This paper by Murad S. Taqqu investigates the weak convergence of normalized partial sums of stationary random variables with long-term non-periodic dependence. The focus is on processes that are not Brownian motion and may not be Gaussian. The author studies the limit process as the number of terms in the sum increases, under the assumption that the variance of the sum grows asymptotically proportional to \(N^{2H}L(N)\), where \(H\) is a constant and \(L\) is a slowly varying function. The paper provides conditions for weak convergence and characterizes the limiting process for different values of \(H\). - **Case \(H = \frac{1}{2}\)**: The limit is Brownian motion. - **Case \(H > \frac{1}{2}\)**: The limit is fractional Brownian motion, which is Gaussian with mean zero and covariance function \(t^{2H}\). - **Case \(H < \frac{1}{2}\)**: The limit is the Rosenblatt process, which is non-Gaussian. The paper also discusses the Hermite rank of the function \(G\) and the correlation kernel of the stationary sequence \(\{X_i\}\), and provides conditions for the limiting process to be semi-stable. The results have practical applications in fields such as hydrology, geophysics, and economics, particularly in the analysis of long-term statistical dependence in time series.This paper by Murad S. Taqqu investigates the weak convergence of normalized partial sums of stationary random variables with long-term non-periodic dependence. The focus is on processes that are not Brownian motion and may not be Gaussian. The author studies the limit process as the number of terms in the sum increases, under the assumption that the variance of the sum grows asymptotically proportional to \(N^{2H}L(N)\), where \(H\) is a constant and \(L\) is a slowly varying function. The paper provides conditions for weak convergence and characterizes the limiting process for different values of \(H\). - **Case \(H = \frac{1}{2}\)**: The limit is Brownian motion. - **Case \(H > \frac{1}{2}\)**: The limit is fractional Brownian motion, which is Gaussian with mean zero and covariance function \(t^{2H}\). - **Case \(H < \frac{1}{2}\)**: The limit is the Rosenblatt process, which is non-Gaussian. The paper also discusses the Hermite rank of the function \(G\) and the correlation kernel of the stationary sequence \(\{X_i\}\), and provides conditions for the limiting process to be semi-stable. The results have practical applications in fields such as hydrology, geophysics, and economics, particularly in the analysis of long-term statistical dependence in time series.