| Abstract [eng] |
The thesis is devoted to limit theorems for stochastic models with long-range dependence. We first consider a random-coefficient AR(1) process, which can have long memory provided the distribution of autoregressive coefficient concentrates near the unit root. We identify three different limit regimes in the scheme of joint temporal-contemporaneous aggregation for independent copies of random-coefficient AR(1) process and for its copies driven by common innovations. Next, we discuss nonparametric estimation of the distribution of the autoregressive coefficient given multiple random-coefficient AR(1) series. We prove the weak convergence of the empirical process based on estimates of unobservable autoregressive coefficients to a generalized Brownian bridge and apply this result to draw statistical inference from panel AR(1) data. In the second part of the thesis we focus on spatial models in dimension 2. We define a nonlinear random field as the Appell polynomial of a long-range dependent linear random field whose moving average coefficients decay at possibly different rate in the horizontal and vertical directions. For the nonlinear random field, we investigate the limit distribution of its normalized partial sums over rectangles and prove the existence of scaling transition. Finally, we study suchlike scaling of the random grain model with long-range dependence and obtain two-change points in its limits. |
