On necessary and sufficient conditions for the local large deviation principle

Abstract: One says that the local large deviation principle (LLDP) is satisfied for a family of random vectors ${ζ<em>T}</em>{T\ge 0}$ in $\mathbb R<sup>d,$ $d\ge 1,$ if there exists a function $D:\mathbb R^d\to [0,\infty],$ $D\not \equiv \infty,$ such that, for any $α\in \mathbb R<sup>d$, [ \lim_{T\to \infty}T^{-1}\ln \mathbf{P} (|ζT -α|<\varepsilon_T)= - D(α)] 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(μ):= \lim{T\to\infty}T^{-1}\ln \mathbf{E} (e^{T\langle μ, ζ_T\rangle}; |ζ_T|\le M_T)\in (-\infty, \infty],\quad μ\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 Gärtner--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.

• The paper demonstrates that the existence of an essentially smooth WSFF is both necessary and sufficient for establishing the LLDP.
• It refines the classical LDP framework by relaxing exponential tightness and global integrability requirements using bounded Laplace transforms.
• The study illustrates practical applications in risk theory and stochastic processes by accurately characterizing local probabilities even with non-convex rate functions.

Necessary and Sufficient Conditions for the Local Large Deviation Principle

Introduction

The paper "On necessary and sufficient conditions for the local large deviation principle" (2604.22257) provides a rigorous analysis of conditions under which the Local Large Deviation Principle (LLDP) holds for families of random vectors in $\mathbb{R}^d$. The study critically examines the relationships between LLDP, the classical Large Deviation Principle (LDP), and the Gärtner–Ellis theorem, introducing the concept of the weak sense fundamental function (WSFF) and establishing its role as both a necessary and sufficient condition for LLDP.

Background: Large Deviations and Local Large Deviations

The classical LDP provides a framework for the asymptotic behavior of the probabilities of rare events by associating a rate function $D$ to a sequence of random vectors $\{\zeta_T\}_{T\ge 0}$ in $\mathbb{R}^d$. The LDP is typically characterized by upper and lower bounds on log-probabilities over Borel sets, involving the rate function $D$ evaluated on the interior and closure of those sets.

The LLDP, introduced to refine the interpretation of the rate function $D$, concerns the asymptotics of local probabilities, specifically those for $\zeta_T$ hitting small neighborhoods around points $\alpha$ in $\mathbb{R}^d$. The LLDP asserts that for $\epsilon\to 0$ slowly enough: $D$0 LLDP can be satisfied in cases where LDP fails due to non-exponential tightness or the presence of non-convex rate functions.

Main Results: WSFF as Necessary and Sufficient for LLDP

The Weak Sense Fundamental Function (WSFF)

The Gärtner–Ellis theorem traditionally provides sufficient conditions for the LDP based on the existence and smoothness of the so-called fundamental function (FF): $D$1 for $D$2 in an open neighborhood of $D$3. The FF is essentially smooth and convex.

The present paper generalizes this concept to the WSFF, which restricts the Laplace transform to bounded regions: $D$4 where $D$5 slowly. The WSFF exists if there is a convex function $D$6 with interior domain non-empty, replacing the restrictive requirement that expectations be finite everywhere.

Theorem Statements

The central theorem states:

(i) If LLDP is satisfied with rate function $D$7, then WSFF $D$8 (the Legendre–Fenchel transform of $D$9) exists.

(ii) If an essentially smooth WSFF $\{\zeta_T\}_{T\ge 0}$0 exists for $\{\zeta_T\}_{T\ge 0}$1, then LLDP holds with rate function $\{\zeta_T\}_{T\ge 0}$2.

These statements constitute necessary and sufficient conditions for LLDP, thereby relaxing the integrability and tightness requirements from the classical Gärtner–Ellis theorem.

Contradictory Claims and Non-Convex Rate Functions

A salient feature of the paper is the demonstration that LLDP can hold even in cases where the FF does not exist (due to divergent Laplace transforms), and where the rate function $\{\zeta_T\}_{T\ge 0}$3 is non-convex. Explicit examples provided include random vectors with probability mass escaping to infinity and Markov chains with non-convex $\{\zeta_T\}_{T\ge 0}$4. In such cases, the WSFF exists and is convex as required, while $\{\zeta_T\}_{T\ge 0}$5; $\{\zeta_T\}_{T\ge 0}$6 becomes the largest convex lower semicontinuous minorant of $\{\zeta_T\}_{T\ge 0}$7.

Implications and Theoretical Developments

Implications for Large Deviation Theory

The findings consolidate the use of WSFF in local large deviation regimes, underscoring that the behavior in "remote tails" is irrelevant for local probabilities. This aligns conditions for LLDP with operational necessities in statistical mechanics, statistical inference, and applied probability, where local probabilities are often of primary interest and integrability assumptions are frequently too restrictive.

By eliminating the need for exponential tightness and global integrability, the results enable broader applicability of LLDP analysis to random vector sequences (including those with heavy tails, non-standard normalization, or escapes to infinity), thus refining the framework for rare event estimation.

Practical Considerations

In applied contexts—e.g., risk theory, queuing, and stochastic processes—the ability to establish LLDP with only local behavior (bounded sets) substantially reduces technical overhead. This allows for more accurate modeling of distributions with pathological behavior at infinity, while still capturing key local asymptotics.

Speculation on Future AI Developments

The relaxation of conditions for LLDP could inform the development of probabilistic inference mechanisms in machine learning models, where understanding the distribution behavior in local neighborhoods proves critical for robust uncertainty quantification. The distinction between global and local deviation principles may become pivotal in analyzing complex generative models, calibration schemes, or simulation-based optimization where strict integrability fails.

Further advancements may include algorithmic exploitation of LLDP-based rate functions (via WSFF) for rapid rare event detection or in designing adaptive sampling methods where only local asymptotics are essential.

Conclusion

The paper rigorously establishes that the existence of an essentially smooth WSFF is both necessary and sufficient for the LLDP for families of random vectors in $\{\zeta_T\}_{T\ge 0}$8, thereby relaxing integrability and tightness constraints inherent to the classical LDP framework. The results clarify the operational boundary between global and local large deviation principles, extend applicability to non-convex and heavy-tailed distributions, and potentially open new avenues in statistical modeling and AI inference designs predicated on local asymptotics (2604.22257).