This paper investigates the weak convergence of normalized partial sums of stationary random variables with long-range dependence to fractional Brownian motion (fBm) and the Rosenblatt process. The study is motivated by geophysical phenomena, where long-range dependence is common. The paper focuses on the case where the dependence structure is strong, leading to non-Gaussian limits. The main result is that under certain conditions, the normalized partial sums converge weakly to a process with properties such as self-similarity and stationary increments. For example, when the Hermite rank $ m = 1 $, the limit is fractional Brownian motion with parameter $ H $, where $ \frac{1}{2} < H < 1 $. When $ m = 2 $, the limit is a non-Gaussian process known as the Rosenblatt process. The paper also discusses the conditions under which such convergence occurs, including the behavior of the correlation function and the Hermite rank of the function $ G $. It provides a reduction theorem that simplifies the analysis by focusing on the first non-zero term in the Hermite expansion of $ G $. The results have practical applications in fields such as hydrology, geophysics, and economics, where long-range dependence is observed in time series data. The study validates the use of the R/S statistic for analyzing long-range dependence. The paper also addresses the convergence of finite-dimensional distributions and the properties of the limiting processes, including their moments and continuity.This paper investigates the weak convergence of normalized partial sums of stationary random variables with long-range dependence to fractional Brownian motion (fBm) and the Rosenblatt process. The study is motivated by geophysical phenomena, where long-range dependence is common. The paper focuses on the case where the dependence structure is strong, leading to non-Gaussian limits. The main result is that under certain conditions, the normalized partial sums converge weakly to a process with properties such as self-similarity and stationary increments. For example, when the Hermite rank $ m = 1 $, the limit is fractional Brownian motion with parameter $ H $, where $ \frac{1}{2} < H < 1 $. When $ m = 2 $, the limit is a non-Gaussian process known as the Rosenblatt process. The paper also discusses the conditions under which such convergence occurs, including the behavior of the correlation function and the Hermite rank of the function $ G $. It provides a reduction theorem that simplifies the analysis by focusing on the first non-zero term in the Hermite expansion of $ G $. The results have practical applications in fields such as hydrology, geophysics, and economics, where long-range dependence is observed in time series data. The study validates the use of the R/S statistic for analyzing long-range dependence. The paper also addresses the convergence of finite-dimensional distributions and the properties of the limiting processes, including their moments and continuity.