assumption, the probability that Z(T)/T lies near any beta, conditioned on the past, has logarithmic asymptotics given by minus the Legendre transform D of A, and the convergence is uniform over the conditioning events. The same limit yields conditional local large deviations for the finite-dimensional distributions of the scaled process z_T(s) = Z(sT)/T. With additional bounds on the oscillations of the trajectories z_T, the result extends to a functional local large deviation principle in the uniform metric on path space.

Core claim

Under the assumption that there exists an essentially smooth function A such that (1/T) ln E[exp(mu (Z(T) - alpha T)) | Z(s), s <= 0] = A(mu) + o(1) uniformly on the event C(T) = {|Z(0)/T - alpha| < eta_T} with eta_T -> 0, the paper proves that for any fixed alpha and beta and for eps_T -> 0 sufficiently slowly, lim (1/T) ln P(Z(T)/T - alpha in (beta - eps_T, beta + eps_T) | Z(s), s <= 0) = -D(beta) uniformly on C(T), where D is the Legendre transform of A. This is then used to obtain a conditional local LDP for finite-dimensional distributions of the scaled process and, under oscillation controls, a functional local LDP for the probability that the scaled path lies in a small uniform ball.

What carries the argument

The essentially smooth function A that serves as the uniform limit of the scaled conditional log-moment generating function, together with its Legendre transform D that supplies the rate function for the large deviations.

Load-bearing premise

There exists an essentially smooth function A such that the scaled conditional log-moment generating function converges uniformly to A(mu) on the events where the initial normalized position is close to alpha.

What would settle it

A specific stochastic process for which the conditional log-moment generating function converges uniformly to some A but the lim (1/T) ln P(Z(T)/T near beta | past) differs from -D(beta) on a set of positive conditional probability, or for which the uniformity over C(T) fails while the pointwise limit holds.

read the original abstract

Let $Z=\{Z(t): t\in \mathbb R\}$ be a stochastic process with trajectories in space $\mathbb D (\mathbb R)$. It is assumed that there exists an essentially smooth function $A:\mathbb R\to (-\infty, \infty] $ such that, for all $\alpha \in \mathbb R, $ $ \mu\in \mbox{dom}\, A$, one has \begin{equation*} \frac1{T} \ln {\mathbf E} \big( e^{\mu (Z(T)-\alpha T)} \big|Z(s), \ s\le 0 \big) = A(\mu) +o(1) \end{equation*} uniformly on the event $C(T):=\{|Z(0)/T - \alpha |< \eta_T \} $, where $ \eta_T \to 0$ as $T\to\infty.$ Under this condition, a uniform conditional local large deviation principle (l.l.d.p.) is established: for any fixed $\alpha, \beta\in \mathbb R$ and a positive function $\eta_T=o(1)$, for $\varepsilon_T \to 0$ sufficiently slowly as $T\to\infty,$ one has \begin{equation*} \lim_{T\to\infty}\frac1T \ln {\mathbf P} \big( {Z(T)}/T-\alpha \in (\beta-\varepsilon_T, \beta +\varepsilon_T) \big| Z(s), \ s\le 0\big) = - D(\beta ) \end{equation*} uniformly on $C(T)$, where $D$ is the Legendre transform of the function $A$. This result is used to establish a conditional l.l.d.p. for the finite-dimen\-sional distributions of the process $ \{ z_T(s) = Z(sT)/T: s\in [0,1]\}$. Under additional conditions on the magnitude of oscillations of the trajectories $z_T$, a functional l.l.d.p. is obtained for the asymptotics of $\ln {\mathbf P} (z_T\in (f)_{\varepsilon_T})$ as $T\to\infty$, where $f\in \mathbb D(0,1),$ $(f)_\varepsilon$ is the $\varepsilon$-neighborhood of $f$ in the space $ \mathbb D(0,1)$ with respect to the uniform metric, and $\varepsilon_T \to 0$ sufficiently slowly. The obtained results can be extended to a more general triangular array scheme where the process itself $Z=Z^{(T)}$ also depends on the parameter $T$.

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

Borovkov and Borovkov establish a uniform conditional local LDP from a uniform mgf assumption, with extensions to functional level, but verification of the assumption remains the main hurdle.

read the letter

This paper's main contribution is a uniform conditional local large deviation principle that holds when the scaled cumulant generating function converges uniformly on a slowly shrinking neighborhood of the typical behavior. They assume an essentially smooth limit A for the mgf, uniform over the event where Z(0) is close to alpha T. From this they extract the local LDP for Z(T)/T being near beta, with rate given by the Legendre transform D of A, and the convergence is uniform on that event. They extend the same idea first to the finite-dimensional distributions of the scaled path z_T and then to a functional LDP in the uniform metric, provided the paths do not oscillate too wildly. The uniformity is the part that feels new. Most conditional LDPs I have seen are pointwise or for fixed conditioning; making it uniform over a neighborhood lets you chain the results more easily when you move to path space. The triangular-array version is also straightforward but useful for time-inhomogeneous cases. On the downside, the assumption is heavy. Requiring the mgf convergence to be uniform right up to the edge of C(T) means you have to control the tails quite precisely, and the paper gives no worked examples to show how this plays out in standard models like random walks with dependence or diffusions. The eps_T has to go to zero slowly, which is typical but means you lose some precision on the scale of the deviations. This work is for people already deep in large-deviation theory who need conditional results for non-Markov processes. A reader who wants ready-to-use theorems for a specific application will probably have to do extra verification work on the mgf condition. The technical development looks careful and the statements are precise. It deserves a serious referee who can check the uniformity arguments in detail. I would recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper assumes there exists an essentially smooth function A such that the conditional log-mgf (1/T) ln E[exp(μ(Z(T) - α T)) | Z(s), s ≤ 0] equals A(μ) + o(1) uniformly on the event C(T) = {|Z(0)/T - α| < η_T} with η_T → 0. Under this assumption it derives a uniform conditional local large deviation principle: lim (1/T) ln P(Z(T)/T - α ∈ (β - ε_T, β + ε_T) | Z(s), s ≤ 0) = -D(β) uniformly on C(T), where D is the Legendre transform of A and ε_T → 0 sufficiently slowly. The result is extended to finite-dimensional distributions of the scaled process z_T(s) = Z(sT)/T and, under additional oscillation controls on z_T, to a functional LDP for ln P(z_T ∈ (f)_ε_T) in the uniform metric on D[0,1]. The framework is also stated for triangular arrays Z = Z^{(T)}.

Significance. If the central derivation holds, the manuscript supplies a general template for obtaining uniform conditional local LDPs directly from an mgf limit, which is potentially useful for processes with dependence or in non-stationary triangular schemes. The uniformity on the shrinking set C(T) is a technically useful feature for applications involving conditioning. The functional extension follows standard lines once oscillation control is granted. However, the assumption is strong and no concrete verification or example is supplied, so the practical reach remains unclear. No machine-checked proofs or reproducible code are present.

major comments (2)

[Main theorem on local LDP] The proof of the uniform conditional local LDP (the statement following the mgf assumption in the abstract and developed in the main theorem) does not supply explicit error bounds relating the o(1) term, the rate η_T, and the choice of ε_T → 0. Because the uniformity claim on C(T) is load-bearing for the entire result, the argument that ε_T may be taken arbitrarily slowly requires a quantitative control on how the mgf error propagates through the Legendre-transform inversion; without it the uniformity statement is not fully substantiated.
[Functional LDP extension] In the passage from the local LDP to the functional LDP, the additional conditions on the magnitude of oscillations of the trajectories z_T are stated only qualitatively. This is load-bearing for the functional claim because the ε_T-neighborhood in the uniform metric on D[0,1] interacts directly with those oscillation bounds; the manuscript should quantify the required oscillation rate relative to ε_T and η_T.

minor comments (2)

The definition of the space D(R) and the uniform metric on D[0,1] should be recalled or referenced explicitly, as the topology is central to the functional statement.
[Introduction] A reference to a standard text on large deviations (e.g., Dembo-Zeitouni) would help readers place the Legendre-transform argument and the essentially-smooth assumption.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address the major comments point by point below and will revise the manuscript to incorporate clarifications that strengthen the substantiation of the uniformity and functional claims.

read point-by-point responses

Referee: [Main theorem on local LDP] The proof of the uniform conditional local LDP (the statement following the mgf assumption in the abstract and developed in the main theorem) does not supply explicit error bounds relating the o(1) term, the rate η_T, and the choice of ε_T → 0. Because the uniformity claim on C(T) is load-bearing for the entire result, the argument that ε_T may be taken arbitrarily slowly requires a quantitative control on how the mgf error propagates through the Legendre-transform inversion; without it the uniformity statement is not fully substantiated.

Authors: We agree that the current presentation would benefit from a more explicit discussion of error propagation to fully substantiate the uniformity on C(T). In the revised manuscript we will expand the proof of the main theorem with a remark that relates the o(1) convergence rate (denoted, say, by δ_T → 0) to the admissible speed of ε_T. Specifically, we will show that whenever ε_T → 0 is chosen so that ε_T / δ_T → ∞ (in the appropriate sense), the Legendre-transform inversion preserves the uniform limit -D(β). Because the assumption is stated for a general o(1) term without a prescribed rate, fully numerical bounds cannot be supplied; the added remark nevertheless makes the dependence on the convergence rate transparent and confirms that ε_T may be taken arbitrarily slowly relative to δ_T. This revision addresses the concern while preserving the generality of the framework. revision: partial
Referee: [Functional LDP extension] In the passage from the local LDP to the functional LDP, the additional conditions on the magnitude of oscillations of the trajectories z_T are stated only qualitatively. This is load-bearing for the functional claim because the ε_T-neighborhood in the uniform metric on D[0,1] interacts directly with those oscillation bounds; the manuscript should quantify the required oscillation rate relative to ε_T and η_T.

Authors: We concur that the oscillation conditions should be stated with explicit rates to ensure compatibility with the ε_T-neighborhood. In the revision we will quantify the required control by adding the assumption that the modulus of continuity of z_T satisfies ω(z_T, δ_T) = o(ε_T) for a sequence δ_T → 0 whose speed is tied to η_T (for instance, δ_T chosen so that the conditioning event C(T) remains consistent with the local LDP). This explicit relation guarantees that the finite-dimensional local LDP lifts to the functional LDP in the uniform metric on D[0,1]. The updated statement will appear in the theorem on the functional large-deviation principle. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation proceeds one-directionally from external assumption

full rationale

The paper posits an external assumption that an essentially smooth A exists such that the conditional scaled cumulant generating function converges uniformly to A(mu) on C(T). It then proves that the conditional local LDP holds with rate -D(beta), where D is the Legendre transform of this A. This is a standard one-way implication (cumulant convergence implies LDP) with no reduction to tautology, no fitted parameters renamed as predictions, and no load-bearing self-citations or ansatzes smuggled in; the core chain is self-contained against the stated hypothesis and does not invoke prior results by the same authors to justify the central step.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and essential smoothness of the limiting function A together with the uniform convergence of the conditional cumulant generating function; no free parameters are fitted and no new entities are postulated.

axioms (1)

domain assumption Existence of an essentially smooth function A: R -> (-inf, inf] such that the conditional log mgf converges uniformly to A(mu) + o(1) on C(T)
This is the sole load-bearing hypothesis from which the LDP is derived; it is stated explicitly in the abstract.

pith-pipeline@v0.9.0 · 5788 in / 1330 out tokens · 53584 ms · 2026-05-07T10:12:18.667173+00:00 · methodology

discussion (0)

Reference graph

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Sur un nouveau th\'eor\`eme–limite de la th\'eorie des probabilit\'es

Cram\'er, H. Sur un nouveau th\'eor\`eme–limite de la th\'eorie des probabilit\'es. In: Actualit\'es Scientifiques et Industrielles , number 736 in Colloque consacr\'e \`a la th\'eorie des probabilit\'es, pp. 5--23. Hermann, Paris, 1938. [English translation: arXiv:1802.05988v4 .]

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http://mi.mathnet.ru/rus/mt470 А. А. Боровков, А. А. Могульский Большие уклонения и проверка статистических гипотез Введение, Гл. 1 Тр. Ин-та математики СО РАН 19 1992 1--63 http://mi.mathnet.ru/rus/mt470 https://mathscinet.ams.org/mathscinet-getitem?mr=1261864 https://mathscinet.ams.org/mathscinet-getitem?mr=1181029 https://zbmath.org/?q=an:0847.62012 ан...

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https://zbmath.org/?q=an:1351.60003 А. А. Боровков Асимптотический анализ случайных блужданий. Быстро убывающие распределения приращений М. Физматлит 2013 447 https://zbmath.org/?q=an:1351.60003 англ. пер.: https://doi.org/10.1017/9781139871303 A. A. Borovkov Asymptotic analysis of random walks. Light-tailed distributions Encyclopedia Math. Appl. 176 Camb...

work page doi:10.1017/9781139871303 2013

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https://doi.org/10.4213/tvp4415 А. А. Боровков, А. А. Могульский Принципы больших уклонений для траекторий случайных блужданий. I Теория вероятн. и ее примен. 2011 56 4 627--655 http://mi.mathnet.ru/rus/tvp4415 https://doi.org/10.4213/tvp4415 https://mathscinet.ams.org/mathscinet-getitem?mr=3137061 https://zbmath.org/?q=an:1267.60029 англ. пер.: https://d...

work page doi:10.4213/tvp4415 2011

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А. А. Боровков Обобщенные процессы восстановления М. РАН 2020 455 англ. пер.: https://zbmath.org/?q=an:1519.60002 A. A. Borovkov Compound renewal processes Encyclopedia Math. Appl. 184 Cambridge Cambridge Univ. Press xvi+364 2022 https://zbmath.org/?q=an:1519.60002

work page arXiv 2020

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Гертнер О больших уклонениях от инвариантной меры Теория вероятн

http://mi.mathnet.ru/rus/tvp3100 Ю. Гертнер О больших уклонениях от инвариантной меры Теория вероятн. и ее примен. 1977 22 1 27--42 http://mi.mathnet.ru/rus/tvp3100 https://mathscinet.ams.org/mathscinet-getitem?mr=471040 https://zbmath.org/?q=an:0375.60033 англ. пер.: https://doi.org/10.1137/1122003 J. G\"artner On large deviations from the invariant meas...

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