A self-normalized law of the iterated logarithm for Rademacher random multiplicative functions

@misc{buraczewski2026selfnormalized,
  title = {{A self-normalized law of the iterated logarithm for Rademacher random multiplicative functions}},
  author = {Dariusz Buraczewski and Alexander Iksanov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40605},
  eprint = {2610.40605},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40605},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - A self-normalized law of the iterated logarithm for Rademacher random multiplicative functions
AU  - Dariusz Buraczewski
AU  - Alexander Iksanov
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40605
UR  - https://arxiv.org/abs/2610.40605
AB  - Let f be a Rademacher random multiplicative function supported on the squarefree integers, and let M(x)=∑_n≤ xf(n). We prove a two-sided law of the iterated logarithm normalized by the full predictable quadratic variation Q(x)=∑_p≤ x(∑_substackm≤ x/p P(m)<pf(m))^2 of the martingale obtained by revealing the prime signs. Almost surely, limsup_x→∞M(x)/(√(2Q(x)loglog x))=1, liminf_x→∞M(x)/(√(2Q(x)loglog x))=−1. The same equalities hold with loglog Q(x) in the denominator. The almost sure cluster set of the normalized sum is [−1,1]. This gives a self-normalized version of the questions of Wintner and Erdős on almost sure fluctuations. We also prove that $Q(x)/q(x)$ cannot converge in probability to $1$ for any positive deterministic function q. Moreover, limsup_x→∞ |M(x)|/(√x(loglog x)^1/4) is either an almost surely finite and positive random variable with nondegenerate distribution or infinite almost surely. We do not determine which alternative holds.
ER  -