Explicit bivariate rate functions for large deviations in AR(1) and MA(1) processes with Gaussian innovations

Figure 2.1.  Regions $ {\cal{D}}_1 $ (left-hand side panel), $ {\cal{D}}_2 $ (middle side panel) and $ {\cal{D}} = {\cal{D}}_1 \cup {\cal{D}}_2 $ (right-hand side panel), for $ {\cal{D}}_1 $ and $ {\cal{D}}_2 $ defined in (2.14) in the particular case when $ \phi = 0.5 $. In the third draw, we plotted the lines $ \lambda_2 = - \phi \pm(1+\phi^2-2\lambda_1)/2 $, for $ \lambda_1 < (1-\phi^2)/2 $ in thick green color to illustrate that this border belongs to the set $ {\cal{D}} $, whereas the curve $ (\phi + \lambda_2)^2 = \phi^2(1 - 2 \lambda_1) $, for $ \dfrac{1 - \phi^2}{2} \leqslant \lambda_1 \leqslant \dfrac{1}{2} $, is plotted in red color to indicate that it does not make part of the set $ {\cal{D}} $.

Figure 2.2.  The two left-hand side panels show the graph of the function $ J(x,y) $, given in (2.19), from two different points of view, with $ \phi = 0.8 $, $ x \in (0,6] $ and $ y \in [-6, 6] $. The white line represents the curve $ {\cal{C}}_{0.8} $, defined in (2.16), where the domains of the two functions $ J_1(\cdot,\cdot) $ and $ J_2(\cdot,\cdot) $ intersect, changing roles in the definition of $ J(\cdot,\cdot) $. Notice that, $ J_1(\cdot,\cdot) $ converges to $ +\infty $ as $ (x,y) $ approaches the curves $ \{(x,y) \in \mathbb{R}^2: x \geqslant 0, |y| = x\} $; this behavior is inherited by $ J(\cdot,\cdot) $. The right-hand side panel displays the domains $ {\cal{A}}_\phi $ (in blue color) and $ {\cal{B}}_\phi $ (in orange color).

Figure 2.3.  In the left-hand side panel, a plot of the region $ F = [2,3]\times[-1,1] $ is presented, showing that it is contained in the set of exposed points $ {\cal{F}}_{0.5}. $ In the right-hand side panel, a graph of the function $ J(x,y) ,$ for (x, y) $ \in F. $

Figure 2.4.  In the left-hand side panel, a plot of the region $ F = [5,6]\times[-4,-1] $ is presented, showing that it is not entirely contained in the set of exposed points $ {\cal{F}}_{0.5} .$ In the right-hand side panel, a graph of the function $ J(x,y) $ for (x, y) $ \in F .$

Figure 3.1.  Graphs of $ J_1(c,y) $ and $ J_2(c,y) $ with $ \phi = 0.8 $ and $ c = 4 $. The green and red dots represent the points $ -\alpha_c $ and $ \alpha_c $, respectively, where $ J_1(c, \cdot) $ and $ J_2(c, \cdot) $ change roles in the law of $ J(\cdot,\cdot) $. The blue dot is the point $ (y_c, J(c,y_c)) $. As this graph shows us, given that $ \phi > 0 $, $ J(c,y) $ is equal to $ J_1(c,y) $ if $ y \in (c,-\alpha_c)\cup(\alpha_c,c) $, and equal to $ J_2(c,y) $ if $ y \in [-\alpha_c,\alpha_c] $. When $ J(c,y) $ changes from $ J_2(c,y) $ to $ J_1(c,y) $ on the red dot $ \alpha_c $, the derivative of $ J_1(c,y) $ being negative for $ y \in (\alpha_c,y_c) $ shows that this function keeps decreasing from left to right until it reaches the global minimum at $ y_c $ (remember from Proposition 2.1 that $ J_1 $ and $ J_2 $ are convex), so that every point on its right and its left is greater than $ J_1(c,y_c) $, including the points in $ [-\alpha_c,\alpha_c] $ where $ J_2(c,y) $ rules. A similar analysis follows by symmetry for the case $ \phi < 0 $.

Figure 3.2.  Graphs of $ I_1(c) $ for $ \phi \in \{0, \, 0.3, \, 0.6, \, 0.9\} $ and $ c \in (0, 10] $.

Figure 3.3.  Graphs of $ J_1(x,c) $ and $ J_2(x,c) $, with $ \phi = 0.8 $. Here $ c = 2 $ is fixed. The red dot represents the point $ \beta_c $ where $ J_1(\cdot,c) $ and $ J_2(\cdot,c) $ change roles in the law of $ J(\cdot,\cdot) $. The blue dot is the point $ (x_c, J(x_c,c)) $. As the graph shows us, $ J(c,y) $ is equal to $ J_1(c,y) $ when $ x \in (|c|,\beta_c) $, and equal to $ J_2(c,y) $ when $ y \in [\beta_c,+\infty) $. When $ J(c,y) $ changes from $ J_2(x,c) $ to $ J_1(x,c) $ on the red dot $ \beta_c $, the derivative of $ J_1(c,y) $ being positive for $ y \in (x_c,+\infty) $ shows that this function keeps decreasing from right to left until it reaches the global minimum at $ x_c $ (remember from Proposition 2.1 that $ J_1 $ and $ J_2 $ are convex), so that every point on its right and its left is greater than $ J_1(x_c,c) $, including the points in $ [\beta_c,+\infty) $ where $ J_2(x,c) $ rules.

Figure 3.4.  Graphs of $ I_2(c) $ for $ \phi \in \{-0.99, -0.6, \ 0, \, 0.6, \, 0.99\} $ and $ c \in [-4,4] $.

Figure 3.5.  Graph of $ I_{\phi}(c) $ for $ \phi \in \{-0.5, 0, 0.5\} $ and $ c \in (-1, 1) $.

Figure 3.6.  Graphs of $ J_1(x,c x) $ and $ J_2(x,c x) $, with $ \phi = 0.8 $. Here $ c = 0.5 $ is fixed. The red dot represents the point $ \delta_c $ where $ J_1(x,c x) $ and $ J_2(x,c x) $ change roles in the law of $ J(\cdot,\cdot) $. The blue dot is the point $ (x_c, J(x_c,c x_c)) $. Notice that $ J(x, c x) $ is equal to $ J_1(x, c x) $ when $ x \in (0,\delta_c) $, and equal to $ J_2(x,c x) $ when $ x \in [\delta_c,+\infty) $. When $ J(x,c x) $ changes from $ J_2(x,c x) $ to $ J_1(x,c x) $ on the red dot $ \delta_c $, the derivative of $ J_1(x,c x) $ being positive for $ x \in (x_c,+\infty) $ shows that this function keeps decreasing from right to left until it reaches the global minimum at $ x_c $ (remember from Proposition 2.1 that $ J_1 $ and $ J_2 $ are convex) so that every point on its right and its left is greater than $ J_1(x_c, c x_c) $, including the points in $ [\delta_c,+\infty) $ where $ J_2(x,c x) $ rules.

Figure 4.1.  Graph of $ I_{\overline{X}}(c) $ for $ \phi \in \{-0.5, 0, 0.5\} $ and $ c \in [-10, 10] $.

Figure 4.2.  Graphs of $ K_\theta(x) $ for $ \theta \in \{0.2, \, 0.4, \, 0.6, \, 0.8\} $ and $ x \in (0, 5] $.