Stochastic processes with the long-range dependency (LRD) property are fundamental to modeling data that exhibit slow power decay of the covariance function. Such  behavior often appears in the analysis of financial data, telecommunications, and various natural phenomena. Thus, introducing new stochastic models and statistical methods that take the LRD into account is of great interest. Based on previous work, we introduce a  new stochastic process called quadratic fractionally integrated moving average, that arises from the Quadratic Ornstein-Uhlenbeck Type (QOUT) process, proposed in the literature. We consider Lévy noises of finite second-order moments and use a construction based on a moving average stochastic process whose kernel is that of a QOUT process. Then, using Riemann-Liouville fractional integrals, we propose a fractionally integrated moving average process, for ...