Chung-Type Laws of the Iterated Logarithm for Lévy Processes

• Chung-type LIL describes almost-sure scaling limits of small-time path fluctuations in Lévy processes using norming functions based on small ball probabilities.
• The approach exploits detailed small deviation estimates to link the local geometry of the Lévy triplet with the construction of effective norming functions.
• This framework unifies classical and exotic cases, enabling rigorous analysis of Brownian, stable, and jump processes with explicit logarithmic corrections.

Chung-type laws of the iterated logarithm (LIL) provide a precise almost-sure asymptotic description of the short-time path fluctuations for Lévy processes. The objective is to identify a norming (scaling) function $b(t)$ such that, almost surely, $\liminf_{t\to0} \left(\|X\|_t / b(t)\right) = 1$, where $\|X\|_t = \sup_{0 \leq s \leq t} |X_s|$. The approach directly exploits small deviation probability estimates (“small ball probabilities”), thereby linking the local oscillatory behavior of paths to the local geometry prescribed by the Lévy triplet. This framework enables derivation of Chung-type LIL results in both classical and exotic settings, with norming functions that may exhibit logarithmic or even more intricate corrections.

1. Definition, Background, and Classical Context

For a Lévy process $X$, the Chung-type LIL at zero seeks a norming function $b(t)$ such that

$$\liminf_{t\to0} \frac{\|X\|_t}{b(t)} = 1 \quad \text{a.s.}$$

This is in contrast to classical Kolmogorov/Hartman–Wintner-type LILs, which involve $\limsup$ scaling on large intervals. For Brownian motion, $b(t) = \sqrt{\pi^2 t / (8\log|\log t|)}$ provides the classical normalization, itself a consequence of time inversion and invariance principles. The small-time Chung-type LIL for general Lévy processes requires new techniques due to non-Gaussianity and the potential presence of jumps and asymmetry. Earlier results (Doney, Bertoin, Sato) provide partial answers for large-time or specific Lévy process classes; the present approach generalizes to broad settings using small deviation asymptotics.

2. Main Theorem and Construction of Norming Functions

The main result establishes direct transfer of sharp small deviation (small ball) estimates to Chung-type LIL analogues. Suppose for a non-decreasing function $F(\varepsilon)$ satisfying $F(\varepsilon)\to\infty$ as $\varepsilon\to 0$ and for $0<\lambda_1\leq \lambda_2<\infty$,

$$\lambda_1 F(\varepsilon) t \leq -\log \mathbb{P}(\|X\|_t < \varepsilon) \leq \lambda_2 F(\varepsilon) t \qquad (0 < \varepsilon \ll 1,\, t < t_0).$$

Define norming functions: $$b_\lambda(t) = F^{-1} \left( \frac{\log|\log t|}{\lambda t} \right),\quad \lambda > 0.$$ If a technical regularity (negligibility of small-jump drift; condition (1.7) of the paper) is satisfied, then almost surely,

$$1 \leq \liminf_{t\to 0} \frac{\|X\|_t}{b_{\lambda_1'}(t)} \quad \text{and} \quad \liminf_{t\to 0} \frac{\|X\|_t}{b_{\lambda_2'}(t)} \leq 1$$

for any $\lambda_1' < \lambda_1$ and $\lambda_2' > \lambda_2$. When $F(\varepsilon)\sim c\varepsilon^{-\alpha}$ (regular variation), one can select $\lambda$ so that $\liminf(\cdots) = 1$ exactly.

Table: Canonical Small Ball Rates and Norming Functions

Process	Small Ball Rate $F(\varepsilon)$	Norming Function $b(t)$
Brownian ($\sigma^2 > 0$)	$c \varepsilon^{-2}$	$\sqrt{\pi^2 t/(8 \log|\log t|)}$
Symmetric $\alpha$-stable	$c \varepsilon^{-\alpha}$	$(c_\alpha t/\log|\log t|)^{1/\alpha}$
Subordinator/Fristedt	Problem-specific (see paper)	Varies according to Lévy measure
Variance-gamma ($\mu=0$)	Exotic; see eqn (1.12) in the paper	$b(t) = \exp\{-\lambda \log|\log t|/t\}$ (trapped between constants)

In asymmetric or non-symmetric cases, different technical conditions (eqn:regularityofb) regulate the scaling, and the liminf is controlled up to multiplicative constants, not necessarily $1$.

3. Methodology and Analytical Structure

The analysis exploits the interplay between small deviation probabilities and path supremum norm asymptotics. Key methods:

• Rigorous small ball estimates via sharp asymptotics for $-\log\mathbb{P}(\|X\|_t<\varepsilon)$, based on recent advances (Aurzada & Dereich, 2009).
• Construction of norming functions as generalized inverses: $b_\lambda(t)=F^{-1}((\log|\log t|)/(\lambda t))$. Regular variation properties of $F$ facilitate invertibility and explicit expressions.
• Lemmas relating small ball decay to truncated second moment: for symmetric processes, $F(\varepsilon) = (1/\varepsilon^2) U(\varepsilon)$ with $$U(\varepsilon) = \sigma^2 + \int_{|x|<\varepsilon} x^2 \Pi(dx)$$ (truncated variance).
• Martingale techniques: Wald’s identity for subordinators, Doob’s maximal inequality, and decomposition of jump processes with truncation at various scales.
• Handling non-symmetric cases by imposing stricter drift regularity to ensure negligible impact of compensation in the small-time regime.

The approach unifies small deviation theory and strong pathwise fluctuation results, overcoming technical obstacles such as dependency and lack of finite-dimensional independence needed for naive Borel–Cantelli arguments.

4. Examples and Domain of Validity

The theoretical framework is illustrated in several canonical cases:

• Brownian motion / Lévy with nonzero $\sigma^2$: recovers classical normalization $b(t)=\sqrt{\,\pi^2t/(8\log|\log t|)}$ and shows $$\liminf_{t\to0} (\|X\|_t/ \sqrt{t/( \log |\log t|)}) = \pi\sigma/\sqrt{8}$$ a.s.
• Symmetric $\alpha$-stable Lévy: scaling $b(t)=(c_\alpha t/\log|\log t|)^{1/\alpha}$; matches asymptotics derived from classic works (Taylor).
• Processes with Lévy measure $\Pi(dx)\sim C|x|^{-(1+\alpha)}$: admits norming functions reflecting stable regime, or, for drift-dominated bounded variation, “linear” scaling.
• Subordinated processes (e.g., variance–gamma): for vanishing drift, norming functions can be “exotic” (e.g., exponentials of iterated logarithms); otherwise, for nonzero drift, linear behavior.

These show that the LIL laws extend to a wide range of jump processes, including those with complex or non-regularly varying Lévy measure tails and Brownian-subordinator mixtures.

5. Key Formulas Relating Small Deviations and LIL

Prominent formulas are:

• Small deviation (small ball) estimate:

$$-\log \mathbb{P}(\|X\|_t < \varepsilon) \asymp F(\varepsilon)t \qquad \text{as }\varepsilon,t\to 0$$

• Norming function (inverse):

$$b_\lambda(t) = F^{-1}\left(\frac{\log|\log t|}{\lambda t}\right)$$

• For symmetric processes:

$$F(\varepsilon) = \frac{1}{\varepsilon^2}U(\varepsilon),\quad U(\varepsilon) = \sigma^2 + \int_{-|x|<\varepsilon} x^2 \Pi(dx)$$

which captures the second moment contributed by small jumps.

• For Brownian component:

$b(t) = \sqrt{\pi^2 t/(8\log|\log t|)}$

These provide an explicit bridge from probability of small sup-norm excursions to the normalization in the LIL.

6. Implications, Generalizations, and Future Directions

• Unified methodology: By transferring small deviation asymptotics to path supremum laws, the paper builds a general machinery applicable to a wide class of Lévy processes.
• Sharp constants and "exotic" scaling: Regularly varying $F$ yield nearly optimal norming functions (with precise constants), but “exotic” normings may arise in subordinated or slowly varying tail cases.
• Explicit conditions: The Lévy triplet conditions needed (drift, Gaussian, jump measure) are made explicit and, in some cases, relax requirements compared to earlier literature, simplifying verification for a given process.
• Broader relevance: Because small deviation theory has applications in coding, geometric functional analysis, and stochastic approximation, these results create new links and potential advances in related fields.
• Extensions: The “small time at zero” regime is fully characterized; possible extensions include large time, multidimensional Lévy processes, and nonstandard jump structures. Further refinement might produce more explicit constants or rates, especially for processes with additional structure.

Conclusion

The Chung-type law of the iterated logarithm for Lévy processes at zero is obtained by exploiting precise small deviation estimates, allowing general transfer of “small ball” asymptotics to supremum path fluctuations. The principal innovation is the construction of scaling functions via inversion of the small deviation rate function, leading to broad unification of Chung-type results for Brownian motion, stable processes, subordinators, and more exotic classes, with explicit, often sharp, normings. The approach clarifies the central role of the Lévy triplet in determining small-time path oscillations and provides a robust and versatile platform for further probabilistic and analytic investigation.