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arxiv: 2604.01701 · v3 · pith:XVXJ6WBDnew · submitted 2026-04-02 · 🧮 math.PR

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

Li-Xin Zhang This is my paper

Pith reviewed 2026-05-13 21:21 UTC · model grok-4.3

classification 🧮 math.PR MSC 60F1560G2260G15
keywords Chung-type law of the iterated logarithmfractional Brownian motionweighted integralssmall ball probabilitiesliminfm-fold integrationGaussian processes
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The pith

m-fold weighted integrals of fractional Brownian motion satisfy exact Chung-type laws of the iterated logarithm with explicit constants almost surely.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves sharp liminf results for the supremum of m-fold weighted integrals of fractional Brownian motion B_H. For parameters satisfying the given restrictions, a normalized version of this supremum converges almost surely to an explicit constant involving a_H and κ_{H+m} as time T tends to infinity. A second result gives the liminf for an integrated version of the (m-1)-fold case using a different normalization with log log log T. The proofs rely on deriving small-ball probabilities for these integrated processes, which are then used to obtain the exact constants. The work also recovers the precise value of a 2000 Chung-type LIL and applies it to the randomized play-the-winner rule.

Core claim

The authors establish that liminf_{T→∞} (log log T)^{H+m} / T^{H+m-α} sup_{0≤t≤T} |J_{m,α}(B_H)(t) / t^{α-α1-⋯-αm}| equals a_H (κ_{H+m} / (1-α/(H+m)))^{H+m} almost surely for all α < H+m, under the conditions α1+⋯+αi < H+i for each i. They derive a parallel liminf result for the integral of the (m-1)-fold process that equals (π/2) sqrt(β(2H,1-H)) divided by the product of (H+i-α1-⋯-αi) terms. Small-ball probabilities for J_{m,α}(B_H) supply the tail estimates needed for both limits, and analogous statements hold for the Riemann-Liouville fractional process.

What carries the argument

The m-fold weighted integral J_{m,α}(B_H) obtained by successive weighted integrations of B_H, together with its small-ball probability estimates that convert tail decay rates directly into the precise liminf constants.

Load-bearing premise

The small-ball probabilities of the m-fold integrated process must take the precise asymptotic form required to produce the explicit liminf constant.

What would settle it

Numerical computation of the normalized supremum for H=1/2, m=1, and a concrete α < 1.5 over successively larger T, checking whether the observed liminf stabilizes at the predicted value a_{1/2} (κ_{1.5} / (1-α/1.5))^{1.5}.

read the original abstract

Let $\{B_H(t);t\ge 0\}$ be a fractional Brownian motion of order $H\in (0,1)$, and $J_{m,\alpha}(B_H)$ be the $m$-fold weighted integrals of $B_H$ defined as $$ J_{m,\bm\alpha}(B_H)(t) =\int_0^ts_m^{-\alpha_m}\int_0^{s_m}\cdots s_2^{-\alpha_2}\int_0^{s_2}s_1^{-\alpha_1}B_H(s_1)d s_1\; ds_2\cdots d s_m, $$ where $\alpha_1+\cdots+\alpha_i<H+i$, $i=1,\ldots,m$, $\bm\alpha=\bm\alpha_m=(\alpha_1,\ldots,\alpha_m)$. We show that \begin{align*} \liminf_{T\to \infty} \frac{(\log\log T)^{H+m}}{T^{H+m-\alpha}}\sup_{0\le t\le T}\left|\frac{ J_{m,\bm\alpha}(B_H)(t)}{t^{\alpha-\alpha_1-\cdots-\alpha_m}}\right| = a_H\left( \frac{\kappa_{H+m}}{1-\alpha/(H+m)}\right)^{H+m}\;\; a.s. \end{align*} for all $\alpha<H+m$, 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, \bm\alpha_{m-1}}(B_H)(s)}{s^{H+m-\alpha_1-\cdots-\alpha_{m-1}}}ds\right| &= \frac{\pi}{2}\frac{\sqrt{\beta(2H,1-H)}}{\prod_{i=1}^{m-1}\big(H+i-\alpha_1-\cdots-\alpha_i\big)}\;\; a.s., \end{align*} where $a_H$ is an explicit constant with $a_{\frac{1}{2}}=1$, $\kappa_{\lambda}$ is a constant which depends only on $\lambda$, and $\beta(a,b)$ 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 \(J_{m, \bm\alpha}(B_H)\) 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes explicit Chung-type laws of the iterated logarithm for the m-fold weighted integrated fractional Brownian motion J_{m,α}(B_H), giving the liminf as T→∞ of (log log T)^{H+m} / T^{H+m-α} times the sup norm of the normalized process equal to a_H (κ_{H+m}/(1-α/(H+m)))^{H+m} a.s. for α < H+m, together with a second liminf result involving sqrt((log log log T)/log T) for the integral of the (m-1)-fold process. The proofs rely on establishing matching small-ball probability asymptotics for these weighted integrals, with extensions to Riemann-Liouville processes and an application to the randomized play-the-winner rule.

Significance. If the small-ball estimates are rigorously verified with identical leading constants in the upper and lower bounds, the work supplies precise almost-sure constants in Chung-type LIL for integrated fractional processes, recovers the exact value from Duker-Li-Linde (2000) as a special case, and provides a concrete application in sequential statistics.

major comments (2)
  1. [Small-ball probability estimates] The small-ball probability section: the claimed rate P(sup_{0≤t≤1} |J_{m,α}(B_H)(t)| ≤ ε) ∼ exp(−c ε^{-(H+m)}) is used to obtain the exact prefactor (κ_{H+m}/(1-α/(H+m)))^{H+m}; it must be shown that the leading constant c is identical for the upper and lower bounds, rather than obtained via a coarser embedding or comparison with the unweighted case.
  2. [Proof of the liminf] The Borel-Cantelli argument converting small-ball probabilities into the liminf (abstract, displayed equation for the m-fold case): explicit control on the error terms arising from the rescaling by T^{H+m-α} and the weighting t^{α-α_1-⋯-α_m} is required to confirm that no additional logarithmic factors appear in the almost-sure statement.
minor comments (2)
  1. [Abstract and definitions] Notation: the vector bmα is introduced with subscript m but used interchangeably with α in the displayed equations; uniform notation would improve readability.
  2. [Second liminf result] The second displayed liminf involves the beta function β(2H,1-H) and a product over (H+i-α_1-⋯-α_i); a brief remark on how these arise from the covariance eigenvalues would clarify the derivation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major comments below.

read point-by-point responses

  1. Referee: The small-ball probability section: the claimed rate P(sup_{0≤t≤1} |J_{m,α}(B_H)(t)| ≤ ε) ∼ exp(−c ε^{-(H+m)}) is used to obtain the exact prefactor (κ_{H+m}/(1-α/(H+m)))^{H+m}; it must be shown that the leading constant c is identical for the upper and lower bounds, rather than obtained via a coarser embedding or comparison with the unweighted case.

    Authors: We appreciate this observation. The small-ball estimates in Section 3 are obtained directly from the Karhunen-Loève expansion of J_{m, bmα}(B_H), where both the upper and lower bounds share the same leading constant derived from the eigenvalue decay rate κ_{H+m} k^{-(H+m)}. The factor (1 - α/(H+m)) arises from the explicit norm of the weighting function in the integral operator, ensuring the constants match exactly without coarser comparisons. We will revise the manuscript to include a brief remark highlighting this equality of constants. revision: partial

  2. Referee: The Borel-Cantelli argument converting small-ball probabilities into the liminf (abstract, displayed equation for the m-fold case): explicit control on the error terms arising from the rescaling by T^{H+m-α} and the weighting t^{α-α_1-⋯-α_m} is required to confirm that no additional logarithmic factors appear in the almost-sure statement.

    Authors: Thank you for pointing this out. In the proof, the rescaling by T^{H+m-α} is managed by considering increments over exponentially growing intervals, and the weighting is normalized out in the definition of the process under consideration. Explicit bounds show that the approximation errors are o( (log log T)^{-1} ), which are negligible for the liminf and do not introduce additional factors. The application of the Borel-Cantelli lemma thus yields the exact constant as stated. We will add more detailed error estimates in the revised version to address this concern. revision: yes

Circularity Check

0 steps flagged

No significant circularity: small-ball estimates derived independently of target LIL

full rationale

The derivation first establishes small-ball probability asymptotics for the m-fold weighted integral process directly from its covariance kernel and eigenvalue analysis, then feeds those rates into a Borel-Cantelli argument to obtain the explicit liminf constant. No equation in the paper reduces the final constant back to a fitted parameter or to the LIL statement itself; the Duker-Li-Linde (2000) result appears only as a recovered special case. Parameter restrictions guarantee local integrability but do not encode the leading prefactor. The overall chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard properties of fractional Brownian motion and on the existence of small-ball probability asymptotics for the integrated processes; no new free parameters or invented entities are introduced.

axioms (2)
  • standard math Fractional Brownian motion B_H exists and satisfies the usual covariance and self-similarity properties for H ∈ (0,1).
    Invoked throughout the definition of J_{m,α}(B_H) and the small-ball estimates.
  • domain assumption Small-ball probabilities for the m-fold weighted integral admit the precise logarithmic asymptotics needed to convert to liminf constants.
    This is the key technical input that produces the explicit constants; its validity is assumed rather than re-derived in the abstract.

pith-pipeline@v0.9.0 · 5837 in / 1551 out tokens · 50650 ms · 2026-05-13T21:21:42.053271+00:00 · methodology

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