Chung-type laws of the iterated logarithm for $m$-fold weighted integrated fractional processes

Abstract: Let ${B_H(t);t\ge 0}$ be a fractional Brownian motion of order $H\in (0,1)$, and $J_{m,α}(B_H)$ be the $m$-fold weighted integrals of $B_H$ defined as $$ J_{m,\bmα}(B_H)(t) =\int_0^ts_m^{-α_m}\int_0^{s_m}\cdots s_2^{-α_2}\int_0^{s_2}s_1^{-α_1}B_H(s_1)d s_1\; ds_2\cdots d s_m, $$ where $α1+\cdots+α_i<H+i$, $i=1,\ldots,m$, $\bmα=\bmα_m=(α_1,\ldots,α_m)$. We show that \begin{align*} \liminf{T\to \infty} \frac{(\log\log T)^{H+m}}{T^{H+m-α}}\sup_{0\le t\le T}\left|\frac{ J_{m,\bmα}(B_H)(t)}{t^{α-α_1-\cdots-α_m}}\right| = a_H\left( \frac{κ{H+m}}{1-α/(H+m)}\right)^{H+m}\;\; a.s. \end{align*} for all $α<H+m$, and \begin{align*} \liminf{T\to \infty} & \sqrt{\frac{\log\log\log T}{\log T}} \sup_{1\le t\le T}\left|\int_1^t \frac{J_{m-1, \bmα{m-1}}(B_H)(s)}{s^{H+m-α_1-\cdots-α{m-1}}}ds\right| &= \fracπ{2}\frac{\sqrt{β(2H,1-H)}}{\prod_{i=1}^{m-1}\big(H+i-α_1-\cdots-α_i\big)}\;\; a.s., \end{align*} where $a_H$ is an explicit constant with $a_{\frac{1}{2}}=1$, $κλ$ is a constant which depends only on $λ$, and $β(a,b)$ is the beta function.In particular, the exact value of a Chung-type law of the iterated logarithm established by Duker, Li and Linde (2000) is found, and as an application, the Chung-type law of the iterated logarithm for the randomized play-the-winner rule is established. The small ball probabilities of (J{m, \bmα}(B_H)) are established to show the liminf behaviors. Similar Chung-type laws of the iterated logarithm and small ball probabilities for a Riemann-Liouville fractional process are also established.

• The paper derives Chung-type laws for m-fold weighted integrated fractional processes, establishing precise liminf limits with explicit constants.
• Advanced Gaussian process techniques and entropy analysis are employed to obtain sharp asymptotic results for small ball probabilities.
• Results resolve longstanding conjectures and extend applications to integrated Brownian motion and randomized urn models in complex stochastic settings.

Chung-Type Laws of the Iterated Logarithm for $m$-Fold Weighted Integrated Fractional Processes

Overview

This paper delivers a comprehensive and rigorous study of Chung-type laws of the iterated logarithm (LIL) for $m$-fold weighted integrated processes derived from fractional Brownian motion (fBm), with Hurst exponent $H\in(0,1)$, and generalized Riemann–Liouville processes. The author presents precise almost sure liminf and limit laws for the suprema and functionals of $m$-fold weighted integrals of $B_H$, including sharp constants, extends previous results, and verifies longstanding conjectures concerning exact asymptotic values and small ball probabilities. The theoretical development employs advanced techniques from Gaussian process theory, self-similarity, small ball asymptotics, and the Gaussian correlation inequality.

Definitions and Main Objects

Let $\{B_H(t), t\ge 0\}$ be standard fractional Brownian motion. Consider the general $m$-fold weighted fractional integral operator acting on $B_H$: $$J_{m,\bm\alpha}(B_H)(t) = \int_0^t s_m^{-\alpha_m} \int_0^{s_m}\cdots s_1^{-\alpha_1}B_H(s_1)ds_1\cdots ds_m\,,$$ where $\bm\alpha = (\alpha_1, ..., \alpha_m)$ and for all $m$0, $m$1. Of particular interest is the supremum of the appropriately normalized process over large intervals, and the small ball probabilities under weighted norms.

The process admits a decomposition in terms of Riemann–Liouville integrals, allowing reduction to self-similar Gaussian processes and facilitating the transfer of fine asymptotic results.

Summary of Main Results

Chung-Type LIL: Liminf Laws for Weighted Integrals

For $m$2, the paper establishes: $m$3 where $m$4 and $m$5 are explicit constants computed in the paper, and $m$6 is the beta function appearing in endpoint regimes. This generalizes and sharpens previous work, furnishing exact limit values, including cases where only existence or logarithmic orders were formerly established.

For the critical case $m$7, the classical (Chung-type) LIL fails, and a nonstandard normalization is shown to yield: $m$8

Small Ball Probabilities

Detailed asymptotics for small ball probabilities under weighted sup and $m$9 norms are derived for $H\in(0,1)$0, extending the scope of existing results for pure fBm and Brownian motion. The analysis utilizes entropy techniques and the Gaussian correlation inequality, leading to optimal rate and constant specifications for both sup and $H\in(0,1)$1 metrics. Precise equivalence between the small ball probabilities for weighted integrated fBm and that for related Riemann–Liouville processes is rigorously shown.

Applications and Special Cases

• The exact value of the Chung-type LIL constant for the integrated Brownian motion with general power weight is determined, resolving a longstanding open problem,
• Explicit Chung-LIL and small ball results for the randomized play-the-winner urn model are given, leveraging the established functional LILs,
• The paper includes refined results for the Riemann–Liouville fractional process, including joint laws for weighted integrals, and covers the endpoint transition scenarios.

Gaussian Approximation and Invariance Principles

Through an almost-sure invariance principle for functionals of stationary Gaussian processes (Pickands' and Shao's theorems), the paper relates the pathwise behavior of the $H\in(0,1)$2-fold weighted integrated processes to those of Brownian motion, allowing the transfer of sharp iterated logarithm constants.

Numerical Constants and Contrasts with Prior Work

The paper explicitly computes all normalizing and limiting constants, such as $H\in(0,1)$3, $H\in(0,1)$4, and variance functionals, in terms of special functions (gamma, beta). Where previous literature [e.g., "Small ball probabilities for integrals of weighted Brownian motion" (DLL 2000), "Existence of small ball constants for fractional Brownian motions" (Li & Linde 1998)] yielded only the order or bounds, this work settles the exact constants and limit values, and extends them to the fully weighted and iterated setting.

Additionally, the results demonstrate that the presence of weights and multiple iterations introduces new behaviors and normalization phenomena at critical parameter values, and the analysis identifies all such transition points.

Theoretical and Practical Implications

The results have several implications:

• The sharp LILs and small ball probabilities provide precise quantification for the modulus of continuity and small deviation properties of long-memory processes, relevant for the fine analysis of stochastic models in probability, functional analysis, and statistics,
• The explicit constants enable direct application to the assessment of error bounds and risk in statistical models based on fBm and iterated integrals,
• The approach reveals the universal structures in Gaussian processes governed by self-similarity and entropy, furnishing templates for further study in related classes such as Volterra and general Gaussian chaos,
• The application to urn models connects abstract probabilistic LILs and invariance principles with practical models in adaptive designs and randomization procedures in clinical trials.

Future Directions

Future research directions prompted by these results include:

• Extension to multi-parameter and operator-weighted fractional processes,
• Investigation of non-Gaussian and heavy-tailed analogs of the iterated LIL and small ball laws,
• Analysis of the extremal and large deviation behavior of similar weighted integral processes in high dimension and on manifolds,
• Application of these precise limit laws as benchmarks in the sharp assessment of regularity and controllability for stochastic PDEs driven by fBm.

Conclusion

This paper establishes definitive Chung-type LILs and small ball laws for $H\in(0,1)$5-fold weighted integrated fractional processes, precisely characterizing their liminf behavior, critical growth rates, and sharp constants in all regimes. Through technical innovations in Gaussian process theory, self-similarity, and entropy analysis, the work unifies and extends the field's understanding of the pathwise structure of weighted Gaussian processes, with direct implications for probabilistic analysis and statistical methodology.