Functional limit theorem for some arithmetical processes (Q1823269)

Let \(f_{nk}(m)\) be additive arithmetical functions. Let \(a_ k(n)\) be some integer valued functions. Let \(y_ n(t)\) be a function on \([0,1]\) with \(y_ n(0)=0\) and \(y_ n(1)\) an integer tending to infinity with \(n\). Define \(X_ n(m,t)=\sum_{k\leq y_ n(t)}f_{nk}(m+a_ k(n))-A_ n(t),\) \(0\leq t\leq 1\), where \(A_ n(t)\) is a normalizing function defined on \([0,1]\). The author establishes limit theorems for \(X_ n(m,t)\) in Skorokhod's topology, under restrictions on \(f_{nk}(m)\) guaranteeing that, for limits, each \(f_{nk}(m)\) can be truncated at \(r(n)\), where \(r(n)\) is a function appearing in Kubilius's class H. The emphasis is on the method of proof: the author utilizes weak limit theorems for semimartingales.