Multiplicative functions and random processes
The authors consider the invariance principle for multiplicative arithmetic functions. Let \(g(m)\) be a real multiplicative function, \(\beta_n\in\mathbb{R}\), \(\beta_n>0\), \(\beta_n\to\infty\). Define \(g_n(p^k)= |g(p^k) |^{1/ \beta_n} \text{sgn} g(p^k)\), where \(p\) is a prime number and \(k\in N\cup \{0\}\). For real \(u\in[0,1]\), define the sequence \(G_n(m,u)\) as the product of \(g_n(p^k)\), where \(p^k\) are factors of the standard form of the natural number \(m\). Let \(m\leq n\) be selected with probability \(1/n\). The authors obtain the sequence of stochastic processes with trajectory in \(D:=D[0,1]\). In the paper necessary and sufficient conditions on \(g(m)\) are found under which the corresponding measures always converge when \(n\to\infty\). This theorem is analogous to the functional limit theorem for the sequence of independent random variables [\textit{G. Bareikis} and \textit{E. Manstavičius}, Liet. Mat. Rink 37, 139-154 (1997; Zbl 0892.60046)].
scientific article; zbMATH DE number 1296179