An arithmetic semigroup is a commutative semigroup \(\mathcal S\) with identity containing a countable subset \(P\), such that any element \(a\in \mathcal S\), \(a\neq 1\), admits a unique factorization into a finite product of powers of elements of \(P\). A mapping \(\partial :\;\mathcal S\to \mathbb Z_+\), such that \(\partial (m_1m_2)=\partial (m_1)+\partial (m_2)\), \(m_1,m_2\in \mathcal S\), called the degree, is given. If \(S(n)\) is the number of elements with degree \(n\) (\(S(n)\) is assumed finite), then \[ \mu_n(A)=\frac{1}{S(n)}\sum _{m\in A;\;\partial (m)=n} 1 \] is a probability measure on subsets \(A\subset \mathcal S\). Under some additional assumptions, the authors assign to a multiplicative real-valued function on \(\mathcal S\) a sequence of stochastic processes on the above probability space and prove its weak convergence to a stochastically continuous process with independent increments. As an example, they consider the case where \(\mathcal S\) is the set of all polynomials over a finite field.