Limit theorems for random processes constructed from sums of independent identically distributed random variables
Let \((\xi_ k\), \(k\geq 1)\) be a sequence of independent random variables with E \(\xi\) \(=0\), E \(\xi\) \({}^ 2<\infty\); let \((a_ n\), \(n\geq 1)\subset {\mathbb{R}}\). \((f_ n(t)\), \(n\geq 1)\) is a sequence of nonrandom real-valued continuous functions on [0,T]. The author considers the sequence of random processes \((X_ n(t)\), \(n\geq 1)\) defined as follows: \[ (*)\quad X_ n(t)=a_ n^{-1}\sum^{\infty}_{k=0}f^ k_ n(t)\xi_ k. \] Under certain natural conditions on \((a_ n)\), \((f_ n(t))\) and \((\xi_ n)\) the weak convergence of \(X_ n(t)\) to a Gaussian process with known correlation function is established in the space of continuous functions on [0,T]. - Note that in proving the assertion, the verification of the convergence of the series (*) in the space of continuous functions is omitted.
scientific article; zbMATH DE number 4209209