A stochastic process defined via the random permutation divisors

TL;DR

This paper introduces a new stochastic process derived from normalized sums of a multiplicative function over divisors, with results on its convergence, path continuity, and moments, based on random permutations.

Contribution

It establishes a functional limit theorem for the process and derives complex formulas for joint moments using number theory techniques.

Findings

Convergence of the process to a continuous limit in Skorokhod space

Paths of the limit process are almost surely continuous

Explicit formulas for joint power moments of the process

Abstract

The normalised partial sums of values of a nonnegative multiplicative function over divisors with appropriately restricted sizes of a random permutation from the symmetric group define trajectories of a stochastic process. We prove a functional limit theorem in the Skorokhod space when the permutations are drawn uniformly at random. Furthermore, we show that the paths of the limit process almost surely belong to the space of continuous functions on the unit interval and, exploiting the results from number-theoretical papers, we obtain rather complex formulas for the limits of joint power moments of the process.