Arithmetic simulation of stochastic processes
Given a sequence of additive functions \(f_ n(m)\), with a suitable choice of the parameter \(y_{tn}\) \((y_{0n}=0\), \(y_{nn}=1)\) one can define a sequence of stochastic processes \(h_ n(m,t)=\sum_{p^ k \| m, p\leq y_{tn}}f_ n(p^ k).\) Here t is the time, the integers \(m\leq n\), with the uniform measure \(\nu_ n\), the elementary events. A particularly important case is \(f_ n(m)=f(m)/\beta (n)\) with a suitable norming function \(\beta\) (n). The author connects this so-called ''arithmetic process'' to an ''accompanying process'' with independent increments \(X_ n(t)=\sum_{p\leq y_{tn}}\xi_{np}\), where \(\xi_{np}'s\) are independent variables with the distribution \(P(\xi_{np}=h_ n(p^ k))=p^{-k}(1-p^{-1}).\) The author proves that for a wide class of arithmetic functions the convergence of the sequence \((h_ n)\) is equivalent to that of the accompanying \((X_ n)\), finds a sharp estimate for the rate of convergence to the Wiener process in terms of the moments \(\sigma_{sn}=\sum_{p\leq n}| h_ n(p)|^{s/p}\) and gives a condition for the convergence to a stable process other than the Wiener process.
scientific article; zbMATH DE number 3900890