Precise Large Deviations

• Precise Large Deviations is a theory providing complete asymptotic tail expansions, incorporating exponential rates, polynomial pre-factors, and additive constants for rare events.
• It is applied to diverse stochastic models such as heavy-tailed sums, random matrix products, and interacting particle systems to offer sharp probability estimates.
• The methodology employs techniques like change-of-measure, Tauberian analysis, and spectral methods to reveal dominant mechanisms such as single big jumps and mod-φ convergence.

A precise large deviation principle gives an asymptotic expansion—not only of the exponential decay rate, but including the explicit polynomial prefactor and additive constants—of the tail probabilities of sums, maxima, functionals, or other observables in stochastic models as some parameter (typically the sample size, time, or system size) tends to infinity. Unlike classical large deviation principles, which primarily identify the exponential rate function, precise large deviation theory specifies the full asymptotic expansion and reveals the mechanisms (e.g., single big jumps, spectral features, or combinatorial structures) that dominate at large deviations. This level of detail is crucial for quantitative sharpness in applications ranging from heavy-tailed processes to random matrix theory and interacting particle systems.

1. Foundational Framework and Definitions

A precise large deviation expansion takes the generic form

$$\mathbb{P}[S_n>x_n] = C\,n\,\overline{F}(x_n)\,[1+o(1)]$$

or, for lattice or exponentially decaying settings,

$$\mathbb{P}[S_n > x_n] = A(n,x_n)\,\exp\{-n\,I(x_n/n)\}\,[1+o(1)]$$

where $C$ and $A(n,x_n)$ are explicit constants and pre-factors, $\overline{F}$ denotes a regularly or subexponentially varying tail, and $I(\cdot)$ is a rate function determined by the (generally tilted) cumulant generating function of the model.

Precise large deviations connect three major frameworks:

• Classical Cramér–Bahadur–Rao–Petrov theory: Precise expansions for i.i.d. or weakly dependent sums, specifying exact constants beyond rate functions (Buraczewski et al., 2016, Xiao et al., 2019).
• Regular variation and big-jump principle: In subexponential or regularly varying settings, maximal terms dominate, and precise expansions are determined through Tauberian theory, revealing the explicit role of the dominating variable (Cristadoro et al., 4 Jul 2024).
• Mod-$\phi$ convergence: Unified expansions for a broad class of random variables, encode not only the leading exponential decay and the quadratic Gaussian regime but the full pre-factor structure via the residue function (Féray et al., 2013).

2. Sharp Asymptotics in Classic and Modern Models

2.1 Sums of Independent and Dependent Heavy-Tailed Variables

For i.i.d. $X_i$ with regularly varying (or more general subexponential) tail $\mathbb{P}(X_1 > x) \sim L(x)\,x^{-\alpha}$, precise large deviations yield exact uniform expansions for $S_n = \sum_{i=1}^n X_i$: $\mathbb{P}(S_n > x) \sim n\,\mathbb{P}(X_1 > x)$ uniformly for $x$ growing faster than $n^{1/\alpha}$, with analogous principles in the presence of weak dependence or Markov structure under anti-clustering and small-jump negligibility conditions (Mikosch et al., 2012, Mikosch et al., 2020).

For dependent structures (e.g., stationary sequences, autoregressive models, Markov chains, or processes with memory kernels), extensions of the Nagaev principle hold. Anti-clustering conditions (to control local dependence in the exceedance structure) and the domination of the "single big jump" apply, leading to

$\mathbb{P}(S_n > x) \sim c\,n\,\mathbb{P}(X_1 > x)$

with a model-dependent $c$ (Mikosch et al., 2012, Mikosch et al., 2020).

2.2 Random Sums and Cluster Processes

For random sums $S_{N(t)} = \sum_{i=1}^{N(t)} X_i$ with a counting process $N(t)$ (possibly with weak dependence, infinite mean, or slowly varying increments), precise large deviation expansions generalize as: $$\mathbb{P}(S_{N(t)} > x) \sim \mathbb{E}[N(t)]\,\mathbb{P}(X_1 > x)$$ uniformly for $x$ above the relevant threshold, even without concentration of $N(t)$ (Cristadoro et al., 4 Jul 2024, Zhang et al., 2015).

Marked Poisson cluster processes and Hawkes processes with multivariate regular variation in the governing components admit explicit, uniform expansions for both maxima and sums: $\mathbb{P}(S_T > x) \sim \nu\,T\,\mathbb{P}(S > x)$ where $\nu\,T$ is the mean number of clusters and the single-cluster tail has a precise asymptotic governed by a big-jump from the underlying mark or cluster size (Baeriswyl et al., 2023, Wang et al., 2023).

2.3 Subcritical Branching with Immigration

For subcritical branching processes with immigration, if either offspring or immigration is regularly varying, the total population sum $S_n$ satisfies: $$\mathbb{P}(S_n - \mathbb{E}S_n > x) \sim c\,n\,\mathbb{P}(\text{dominating variable} > x)$$ with $c$ depending explicitly on the reproductive and immigration means, uniformly for $x$ in a diverging regime. The distributional mechanism is a single exceptional reproduction or immigration event in one generation, confirming the extended one-big-jump principle (Guo et al., 7 May 2024).

3. Precise Large Deviations in Exponential and Mod-$\phi$ Settings

Classical Cramér/Petrov expansions,

$$\mathbb{P}\left(S_n > n\rho\right) \sim \frac{1}{\alpha\,\sigma(\alpha)\sqrt{2\pi n}}\,e^{-nI(\rho)}$$

give not only the exponential rate $I(\rho)$ (Legendre–Fenchel dual of the cumulant-generating function) but also the pre-exponential prefactor, and similarly for first-passage problems and products of random matrices (Buraczewski et al., 2016, Xiao et al., 2019).

The mod-$\phi$ convergence framework unifies precise large deviations and moderate deviations, providing sharp expansions for both lattice and non-lattice distributions: $$\mathbb{P}[X_n \geq t_n x] = \exp(-t_n F(x))\,\frac{\psi(h)^{-1}}{h\sqrt{2\pi t_n \eta''(h)}}[1+o(1)]$$ where $h$ is the solution to $\eta'(h) = x$ and $\psi(h)$ is the analytic residue function characterizing deviation from the infinitely divisible reference (Féray et al., 2013, Fenzl et al., 2020, Wang et al., 2023, Peng et al., 13 Dec 2025). This framework applies across deterministic sums, random combinatorial structures, and determinantal point processes.

4. Technical Frameworks: Tauberian and Spectral Methods

Uniform Tauberian theorems: The Laplace-Stieltjes transform's behavior for a family of distributions, under regular variation and a uniform Karamata-type assumption, yields uniform precise asymptotics for a broad class of sums—including stable basins of attraction, sums with memory kernels, and randomly stopped sums without expectation constraints (Cristadoro et al., 4 Jul 2024).

Spectral methods for random matrix products and branching processes in random environments employ change of measure, spectral gaps for tilted operators, and saddle-point approximations to derive Petrov-type expansions with explicit rate functions and polynomial prefactors (Xiao et al., 2019, Buraczewski et al., 2017).

Cluster expansions in statistical mechanics and interacting particle systems deliver explicit expansions for probabilities of the number of particles, including all variance corrections and error terms (Scola, 2020).

5. Universality, Extensions, and Applications

Universality: The structure of precise large deviations—exponential decay, explicit pre-factors, and residue/correction terms—emerges in diverse models, including random partitions, combinatorial models (Ewens-Pitman and subgraph counts), determinantal processes, and random walk in random environments (Féray et al., 2013, Peng et al., 13 Dec 2025, Eichelsbacher et al., 2016, Byun et al., 20 Oct 2025, Buraczewski et al., 2017).

Insurance, risk theory, and finance: Precise large deviations quantify ruin probabilities, aggregate claims, and reinsurance risks in heavy-tailed or dependent models, extending Cramér–Lundberg theory to models with extended negative dependence and arbitrary dependence structures (Cui et al., 2021, Chen et al., 2023).

Statistical and random matrix applications: The behavior of sample covariance maxima, entanglement entropy, and spectral extremes in large dimensions or high-density regimes depends crucially on the precise large deviation expansions (Mikosch et al., 2020, Eichelsbacher et al., 2016, Byun et al., 20 Oct 2025).

Diffusion processes: Large deviations of empirical measures for diffusions in unbounded domains are governed by detailed spectral gap, Lyapunov, and duality methods, with precise rate functionals capturing both symmetric (Fisher information) and antisymmetric (irreversible) dynamics (Ferré et al., 2019).

6. Mechanisms and Proof Techniques

A unified view of proof strategies encompasses:

• One-big-jump principle: In heavy-tailed or subexponential settings, the dominant contribution arises from a single extreme summand; anti-clustering guarantees the tail equivalence to a single-site probability.
• Change-of-measure and exponential tilting: In exponential-tail or Cramér settings, the tilted probability measure renders the rare event typical, enabling Laplace method and saddle-point analysis for both sums and first-passage times.
• Tauberian and Karamata theory: Transforms (Laplace or generating functions) encode heavy-tail behavior, and precise regular variation translates through inversion to sharp asymptotics (Cristadoro et al., 4 Jul 2024, Baeriswyl et al., 2023).
• Spectral theory: Tilted or twisted transfer operators (Perron–Frobenius framework) reveal exponential rates and subleading terms for products of random matrices and functionals with Markovian structure (Xiao et al., 2019, Buraczewski et al., 2017).
• Mod-$\phi$ and cumulant approaches: Unified expansions from moment/cumulant control via dependency graphs, establishing normality zones and the transition to large deviation tails (Féray et al., 2013).

7. Outlook and Open Directions

The precise large deviation paradigm continues to evolve in several directions:

• Generalization to infinite mean, non-concentrated random index sums, and models with renewal, Markovian, or more complex dependent structure (including branching in random environment and Hawkes processes) (Cristadoro et al., 4 Jul 2024, Buraczewski et al., 2017, Wang et al., 2023).
• Development of precise local large deviation and moderate deviation results in high-dimensional combinatorial and statistical mechanics models via cluster expansions (Peng et al., 13 Dec 2025, Scola, 2020).
• Mechanistic insights into tail concentration: the dichotomy between big-jump regimes (heavy-tail) and collective deviations (light-tail/exponential), and explicit phase transitions in rate function curvature (e.g., in Ewens–Pitman model at $\alpha=1/2$) (Peng et al., 13 Dec 2025).
• Comprehensive understanding of the universality class for second-order and precise deviation phenomena arising from mod-$\phi$ convergence and related analytic frameworks (Féray et al., 2013).

In summary, precise large deviation theory supplies fully sharp asymptotic expansions for probabilities of rare events in a wide class of stochastic models, quantifying the transition from Gaussian/typical fluctuations to the domain governed by rare, extreme deviations, with broad applications and a unifying set of techniques that continue to expand the frontiers of large deviation analysis.