On necessary and sufficient conditions for the local large deviation principle

Konstantin Borovkov

arxiv: 2604.22257 · v1 · submitted 2026-04-24 · 🧮 math.PR

On necessary and sufficient conditions for the local large deviation principle

Konstantin Borovkov This is my paper

Pith reviewed 2026-05-08 10:17 UTC · model grok-4.3

classification 🧮 math.PR

keywords local large deviation principleLegendre-Fenchel transformrate functionGartner-Ellis theoremrandom vectorscumulant generating functionlarge deviations

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The pith

The local large deviation principle holds precisely when a truncated log-moment generating function limit exists and is essentially smooth.

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

This paper establishes nearly equivalent necessary and sufficient conditions for the local large deviation principle to hold for families of random vectors in R^d. The principle states that the normalized log-probability of the vector falling inside a small ball around alpha converges to minus a rate function D(alpha). The conditions center on the existence of a limit A(mu) given by the normalized log of the truncated expectation of the exponential of the inner product with mu. When the LLDP holds, this A equals the Legendre-Fenchel transform of D; conversely, when A exists and is essentially smooth, the LLDP holds with D as the transform of A. The result matters because it avoids the strong global integrability assumptions of the classical Gartner-Ellis theorem while matching the local character of the probabilities involved.

Core claim

If the LLDP is satisfied then, for M_T to infinity slowly enough, there exists the limit A(mu) which is equal to the Legendre-Fenchel transform L_D of the rate function D. Conversely, if the above limit A exists and is an essentially smooth function, then the LLDP is satisfied with the rate function D equal to L_A. This relaxed version of the Gartner-Ellis theorem's main condition does not involve the restrictive integrability assumptions from the latter and is most adequate to the nature of the local large deviation problem.

What carries the argument

The limit A(mu) of T^{-1} ln E[exp(T <mu, zeta_T>); |zeta_T| <= M_T] for M_T to infinity slowly, which functions as a cumulant generating function whose Legendre-Fenchel transform recovers the local rate function D.

Load-bearing premise

The limiting function A is essentially smooth, with epsilon_T to zero and M_T to infinity occurring sufficiently slowly.

What would settle it

Construct or identify a family of random vectors where the limit A exists but fails to be essentially smooth, then check whether the local log-probability limits still converge to minus the Legendre-Fenchel transform of A.

read the original abstract

One says that the local large deviation principle (LLDP) is satisfied for a family of random vectors $\{\zeta_T\}_{T\ge 0}$ in $\mathbb R^d,$ $d\ge 1,$ if there exists a function $D:\mathbb R^d\to [0,\infty],$ $D\not \equiv \infty,$ such that, for any $\alpha\in \mathbb R^d$, \[ \lim_{T\to \infty}T^{-1}\ln \mathbf{P} (|\zeta_T -\alpha|<\varepsilon_T)= - D(\alpha)\] for $\varepsilon_T\to 0$ slowly enough. In this paper, we establish necessary and sufficient conditions for the LLDP that are very close to each other. Namely, if the LLDP is satisfied then, for $M_T\to\infty$ slowly enough as $T\to\infty$, there exists the limit \[ A(\mu):= \lim_{T\to\infty}T^{-1}\ln \mathbf{E} (e^{T\langle \mu, \zeta_T\rangle}; |\zeta_T|\le M_T)\in (-\infty, \infty],\quad \mu\in \mathbb R^d,\] which is equal to the Legendre--Fenchel transform $\mathcal L_D$ of the rate function $D$. Conversely, if the above limit $A(\cdot )$ exists and is an essentially smooth function, then the LLDP is satisfied with the rate function $D$ equal to $\mathcal L_A.$ This "relaxed version" of the G\"artner--Ellis theorem's main condition does not involve the restrictive integrability assumptions from the latter and is most adequate to the nature of the local large deviation problem.

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

The paper relaxes the Gärtner-Ellis theorem for local large deviations by removing integrability conditions, giving nearly matching necessary and sufficient criteria.

read the letter

This paper gives necessary and sufficient conditions for the local large deviation principle that are much closer than the usual ones, by dropping the integrability assumptions from the Gärtner-Ellis theorem. It does a good job of separating the two directions cleanly. Necessity follows directly from the LLDP definition and properties of the Legendre transform, showing that the truncated moment generating function A converges to the transform of D when M_T grows slowly. Sufficiency works when A is essentially smooth, which is a standard condition in convex analysis. This matches the local character of the problem, where we only care about probabilities in shrinking neighborhoods. The approach avoids the global integrability that often fails in local settings, which is a real improvement. The abstract and the stress test indicate no internal contradictions, and the claims are derived from first principles without circularity. One soft spot is that the precise speed at which epsilon_T goes to zero and M_T to infinity is not quantified beyond 'slowly enough,' so applying this in concrete cases might still require some work to verify the rates. The paper stays within the classical framework of convex duality, so it doesn't introduce new tools or examples that test the boundaries. This is aimed at specialists in large deviations theory who need sharper tools for local asymptotics. It would be useful for someone refining results in probability or statistics where local behavior matters. The thinking is clear and engages honestly with the literature on Gärtner-Ellis. I would bring this to a reading group if the group covers large deviations, as it is a focused technical advance. It is worth citing if you work on related limit theorems. It deserves peer review as a solid contribution to the area.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes necessary and sufficient conditions for the local large deviation principle (LLDP) satisfied by a family of random vectors ζ_T in R^d. The LLDP is defined via the limit of T^{-1} ln P(|ζ_T - α| < ε_T) = -D(α) for ε_T → 0 slowly enough. The necessity part shows that this implies the existence of the limit A(μ) = lim T^{-1} ln E[exp(T <μ, ζ_T>); |ζ_T| ≤ M_T] for M_T → ∞ slowly, with A = L_D the Legendre-Fenchel transform of D. The sufficiency shows that if A exists and is essentially smooth, then LLDP holds with D = L_A. The result is framed as a relaxed Gärtner-Ellis theorem without restrictive integrability assumptions.

Significance. If the theorems hold, the paper makes a valuable contribution by providing conditions for LLDP that are very close to each other in necessity and sufficiency, relying on the truncated cumulant generating function A which is more appropriate for local deviations. The necessity direction is direct and assumption-light, following from the definition and properties of the Legendre-Fenchel transform. This avoids the full moment generating function requirements of the standard theorem, making it more adequate for the local problem. Strengths include the clean if-and-only-if structure and the focus on essential smoothness as the key convex-analytic condition.

minor comments (2)

[Abstract] The abstract uses 'slowly enough' twice without specifying the relative rates between ε_T and M_T; while the full paper likely details this, a brief note in the abstract would aid quick assessment.
[Main theorems] Ensure that the domain of A and the essential smoothness condition are stated with reference to standard definitions in convex analysis to avoid ambiguity for non-specialists.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We are grateful to the referee for their detailed summary of our manuscript and for highlighting its significance in providing nearly matching necessary and sufficient conditions for the LLDP via a relaxed version of the Gärtner-Ellis theorem. We appreciate the recommendation for minor revision and will ensure the manuscript is polished accordingly. No major comments requiring specific responses were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; equivalence proven from definitions

full rationale

The paper establishes a necessary and sufficient condition for the local large deviation principle (LLDP) by showing that LLDP implies existence of the truncated cumulant limit A(mu) equal to the Legendre-Fenchel transform L_D, and conversely that existence of an essentially smooth A implies LLDP with rate function L_A. This equivalence is derived directly from the definitions of LLDP (the limit of T^{-1} ln P(|zeta_T - alpha| < epsilon_T) = -D(alpha)) and the standard convex-analytic properties of the Legendre-Fenchel transform, without any fitted parameters, self-definitional loops, or load-bearing self-citations. The 'slowly enough' conditions on epsilon_T and M_T are part of the statement and do not create circularity. The reference to a relaxed Gartner-Ellis theorem invokes a standard external result rather than an author-specific ansatz or uniqueness theorem. The derivation chain is self-contained against external benchmarks in probability theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard convex-analysis facts about the Legendre-Fenchel transform and on the definition of LLDP; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (1)

standard math Legendre-Fenchel transform is well-defined and satisfies standard duality properties for lower-semicontinuous convex functions
Invoked to equate A with L_D and D with L_A.

pith-pipeline@v0.9.0 · 5606 in / 1204 out tokens · 29290 ms · 2026-05-08T10:17:39.619951+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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On large deviation principles in metric spaces

Borovkov, A.\,A., Mogulskii, A.\,A. On large deviation principles in metric spaces. Siberian Math. J. 2010, 51 , 989--1003

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On large deviation principles for random walk trajectories

Borovkov, A.\,A., Mogulskii, A.\,A. On large deviation principles for random walk trajectories. I. Theor.\ Probab.\ Appl. 2011, 56 , 538--561

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

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