Chung’s Law of the Iterated Logarithm

• Chung’s Law of the Iterated Logarithm is a precise formulation that defines the almost-sure lower bounds for the oscillatory behavior of various stochastic processes.
• It generalizes the classical Brownian motion case by leveraging small-ball probability estimates, chaining arguments, and entropy methods to address processes like Lévy and Gaussian fields.
• The law underpins the analysis of sample path regularity and multifractal structures, offering actionable insights for assessing oscillation scales in SPDEs and other non-Gaussian settings.

Chung’s Law of the Iterated Logarithm (LIL) identifies the almost-sure lower envelope for the pathwise oscillations of stochastic processes, giving precise rates at which the supremum of the process—over shrinking intervals or at different time scales—decays, modulo log-logarithmic correction terms. The “Chung-type” LIL generalizes the classical formulation from Brownian motion to a wide spectrum of stochastic processes, including Lévy processes, Feller processes, Gaussian fields, random fields on manifolds, and solutions to stochastic partial differential equations (SPDEs). These sharper LILs are inextricably linked to precise small-ball probability estimates, metric entropy theory, and measure-concentration tools, and they anchor the assessment of sample path regularity, modulus of continuity, and multifractal structures.

1. Classical Statement and Historical Context

For standard one-dimensional Brownian motion $B_t$, the original Chung’s LIL describes the minimal fluctuation scale of $|B_s|$ as $s$ approaches infinity or zero:

$$\liminf_{t\to\infty} \frac{ \sup_{0 \leq s \leq t} |B_s| }{ \sqrt{ 2 t \ln\ln t } } = \frac{1}{\sqrt{\pi}} \quad\text{a.s.}$$

and, equivalently for small $t$,

$$\liminf_{t\downarrow0} \frac{ \sup_{0 \le s \le t}|B_s| }{ \sqrt{\frac{\pi^2 \, t}{8\, \log |\log t|}} } = 1 \quad\text{a.s.}$$

This form highlights the presence of the iterated logarithm, $\log\log t$, as the critical second-order correction, distinguishing Chung’s law from the classical Hartman–Wintner LIL, which features a $\log t$ correction but describes the upper path envelope (Chen, 2023, Tripathi, 2024, Carfagnini et al., 2020).

2. Foundational Techniques: Small Ball Probabilities and LIL

Chung-type LILs fundamentally rely on precise asymptotics for small-ball probabilities:

$$\Pr\left\{ \sup_{0\leq s \leq t} |X_s| < \varepsilon \right\} \sim C\, \exp\left( -\Lambda\, \frac{ t }{ \varepsilon^{\vartheta} } \right) \quad (\varepsilon \downarrow 0)$$

The exponent $\vartheta$ and constant $\Lambda$ are process-specific and determined by the infinitesimal generator or covariance structure. Key probabilistic tools in establishing the LIL are:

• Chaining and entropy arguments for upper small-ball bounds.
• Local nondeterminism and covering theorems for lower bounds.
• Two-scale Borel–Cantelli arguments for almost-sure liminf behavior.
• Zero–one (tail) laws to ensure almost-sure constancy of limiting objects (Aurzada et al., 2010, Knopova et al., 2013, Richard, 2014, Wang et al., 2021, Carfagnini et al., 2024, Lee et al., 2021).

3. Chung-Type LILs for Lévy-Type and Feller Processes

For pure-jump Lévy processes or spatially inhomogeneous Feller processes, the norming function in the Chung-LIL is explicitly constructed from the symbol $p(x, \xi)$ of the infinitesimal generator:

• For a Feller process $(X_t)$ on $\mathbb{R}$ with symbol $p(x, \xi)$,

$$u(x, R) := \left[\inf_{|y-x|\le 3R} p^{U}(y, 1/R)\right]^{-1}$$

where $p^U$ reflects the jump intensity.

• The scale function is

$R(t) = u^{-1}(x, t / \log|\log t| )$

• The Chung-LIL at zero then reads

$$\limsup_{t\to 0} \frac{ \sup_{0\le s \le t} |X_s - x| }{ u^{-1}(x, t / \log|\log t|) } = C(x)$$

For symmetric stable processes with index $\alpha \in (0,2)$, $u(R) \sim R^{\alpha}$, so the normalization is $(t / \log|\log t|)^{1/\alpha}$ (Knopova et al., 2013, Aurzada et al., 2010).

In the case of general Lévy processes, the approach of Aurzada–Döring–Savov provides a direct translation of small-deviation asymptotics into Chung-LILs. Given

$$-\log \Pr( \| X \|_t < \varepsilon ) \sim \lambda F(\varepsilon) t$$

for a regularly varying $F$, Chung-LIL uses the scaling

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

and almost surely,

$$\liminf_{t\to0} \frac{ \|X\|_t }{ b_{\lambda}(t) } = 1$$

(Aurzada et al., 2010, Knopova et al., 2013).

4. Chung-Type LIL for Gaussian and Anisotropic Fields

Chung-LILs have been rigorously established for broad classes of Gaussian random fields—including anisotropic fields, fractional Brownian motion variants, and fields indexed by space-time or manifolds.

• For anisotropic Gaussian fields $v(x)$ indexed by $T \subset \mathbb{R}^k$ with quasi-metric $A(x,y) = \sum_j |x_j - y_j|^{\alpha_j}$ and overall index $Q = \sum_j 1/\alpha_j$, the normalization is

$h(r) = r ( \log\log(1/r) )^{-1/Q}$

and

$$\liminf_{r \to 0} \sup_{A(x,x_0)\le r} \frac{ |v(x) - v(x_0)| }{ h(r) } = C^{1/Q}$$

for explicit $C$ (Lee et al., 2021).

• In multiparameter fractional Brownian motion of Hurst index $h$ and parameter-dimension $v$, with covariance as in Richard (Richard, 2014),

$q^{(e)}(r) = r^{vh} ( \log\log(1/r) )^{-h/v }$

and

$$\liminf_{r \downarrow 0} \frac{ M(r) }{ q^{(e)}(r) } \in [ K_2^{h/v}, K_1^{h/v} ]$$

Small-ball exponents depend on both the spatial Hausdorff dimension and local roughness.

• Isotropic space-time Gaussian fields on $\mathbb{S}^2 \times \mathbb{R}$, with covariance dictated by the decay of the angular power spectrum and memory parameter, yield

$$\phi(r) := \left( \frac{ \log |\log r| }{ r^3 } \right)^{p }, \quad p = \left( \frac{2}{\beta} + \frac{4}{\alpha-2} \right)^{-1}$$

so that

$$\liminf_{ r \to 0 } \phi(r) M_r(x, t ) = \kappa_{\alpha, \beta } \quad \text{a.s.}$$

(Carfagnini et al., 2024).

5. Chung-Type LIL for SPDEs and Non-Gaussian Settings

Chung-LILs extend beyond Gaussian processes, with significant results for solutions to stochastic partial differential equations:

• For stochastic heat equations with multiplicative noise, under global conditions on the nonlinearity $\sigma$ and for the mild solution $u(t,x)$, via “truncation and freezing” techniques, one obtains

$$\liminf_{ r \downarrow 0 } \sup_{ 0\leq t < r^4,\, x \in [0, r^2] } \frac{ |u(t,x)| }{ r [ \ln\ln (1/r)]^{1/6} } = K$$

where $K = \sigma(0) K_0$, $K_0$ is the Gaussian constant from the linearized equation (Chen, 2023).

• For SPDEs with fractional/noisy drivers, the anisotropy and fractional indices tune the normalization rates (see (Lee et al., 2021) for the fractional-colored noise model).

6. Generalized Fractional Brownian Motion, Functional Extensions, and Integral Criteria

Sophisticated functional and lower-class Chung-type LILs have been proved for generalized/fractional Brownian motions and their integrals:

• For generalized (Riemann-type or Pang–Taqqu) fBm/integrals, under self-similarity index $H$ and integral order $\theta$,

$$\liminf_{ t \to 0+ } \frac{ \sup_{0 \leq s \leq t} |Y(s)| }{ t^{H+\theta} / ( \ln \ln (1/t) )^\theta } = K_1$$

where $K_1$ is determined by small-ball exponents via precise integral criteria (Lyu et al., 2024, Wang et al., 2021).

• Functional Chung-LILs describe entire appropriately normalized paths (not just pointwise maxima) converging to extremal "border" functions or into balls in reproducing kernel Hilbert spaces (Varron, 2012, Richard, 2014).

Integral tests (in the sense of Talagrand) characterize the precise almost-sure lower (and upper) class for functionals of such processes, sharpening the liminf-type statements to exact thresholds on the normalization function.

7. Variants and Applications: Nonlinear Expectations, Groups, and Diffusions

• Under sublinear (nonlinear) expectations, such as in $G$-Brownian motion, there is no unique variance; Chung-type LIL yields capacity-one results for both upper and lower variances, explicitly quantifying the impact of model ambiguity (Zhang, 2015).
• For hypoelliptic Brownian motion on the Heisenberg group or iterated Kolmogorov diffusions, the liminf scaling and the associated constants in the LIL are governed by Dirichlet eigenvalues of the corresponding Laplacian, or follow from the dominance of specific (e.g., “lowest order”) process coordinates (Carfagnini et al., 2020, Carfagnini, 2021).
• Empirical and quantile processes, and their functionals, admit functional Chung-LILs characterizing clustering rates and moduli of the deviation from boundary (Strassen) sets, with rates determined by Talagrand-type, small-ball probability, and strong approximation tools (Varron, 2012).

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The general paradigm is summarized in the following table:

Process/Field Type	Normalization (LIL)	Reference
Brownian motion	$\sqrt{2t\ln\ln t}$	(Chen, 2023)
Symmetric $\alpha$-stable	$( t/\log|\log t| )^{1/\alpha}$	(Knopova et al., 2013)
General Lévy process	$F^{-1}( \log|\log t| / \lambda t )$	(Aurzada et al., 2010)
Multiparameter fBm	$r^{vh}/(\log\log (1/r))^{h/v}$	(Richard, 2014)
Anisotropic Gaussian field	$r(\log\log(1/r))^{-1/Q}$	(Lee et al., 2021)
Generalized fBm/fractional int	$t^{H+\theta}/(\ln\ln(1/t))^\theta$	(Lyu et al., 2024)
Spherical isotropic field	$(\ln|\ln r|/r^3)^p$ (with $p$ from spectral decay)	(Carfagnini et al., 2024)
SPDE mild solution	$r\,(\ln\ln(1/r))^{1/6}$	(Chen, 2023)
Heisenberg group BM	$\sqrt{ t/\log\log t }$	(Carfagnini et al., 2020)

Chung’s Law of the Iterated Logarithm thus provides a robust machinery for linking sample path fluctuations, entropy, and small-ball probabilities across a broad variety of probabilistic models, with extensions spanning Gaussian fields, Lévy and Feller processes, SPDEs, empirical processes, and domains with non-Euclidean or group-valued structure. This machinery is essential for the fine analysis of stochastic regularity, multifractality, and pathwise phenomena in probability theory and stochastic analysis.