Martingale central limit theorem for random multiplicative functions

2024/05/30 by Gorodetsky, Ofir, Wong, Mo Dick · 3 citations
#FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2405.20311

Abstract

Let α be a Steinhaus or a Rademacher random multiplicative function. For a wide class of multiplicative functions f we show that the sum ∑_{n ≤ x}α(n) f(n), normalised to have mean square 1, has a non-Gaussian limiting distribution. More precisely, we establish a generalised central limit theorem with random variance determined by the total mass of a random measure associated with αf. Our result applies to d_z, the z-th divisor function, as long as z is strictly between 0 and \tfrac1√(2). Other examples of admissible f-s include any multiplicative indicator function with the property that f(p)=1 holds for a set of primes of density strictly between 0 and \tfrac12.

Cited by

• A few notes on the asymptotic behavior of Rademacher random multiplicative functions
• The Fourier coefficients of the critical holomorphic multiplicative chaos
• On the limiting distribution of sums of random multiplicative functions

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