Sharp LDPs: Asymptotic Estimates

• Sharp LDPs are refined large-deviation principles that quantify not only exponential decay rates but also explicit polynomial or complex correction terms in rare-event probabilities.
• They employ advanced tools such as saddle-point analysis, local CLT, and spectral methods to derive precise prefactor corrections across various models including i.i.d. sums and dynamical systems.
• Applications include improved tail risk assessment, efficient computation in rare-event simulation, and sharper confidence interval calibration in statistical estimation.

Sharp large-deviation asymptotic estimates ("sharp LDPs") provide refined probabilistic descriptions of rare events, quantifying not only the exponential rate functions but also the crucial polynomial (or other explicit) prefactors arising in large deviations theory. These results represent a fundamental tool for analyzing tail probabilities, extremal statistics, risk measures, and statistical confidence levels in both classical and high-dimensional probability, stochastic processes, and dynamical systems.

1. Fundamental Principles and Formalisms

The prototypical large-deviation principle (LDP) for a sequence of random variables $\{X_n\}$ has the form:

$\mathbb{P}\{X_n \in A\} \sim e^{-n I(A)},$

where $I(A)$ encodes the exponential rate of decay, often found as the Legendre transform of a cumulant generating function.

Sharp LDPs refine this by supplying leading-order polynomial or more intricate corrections:

$$\mathbb{P}\{X_n \in A\} = C_n(A) \, e^{-n I(A)} [1 + o(1)],$$

where the prefactor $C_n(A)$ depends on the underlying geometry, local curvature of the rate function, and, in many cases, deeper structural properties (e.g., directionality or nonlattice/Markov/dynamical features). In classical cases, $C_n(A)$ often scales as $n^{-1/2}$, but more intricate settings exhibit polylogarithmic or nonuniversal corrections.

2. Classical and Higher-order Expansions (Cramér–Bahadur–Rao Type)

In the i.i.d. nonlattice setting, sharp expansions have the canonical form:

$$\mathbb{P}\left\{S_n \geq a n\right\} \sim \frac{c(a)}{\sqrt{2\pi n}} \exp(-n I(a)),$$

where $I(a)$ is as above and $c(a)$ is expressed via derivatives of the log mgf at the saddle point. These expansions are universal for i.i.d. sums, finite-state Markov chains, and expanding dynamical systems admitting spectral gaps, provided suitable analyticity and nondegeneracy conditions (“Nagaev–Guivarc’h hypotheses”) are met (Fernando et al., 2018).

Incomplete table of model settings and sharp prefactors:

Model Type	Rate Function %%%%10%%%%	Prefactor Structure
i.i.d. nonlattice	Legendre of log mgf	$n^{-1/2}$ Bahadur–Rao
Finite-state Markov chain	Spectral log-eigenvalue max	$n^{-1/2}$ + higher-order
Expanding dynamical system	Spectral coding via Ruelle operator	$n^{-k-1/2}$ polynomial

More generally, the strong expansion takes the form:

$$\mathbb{P}(S_n \geq a n) e^{I(a) n} = \sum_{k=0}^{r/2} D_k(a) n^{-(k+1/2)} + O(n^{-(r+1)/2}),$$

where $D_k(a)$ are explicit polynomials in cumulant and spectral derivatives (Fernando et al., 2018).

3. Sharp LDPs for Sums with Constraints: Bounds and Prefactor Effects

For sums of independent, bounded random variables $\{\xi_{i,n}\}$, the Fan–Grama–Liu theorem establishes:

$$\mathbb{P}\left(S_n \geq x \sigma_n\right) = \left[\Theta(x) + \theta \varepsilon_n(x)\right] \exp(-n \Lambda_n^*(x \sigma_n / n)),$$

where $\Theta(x) = (1 - \Phi(x)) e^{x^2/2}$ (with $\Phi$ standard normal cdf), and $\varepsilon_n(x)$ admits fully explicit upper bounds given Lyapunov-type conditions. These bounds sharpen Talagrand’s inequalities and generalize Bahadur–Ranga Rao, producing nearly optimal error estimates (Fan et al., 2012).

4. Heavy-tailed and One-Big-Jump Regimes

For heavy-tailed random variables ($\alpha$-index, slowly-varying $L$), sharp expansions resolve the “one-big-jump” phenomenon. Vogel’s results show:

$$P(S_n > x) = n P(X_1 > x)\left[1 + \varepsilon_n(x)\right],$$

with $\varepsilon_n(x)$ explicitly controlled via local deviation, the maximum, and slow variation errors. Conditioned on $\{S_n > x\}$, the law is close (in total variation) to a single large summand plus $n-1$ typical ones (Vogel, 2022).

5. Large Deviations for Parameter Estimators in Stochastic Processes

Maximum likelihood estimators (MLEs) for parameters in stochastic processes, notably the Ornstein-Uhlenbeck (OU) drift estimator, exhibit regime-dependent sharp LDPs.

For the OU diffusion (Bercu et al., 2011), three regimes arise:

• Stable ($\theta < 0$), Unstable ($\theta = 0$), Explosive ($\theta > 0$):
  • Rate function and polynomial prefactor ($T^{-1/2}$) differing by regime.
  • Explosive case features a flat valley in $I(c)$ and regime-dependent corrections (e.g., $T^{-1/4}$ at critical points).
• Analogous expansions hold for the shifted-OU drift and shift MLEs, with the shift estimator having an implicit rate function via contraction principles and saddle-point analysis (Bercu et al., 2013).

Explicit formulae for the drift estimator (Bercu et al., 2013):

$$\Pr\{\theta_T \geq c\} = \frac{e^{-T I_{\rm drift}(c) + J(c)}}{a_c \sigma_c \sqrt{2\pi T}} [1 + o(1)],$$

with regime-split expressions for $a_c$, $\sigma_c^2$, and $J(c)$.

6. Sharp LD for Extrema and High-dimensional Projections

For extremal statistics (Gaussian maxima) (Zapata, 20 Dec 2025):

$$\mathbb{P}(Z_n > x) = n^{-I(x) + o(1)}, \quad I(x) = x + x^2/4,$$

where $Z_n$ is double-normalized maximum, yielding significantly better tail approximations than Gumbel-type expansions.

For high-dimensional random projections (e.g., $\ell_p^n$ spheres and balls), sharp LDPs feature direction-dependent prefactors and distinguish convex bodies beyond the universal exponential rate (Liao et al., 2020). The full asymptotic form involves curvature of the rate-function surface and higher-order corrections, leading to geometric identification in empirical scenarios.

7. Rare-event Analysis in Dynamical Systems and Portfolio Models

Sharp LD estimates extend to dynamical systems (Axiom A flows), with exact polynomial prefactors via Tauberian and spectral methods for exponentially small windows in time-integrated observables (Petkov et al., 2020). In portfolio credit risk, threshold models reveal multiple universality classes (Gaussian/exponential, power-law, bounded-support), precise Gibbs conditioning phenomena, and prefactor scaling determining the sharpness and effective risk measures (e.g., $n^{-1/2}$, $n^{-3/2}$, index-driven scaling) (Deng et al., 23 Sep 2025).

8. Infinitely Divisible and Poisson/Levy Structures, Dickman-type Asymptotics

Sharp LDPs for infinitely-divisible laws with Levy measures on $[0,1]$ reduce to saddle-point + local CLT + untilt mechanisms, yielding

$$f(x) \sim (2\pi \sigma_\beta^2)^{-1/2} \exp\{K(\beta) - \beta x\}.$$

This framework encompasses Dickman function asymptotics, with $O(1/x)$ or $O(1/\sqrt{x})$ relative errors depending on domain regularity and arrival intensity (Arratia et al., 2016).

9. Applications and Statistical Implications

• Tail risk assessment: Exponential decay and polynomial prefactor enable precise quantification of extreme-event probabilities in estimation and risk models.
• Efficiency and confidence intervals: Prefactor corrections refine classical bounds, e.g., coverage error calibration for parameter intervals (Bercu et al., 2013).
• Numerical schemes: Importance sampling and saddle-point methods utilize sharp LDP structure for efficient rare-event computation (Liao et al., 2020).
• Conditioning principles: Gibbs-type total-variation convergence under rare events clarifies loss distributions and default behaviors (Deng et al., 23 Sep 2025).

10. Outlook and Transferable Techniques

Sharp LDPs across disparate domains rely on measure tilting (Esscher/Cramér transforms), saddle-point analysis, spectral asymptotics, and uniform control of local fluctuations (CLT, Berry–Esseen, Edgeworth). Their explicit prefactors are essential for practical risk assessment, extreme-value theory, and statistical estimation at rare-event scales. Extensions to multivariate regimes, time-dependent processes, and systems with intricate dependence continue to expand the reach and utility of sharp large-deviation techniques.

Key references: (Bercu et al., 2013, Bercu et al., 2011, Fan et al., 2012, Vogel, 2022, Fernando et al., 2018, Zapata, 20 Dec 2025, Liao et al., 2020, Arratia et al., 2016, Buraczewski et al., 2014, Petkov et al., 2020, Deng et al., 23 Sep 2025, Fill et al., 2019, Grafke et al., 2021, Arguin et al., 2022).