In this work, we thoroughly study the seasonal fractionally integrated ARIMA time series with stable innovations. In the first chapter, we make an overview of the various properties of the univariate -stable distribution (stability, calculus of moments, simulation, : : :). Then we introduce two models with infinite variance which are widely used in the statistic literature: the stable ARMA model and the stable ARFIMA model developed respectively by Mikosch et al. [57] and Kokoszka and Taqqu [45]. In the second chapter, noting that these models admit some limit, we build a new model called the ARFISMA symmetric -stable model. These models help us to take into account, in a modelisation, the following three stylized facts: long-memory, seasonality and infinite variance, which are often encountered at finance, telecommunication, or hydrology. After having concluded the chapter with the study of the asymptotic behaviour of the model by some simulations, we focus, in the third chapter, on the parameter estimation problem of the stable ARFISMA model. We present various estimation procedures: the semiparametric estimation methods proposed by Reisen et al.[67], the classical Whittle estimation method used by Mikosch et al. [57] and Kokoszka and Taqqu [45], and another approach of Whittle's method based on the approximation of the Whittle's likelihood function by Markov Chains Monte Carlo (MCMC) method. Many simulations, carried out in R [64], allow to compare these estimation methods. However, these methods do not permit to estimate the innovation parameter. Thus, in the fourth chapter, we introduce two estimation methods: the empirical characteristic function method developed by Knight et Yu [43] and the generalized method of moments with a continum of moment conditions, suggested by Carrasco et Florens [16]. Moreover, the asymptotic properties are validated and compared via Monte Carlo simulations. To finish, in the fifth chapter, we apply this model to monthly mean level observations in Bakel (Sénégal). For comparison purpose, we consider the classical linear model of Box et Jenkins [11], and a comparison is made between their forecasting abilities.
