2025/09/23 by Yeor Hafouta, Hafouta, Yeor
Computer Science · Decision Sciences · Mathematics · #Chaos-based Image/Signal Encryption #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Probability (math.PR) #Probability and Risk Models
paper · pdf · doi:10.48550/arxiv.2509.19067
openalex publication_date 2025/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Let Xp, p∈\cP be a sequence of independent random variables s.t. \bbP(Xp=± 1)=1/2. Let \tej=∏p|jXp if j is square free and \tej=0 otherwise. Denote Sn=∑ℓ=1n\te_ℓ. The from this point of view proving limit theorems for Sn is natural problem, since Sn mimics the behavior of e√(ln(β)). It is a natural guiding conjecture that Sn/√ n obeys the central limit theorem (CLT). However, S. Chatterjee conjectured (as expressed in \cite[25]) that the CLT should not hold. Chatterjee's conjecture was proved by Harper \cite[17], and by now it is a direct consequence of a more recent breakthrough by Harper \citeHar20 that (Sn)/(bn)→ 0 in L1, where bn=(n1/2(ln(ln(n)))-1/4)un, un→∞. In particular Sn/√ n→ 0. Nevertheless, the question whether there exists a sequence an=o(bn) such that Sn/an converges to some limit remains a mystery. Note that the corresponding problem in the Steinhaus Setting was recently resolved by \citeGor1. In this paper make an attempt to shed some light on the convergence of Sn/an. Additionally, we obtain explicit estimates on hight moments of Sn without restrictions on the size of the moment compared to n like in \cite[Theorem 1.2]Har19, which is of independent interest. This is achieved by a martingale argument together with the Burkholder inequality, and it has applications in a natural number theoretic combinatorial problem. Using martingale techniques we will also obtain exponential concentration inequalities for Sn (in the large deviations regime)