Pith Number

pith:XVXJ6WBD

pith:2026:XVXJ6WBDMG5OTUS7XBO4OZS53V

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

Li-Xin Zhang

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

arxiv:2604.01701 v3 · 2026-04-02 · math.PR

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Claims

C1strongest claim

liminf_{T→∞} (log log T)^{H+m} / T^{H+m-α} sup_{0≤t≤T} |J_{m,α}(B_H)(t) / t^{α - α_1 - ⋯ - α_m}| = a_H (κ_{H+m} / (1 - α/(H+m)))^{H+m} a.s. for all α < H+m, with a similar explicit liminf for the (m-1)-fold case.

C2weakest assumption

The parameter restrictions α_1 + ⋯ + α_i < H + i for each i, together with the existence and precise form of small-ball probabilities for the m-fold integrated process; if these tail estimates fail or require additional regularity not stated, the conversion from small-ball probabilities to the liminf constant breaks.

C3one line summary

Chung-type laws of the iterated logarithm are established for m-fold weighted integrals of fractional Brownian motion, yielding explicit liminf expressions and resolving an exact constant from prior work.

References

26 extracted · 26 resolved · 0 Pith anchors

[1] Anderson, T.W. (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6 :170-176. 0069229 1955

[2] Bai, Z. D. , Hu, F. and Zhang, L. X. (2002). The Gaussian approximation theorems for urn models and their applications. Ann. Appl. Probab. , 12 : 1149-1173. 1936587 2002

[3] Chen, X. (2015). The limit law of the iterated logarithm. J. Theor. Probab. 28 : 721–725. 3370672 2015

[4] Chen, X. and Li, W. V. (2003). Quadratic functionals and small ball probabilities for the m -fold integrated Brownian motion. Ann. Probab. 31 (2):1052–1077. 196495 2003

[5] Creutzig, J. (1999). Gau ma e kleiner Kugeln und metrische Entropie . Diplomarbeit, FSU Jena 1999

Formal links

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Receipt and verification

First computed	2026-05-20T00:00:37.461026Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

bd6e9f582361bae9d25fb85dc7665ddd7edc44b440e8a081ea7767cd49aaed88

Aliases

arxiv: 2604.01701 · arxiv_version: 2604.01701v3 · doi: 10.48550/arxiv.2604.01701 · pith_short_12: XVXJ6WBDMG5O · pith_short_16: XVXJ6WBDMG5OTUS7 · pith_short_8: XVXJ6WBD

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/XVXJ6WBDMG5OTUS7XBO4OZS53V \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: bd6e9f582361bae9d25fb85dc7665ddd7edc44b440e8a081ea7767cd49aaed88

Canonical record JSON

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  "metadata": {
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    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-04-02T06:59:07Z",
    "title_canon_sha256": "d09d73834399056d9a1c41109fee806385cdff18487f1afd41165f7f9351b252"
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  "source": {
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