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Theorem 4.1.18 Suppose that \( \\left\\{ {\\mu }_{\\epsilon }\\right\\} \) satisfies the LDP in a regular topological space \( \\mathcal{X} \) with rate function \( I \) . Then, for any base \( \\mathcal{A} \) of the topology of \( \\mathcal{X} \), and for any \( x \\in \\mathcal{X} \) ,\n\n\\[ \nI\\left( x\\right) = \... | Proof: Fix \( x \\in \\mathcal{X} \) and let\n\n\\[ \n\\ell \\left( x\\right) = \\mathop{\\sup }\\limits_{{\\{ A \\in \\mathcal{A} : x \\in A\\} }}\\mathop{\\inf }\\limits_{{y \\in \\bar{A}}}I\\left( y\\right) .\n\\]\n\n(4.1.20)\n\nSuppose that \( I\\left( x\\right) > \\ell \\left( x\\right) \) . Then, in particular, \... | Yes |
Lemma 4.1.21 Let \( \mathcal{A} \) be a base for a Hausdorff topological vector space \( \mathcal{X} \) , such that in addition to condition (4.1.14), for every \( {A}_{1},{A}_{2} \in \mathcal{A} \) ,\n\n\[ \mathop{\limsup }\limits_{{\epsilon \rightarrow 0}}\epsilon \log {\mu }_{\epsilon }\left( \frac{{A}_{1} + {A}_{2}... | Proof: It suffices to show that the condition (4.1.22) yields the convexity of the rate function \( I \) of (4.1.13). To this end, fix \( {x}_{1},{x}_{2} \in \mathcal{X} \) and \( \delta > 0 \) . Let \( x = \left( {{x}_{1} + {x}_{2}}\right) /2 \) and let \( {I}^{\delta } \) denote the \( \delta \) -rate function. Then,... | Yes |
Lemma 4.1.23 Suppose the topological space \( \mathcal{X} \) has a countable base. For any family of probability measures \( \left\{ {\mu }_{\epsilon }\right\} \), there exists a sequence \( {\epsilon }_{k} \rightarrow 0 \) such that \( \left\{ {\mu }_{{\epsilon }_{k}}\right\} \) satisfies the weak LDP in \( \mathcal{X... | Proof: Fix a countable base \( \mathcal{A} \) for the topology of \( \mathcal{X} \) and a sequence \( {\epsilon }_{n} \rightarrow 0 \) . By Tychonoff’s theorem (Theorem B.3), the product topology makes \( \mathcal{Y} = \) \( {\left\lbrack 0,1\right\rbrack }^{\mathcal{A}} \) into a compact metrizable space. Since \( \ma... | Yes |
Lemma 4.1.24 Suppose \( \left\{ {\mu }_{\epsilon }\right\} \) is a family of tight (Borel) probability measures on a metric space \( \left( {\mathcal{X}, d}\right) \), such that the upper bound (1.2.12) holds for all compact sets and some rate function \( I\left( \cdot \right) \) . Then, for any base \( \mathcal{A} \) ... | Proof: We argue by contradiction, fixing a base \( \mathcal{A} \) of the metric topology and \( x \in \mathcal{X} \) for which (4.1.25) fails. For any \( m \in {\mathbb{Z}}_{ + } \), there exists some \( A \in \mathcal{A} \) such that \( x \in A \subset {B}_{x,{m}^{-1}} \) . Hence, for some \( \delta > 0 \) and any \( ... | Yes |
Theorem 4.2.1 (Contraction principle) Let \( \mathcal{X} \) and \( \mathcal{Y} \) be Hausdorff topological spaces and \( f : \mathcal{X} \rightarrow \mathcal{Y} \) a continuous function. Consider a good rate function \( I : \mathcal{X} \rightarrow \left\lbrack {0,\infty }\right\rbrack \) .\n\n(a) For each \( y \in \mat... | Proof: (a) Clearly, \( {I}^{\prime } \) is nonnegative. Since \( I \) is a good rate function, for all \( y \in f\left( \mathcal{X}\right) \) the infimum in the definition of \( {I}^{\prime } \) is obtained at some point of \( \mathcal{X} \) . Thus, the level sets of \( {I}^{\prime },{\Psi }_{{I}^{\prime }}\left( \alph... | Yes |
Theorem 4.2.4 (Inverse contraction principle) Let \( \mathcal{X} \) and \( \mathcal{Y} \) be Hausdorff topological spaces. Suppose that \( g : \mathcal{Y} \rightarrow \mathcal{X} \) is a continuous bijection, and that \( \left\{ {\nu }_{\epsilon }\right\} \) is an exponentially tight family of probability measures on \... | Proof: Note first that for every \( \alpha < \infty \), by the continuity of \( g \), the level set \( \left\{ {y : {I}^{\prime }\left( y\right) \leq \alpha }\right\} = {g}^{-1}\left( {{\Psi }_{I}\left( \alpha \right) }\right) \) is closed. Moreover, \( {I}^{\prime } \geq 0 \), and hence \( {I}^{\prime } \) is a rate f... | Yes |
Corollary 4.2.6 Let \( \\left\\{ {\\mu }_{\\epsilon }\\right\\} \) be an exponentially tight family of probability measures on \( \\mathcal{X} \) equipped with the topology \( {\\tau }_{1} \). If \( \\left\\{ {\\mu }_{\\epsilon }\\right\\} \) satisfies an LDP with respect to a Hausdorff topology \( {\\tau }_{2} \) on \... | Proof: The proof follows from Theorem 4.2.4 by using as \( g \) the natural embedding of \( \\left( {\\mathcal{X},{\\tau }_{1}}\\right) \) onto \( \\left( {\\mathcal{X},{\\tau }_{2}}\\right) \), which is continuous because \( {\\tau }_{1} \) is finer than \( {\\tau }_{2} \). Note that, since \( g \) is continuous, the ... | Yes |
Theorem 4.2.13 If an LDP with a good rate function \( I\left( \cdot \right) \) holds for the probability measures \( \left\{ {\mu }_{\epsilon }\right\} \), which are exponentially equivalent to \( \left\{ {\widetilde{\mu }}_{\epsilon }\right\} \), then the same LDP holds for \( \left\{ {\widetilde{\mu }}_{\epsilon }\ri... | Proof: This theorem is a consequence of the forthcoming Theorem 4.2.16. To avoid repetitions, a direct proof is omitted. | No |
Theorem 4.2.16 Suppose that for every \( m \), the family of measures \( \left\{ {\mu }_{\epsilon, m}\right\} \) satisfies the LDP with rate function \( {I}_{m}\left( \cdot \right) \) and that \( \left\{ {\mu }_{\epsilon, m}\right\} \) are exponentially good approximations of \( \left\{ {\widetilde{\mu }}_{\epsilon }\r... | Proof: (a) Throughout, let \( \left\{ {Z}_{\epsilon, m}\right\} \) be the exponentially good approximations of \( \left\{ {\widetilde{Z}}_{\epsilon }\right\} \), having the joint laws \( \left\{ {P}_{\epsilon, m}\right\} \) with marginals \( \left\{ {\mu }_{\epsilon, m}\right\} \) and \( \left\{ {\widetilde{\mu }}_{\ep... | Yes |
Corollary 4.2.21 Suppose \( f : \mathcal{X} \rightarrow \mathcal{Y} \) is a continuous map from a Hausdorff topological space \( \mathcal{X} \) to the metric space \( \left( {\mathcal{Y}, d}\right) \) and that \( \left\{ {\mu }_{\epsilon }\right\} \) satisfy the LDP with the good rate function \( I : \mathcal{X} \right... | Proof: The contraction principle (Theorem 4.2.1) yields the desired LDP for \( \left\{ {{\mu }_{\epsilon } \circ {f}^{-1}}\right\} \) . By (4.2.22), these measures are exponentially equivalent to \( \left\{ {{\mu }_{\epsilon } \circ {f}_{\epsilon }^{-1}}\right\} \), and the corollary follows from Theorem 4.2.13. | Yes |
Theorem 4.2.23 Let \( \left\{ {\mu }_{\epsilon }\right\} \) be a family of probability measures that satisfies the LDP with a good rate function I on a Hausdorff topological space \( \mathcal{X} \) , and for \( m = 1,2,\ldots \), let \( {f}_{m} : \mathcal{X} \rightarrow \mathcal{Y} \) be continuous functions, with \( \... | Proof: By assumption, the functions \( {f}_{m} : \mathcal{X} \rightarrow \mathcal{Y} \) are continuous. Hence, by the contraction principle (Theorem 4.2.1), for each \( m \in {\mathbb{Z}}_{ + } \), the family of measures \( \left\{ {{\mu }_{\epsilon } \circ {f}_{m}^{-1}}\right\} \) satisfies the LDP on \( \mathcal{Y} \... | Yes |
Lemma 4.3.4 If \( \phi : \mathcal{X} \rightarrow \mathbb{R} \) is lower semicontinuous and the large deviations lower bound holds with \( I : \mathcal{X} \rightarrow \left\lbrack {0,\infty }\right\rbrack \), then | \[ \mathop{\liminf }\limits_{{\epsilon \rightarrow 0}}\epsilon \log E\left\lbrack {e}^{\phi \left( {Z}_{\epsilon }\right) /\epsilon }\right\rbrack \geq \mathop{\sup }\limits_{{x \in \mathcal{X}}}\{ \phi \left( x\right) - I\left( x\right) \} . \] | Yes |
Lemma 4.3.6 If \( \phi : \mathcal{X} \rightarrow \mathbb{R} \) is an upper semicontinuous function for which the tail condition (4.3.2) holds, and the large deviations upper bound holds with the good rate function \( I : \mathcal{X} \rightarrow \left\lbrack {0,\infty }\right\rbrack \), then | \[ \mathop{\limsup }\limits_{{\epsilon \rightarrow 0}}\epsilon \log E\left\lbrack {e}^{\phi \left( {Z}_{\epsilon }\right) /\epsilon }\right\rbrack \leq \mathop{\sup }\limits_{{x \in \mathcal{X}}}\{ \phi \left( x\right) - I\left( x\right) \} . \] | Yes |
Lemma 4.3.8 Condition (4.3.3) implies the tail condition (4.3.2). | Proof of Lemma 4.3.4: Fix \( x \in \mathcal{X} \) and \( \delta > 0 \) . Since \( \phi \left( \cdot \right) \) is lower semicontinuous, it follows that there exists a neighborhood \( G \) of \( x \) such that \( \mathop{\inf }\limits_{{y \in G}}\phi \left( y\right) \geq \phi \left( x\right) - \delta \) . Hence,\n\n\[ \... | Yes |
Theorem 4.4.2 (Bryc) Suppose that the family \( \left\{ {\mu }_{\epsilon }\right\} \) is exponentially tight and that the limit \( {\Lambda }_{f} \) in (4.4.1) exists for every \( f \in {C}_{b}\left( \mathcal{X}\right) \) . Then \( \left\{ {\mu }_{\epsilon }\right\} \) satisfies the LDP with the good rate function \[ I... | Proof: Since \( {\Lambda }_{0} = 0 \), it follows that \( I\left( \cdot \right) \geq 0 \) . Moreover, \( I\left( x\right) \) is lower semicontinuous, since it is the supremum of continuous functions. Due to the exponential tightness of \( \left\{ {\mu }_{\epsilon }\right\} \), the LDP asserted follows once the weak LDP... | Yes |
Lemma 4.4.6 (Lower bound) If \( {\Lambda }_{f} \) exists for each \( f \in {C}_{b}\left( \mathcal{X}\right) \), then, for every open \( G \subset \mathcal{X} \) and each \( x \in G \), \[ \mathop{\liminf }\limits_{{\epsilon \rightarrow 0}}\epsilon \log {\mu }_{\epsilon }\left( G\right) \geq - I\left( x\right) . \] | Proof of Lemma 4.4.6: Fix \( x \in \mathcal{X} \) and a neighborhood \( G \) of \( x \) . Since \( \mathcal{X} \) is a completely regular topological space, there exists a continuous function \( f : \mathcal{X} \rightarrow \left\lbrack {0,1}\right\rbrack \), such that \( f\left( x\right) = 1 \) and \( f\left( y\right) ... | Yes |
Lemma 4.4.8 Let \( \mathcal{X} \) be a locally convex, Hausdorff topological vector space. Then the class \( \mathcal{G} \) of all continuous, bounded above, concave functions on \( \mathcal{X} \) is well-separating. | Proof: Let \( {\mathcal{X}}^{ * } \) denote the topological dual of \( \mathcal{X} \), and let \( {\mathcal{G}}_{0} \triangleq \{ \lambda \left( x\right) + c \) : \( \lambda \in {\mathcal{X}}^{ * }, c \in \mathbb{R}\} \) . Note that \( {\mathcal{G}}_{0} \) contains the constant functions, and by the Hahn-Banach theorem... | Yes |
Theorem 4.5.10 Let \( \mathcal{X} \) be a locally convex Hausdorff topological vector space. Assume that \( {\mu }_{\epsilon } \) satisfies the LDP with a good rate function I. Suppose in addition that\n\n\[ \bar{\Lambda }\left( \lambda \right) \triangleq \mathop{\limsup }\limits_{{\epsilon \rightarrow 0}}\epsilon {\La... | Proof: (a) Fix \( \lambda \in {\mathcal{X}}^{ * } \) and \( \gamma > 1 \) . By assumption, \( \bar{\Lambda }\left( {\gamma \lambda }\right) < \infty \), and Varadhan's lemma (Theorem 4.3.1) applies for the continuous function\n\n\( \lambda : \mathcal{X} \rightarrow \mathbb{R} \) . Thus, \( \Lambda \left( \lambda \right... | Yes |
Corollary 4.5.13 Suppose that both condition (4.5.11) and the assumptions of Lemma 4.1.21 hold for the family \( \left\{ {\mu }_{\epsilon }\right\} \), which is exponentially tight. Then \( \left\{ {\mu }_{\epsilon }\right\} \) satisfies in \( \mathcal{X} \) the LDP with the good, convex rate function \( {\Lambda }^{ *... | Proof: By Lemma 4.1.21, \( \left\{ {\mu }_{\epsilon }\right\} \) satisfies a weak LDP with a convex rate function. As \( \left\{ {\mu }_{\epsilon }\right\} \) is exponentially tight, it is deduced that it satisfies the full LDP with a convex, good rate function. The corollary then follows from parts (a) and (b) of Theo... | Yes |
Theorem 4.5.14 Suppose that \( \left\{ {\mu }_{\epsilon }\right\} \) satisfies a weak LDP with a convex rate function \( I\left( \cdot \right) \), and that \( \mathcal{X} \) is a locally convex, Hausdorff topological vector space. Assume that for each \( \lambda \in {\mathcal{X}}^{ * } \), the limits \( {\Lambda }_{\la... | Proof: Fix \( \lambda \in {\mathcal{X}}^{ * } \) . By the inequality (4.5.15),\n\n\[ \mathop{\sup }\limits_{{x \in \mathcal{X}}}\{ \langle \lambda, x\rangle - I\left( x\right) \} = \mathop{\sup }\limits_{{a \in \mathbb{R}}}\mathop{\sup }\limits_{{\{ x : \left( {\langle \lambda, x\rangle - a}\right) > 0\} }}\{ \langle \... | Yes |
Theorem 4.5.20 (Baldi) Suppose that \( \left\{ {\mu }_{\epsilon }\right\} \) are exponentially tight probability measures on \( \mathcal{X} \) . (a) For every closed set \( F \subset \mathcal{X} \) , \[ \mathop{\limsup }\limits_{{\epsilon \rightarrow 0}}\epsilon \log {\mu }_{\epsilon }\left( F\right) \leq - \mathop{\in... | Proof: (a) The upper bound is a consequence of Theorem 4.5.3 and the assumed exponential tightness. (b) If \( \bar{\Lambda }\left( \lambda \right) = - \infty \) for some \( \lambda \in {\mathcal{X}}^{ * } \), then \( {\bar{\Lambda }}^{ * }\left( \cdot \right) \equiv \infty \) and the large deviations lower bound trivia... | Yes |
Theorem 4.6.1 (Dawson-Gärtner) Let \( \left\{ {\mu }_{\epsilon }\right\} \) be a family of probability measures on \( \mathcal{X} \), such that for any \( j \in J \) the Borel probability measures \( {\mu }_{\epsilon } \circ {p}_{j}^{-1} \) on \( {\mathcal{Y}}_{j} \) satisfy the LDP with the good rate function \( {I}_{... | Proof: Clearly, \( I\left( \mathbf{x}\right) \) is nonnegative. For any \( \alpha \in \lbrack 0,\infty ) \) and \( j \in J \), let \( {\Psi }_{{I}_{j}}\left( \alpha \right) \) denote the compact level set of \( {I}_{j} \), i.e., \( {\Psi }_{{I}_{j}}\left( \alpha \right) \triangleq \left\{ {{y}_{j} : {I}_{j}\left( {y}_{... | Yes |
Lemma 4.6.5 If \( I\left( \cdot \right) \) is a good rate function on \( \mathcal{X} \) such that\n\n\[ \n{I}_{j}\left( y\right) = \inf \left\{ {I\left( \mathbf{x}\right) : \mathbf{x} \in \mathcal{X},\;y = {p}_{j}\left( \mathbf{x}\right) }\right\} ,\n\]\n\nfor any \( y \in {\mathcal{Y}}_{j}, j \in J \), then the identi... | Proof: Fix \( \alpha \in \lbrack 0,\infty ) \) and let \( A \) denote the compact level set \( {\Psi }_{I}\left( \alpha \right) \) . Since \( {p}_{j} : \mathcal{X} \rightarrow {\mathcal{Y}}_{j} \) is continuous for any \( j \in J \), by (4.6.6) \( {A}_{j} \triangleq {\Psi }_{{I}_{j}}\left( \alpha \right) = {p}_{j}\left... | Yes |
Theorem 4.6.9 Let Assumption 4.6.8 hold. Further assume that for every \( d \in {\mathbb{Z}}_{ + } \) and every \( {\lambda }_{1},\ldots ,{\lambda }_{d} \in \mathcal{W} \), the measures \( \left\{ {{\mu }_{\epsilon } \circ {p}_{{\lambda }_{1},\ldots ,{\lambda }_{d}}^{-1},\epsilon > 0}\right\} \) satisfy the LDP with th... | Proof: Let \( \mathcal{V} \) be the system of all finite dimensional linear subspaces of \( \mathcal{W} \), equipped with the partial ordering defined by inclusion. To each \( V \in \mathcal{V} \), attach its (finite dimensional) algebraic dual \( {V}^{\prime } \) equipped with the \( V \) -topology. The latter are cle... | No |
Corollary 4.6.11 Let Assumption 4.6.8 hold.\n\n(a) Suppose that for each \( \lambda \in \mathcal{W} \), the limit\n\n\[ \Lambda \left( \lambda \right) = \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}\epsilon \log {\int }_{\mathcal{X}}{e}^{{\epsilon }^{-1}\langle \lambda, x\rangle }{\mu }_{\epsilon }\left( {dx}\right)... | Proof: (a) Fix \( d \in {\mathbb{Z}}_{ + } \) and \( {\lambda }_{1},\ldots ,{\lambda }_{d} \in \mathcal{W} \) . Note that the limiting logarithmic moment generating function associated with \( \left\{ {{\mu }_{\epsilon } \circ {p}_{{\lambda }_{1},\ldots ,{\lambda }_{d}}^{-1},\epsilon > 0}\right\} \) is \( g\left( \left... | Yes |
Corollary 4.6.14 Let \( \left\{ {\mu }_{\epsilon }\right\} \) be an exponentially tight family of Borel probability measures on the locally convex Hausdorff topological vector space \( \mathcal{E} \) . Suppose \( \Lambda \left( \cdot \right) = \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}\epsilon {\Lambda }_{{\mu }_... | Proof: Let \( \mathcal{W} \) be the topological dual of \( \mathcal{E} \) . Suppose first that \( \mathcal{W} \) is an infinite dimensional vector space, and define \( \mathcal{X} \) according to Assumption 4.6.7. Let \( i : \mathcal{E} \rightarrow \mathcal{X} \) denote the map \( x \mapsto i\left( x\right) \), where \... | Yes |
Lemma 4.7.4 \( \mathcal{Q}\left( \mathcal{X}\right) \) contains all sup-measures and all set functions of the form \( {\mu }^{\epsilon } \) for \( \mu \) a probability measure on \( \mathcal{X} \) and \( \epsilon \in (0,1\rbrack \) . | Proof: Conditions (a) and (c) trivially hold for any sup-measure. Since any point \( y \) in an open set \( G \) is also in \( {G}^{-\delta } \) for some \( \delta = \delta \left( y\right) > 0 \), all sup-measures satisfy condition (d). For (b), let \( \nu \left( {\{ y\} }\right) = {e}^{-I\left( y\right) } \) . Fix \( ... | Yes |
Lemma 4.7.9 If \( \nu \in \mathcal{Q}\left( \mathcal{X}\right) \) is tight, then for any \( \Gamma \in {\mathcal{B}}_{\mathcal{X}} \) , \n\n\[ \nu \left( \bar{\Gamma }\right) = \mathop{\lim }\limits_{{\delta \rightarrow 0}}\nu \left( {\Gamma }^{\delta, o}\right) . \] | Proof: Fix a non-empty set \( \Gamma \in {\mathcal{B}}_{\mathcal{X}},\eta > 0 \) and a compact set \( K = {K}_{\eta } \) for which \( \nu \left( {K}_{\eta }^{c}\right) < \eta \) . For any open set \( G \subset \mathcal{X} \) such that \( \bar{\Gamma } \subset G \), either \( K \subset G \) or else the non-empty compact... | Yes |
Lemma 5.1.8 Let \( J \) denote the collection of all ordered finite subsets of \( (0,1\rbrack \) . For any \( j = \left\{ {0 < {t}_{1} < {t}_{2} < \cdots < {t}_{\left| j\right| } \leq 1}\right\} \in J \) and any \( f : \left\lbrack {0,1}\right\rbrack \rightarrow \) \( {\mathbb{R}}^{d} \), let \( {p}_{j}\left( f\right) ... | Proof: Fix \( j \in J \) and observe that \( {\mu }_{n} \circ {p}_{j}^{-1} \) is the law of the random vector \[ {Z}_{n}^{j} \triangleq \left( {{Z}_{n}\left( {t}_{1}\right) ,{Z}_{n}\left( {t}_{2}\right) ,\ldots ,{Z}_{n}\left( {t}_{\left| j\right| }\right) }\right) . \] Let \[ {Y}_{n}^{j} \triangleq \left( {{Z}_{n}\left... | Yes |
Lemma 5.1.14 Let \( X \) be a real valued random variable distributed according to the law \( \nu \) . Then \( E\left\lbrack {e}^{\delta {\Lambda }_{\nu }^{ * }\left( X\right) }\right\rbrack < \infty \) for all \( \delta < 1 \) . | Proof: Let \( {\Lambda }_{\nu } \) denotes the logarithmic moment generating function of \( X \) . If \( {\Lambda }_{\nu }\left( \lambda \right) = \infty \) for all \( \lambda \neq 0 \), then \( {\Lambda }_{\nu }^{ * } \) is identically zero and the lemma trivially holds. Assume otherwise and recall that then \( \bar{x... | Yes |
Theorem 5.1.19 The probability measures \( {\nu }_{\epsilon } \) induced on \( {L}_{\infty }\left( \left\lbrack {0,1}\right\rbrack \right) \) by \( {Y}_{\epsilon }\left( \cdot \right) \) satisfy the LDP with the good rate function \( I\left( \cdot \right) \) of (5.1.3). | Proof: For any sequence \( {\epsilon }_{m} \rightarrow 0 \) such that \( {\epsilon }_{m}^{-1} \) are integers, Theorem 5.1.19 is a consequence of Theorem 5.1.2. Consider now an arbitrary sequence \( {\epsilon }_{m} \rightarrow 0 \) and let \( {n}_{m} \triangleq \left\lbrack {\epsilon }_{m}^{-1}\right\rbrack \) . By The... | No |
Theorem 5.2.3 (Schilder) \( \left\{ {\nu }_{\epsilon }\right\} \) satisfies, in \( {C}_{0}\left( \left\lbrack {0,1}\right\rbrack \right) \), an LDP with good rate function\n\n\[ \n{I}_{w}\left( \phi \right) = \left\{ \begin{matrix} \frac{1}{2}{\int }_{0}^{1}{\left| \dot{\phi }\left( t\right) \right| }^{2}{dt}, & \phi \... | Proof: Observe that the process\n\n\[ \n{\widehat{w}}_{\epsilon }\left( t\right) \triangleq {w}_{\epsilon }\left( {\epsilon \left\lbrack \begin{matrix} t \\ \epsilon \end{matrix}\right\rbrack }\right) \n\]\n\nis merely the process \( {Y}_{\epsilon }\left( \cdot \right) \) of Section 5.1, for the particular choice of \(... | Yes |
Theorem 5.3.1 The sequence \( \left\{ {\mu }_{n}\right\} \) satisfies in \( {L}_{\infty }\left( {\left\lbrack 0,1\right\rbrack }^{d}\right) \) the LDP with the good rate function\n\n\[ I\left( \phi \right) = \left\{ \begin{matrix} {\int }_{{\left\lbrack 0,1\right\rbrack }^{d}}{\Lambda }^{ * }\left( \frac{d{\mu }_{\phi ... | Proof: The proof of the theorem follows closely the proof of Theorem 5.1.2. Let \( {\widetilde{\mu }}_{n} \) denote the law on \( {L}_{\infty }\left( {\left\lbrack 0,1\right\rbrack }^{d}\right) \) induced by the natural polygonal interpolation of \( {Z}_{n}\left( t\right) \) . E.g., for \( d = 2 \), it is induced by th... | No |
Lemma 5.3.3 Let \( \mathcal{X} \) consist of all the maps from \( {\left\lbrack 0,1\right\rbrack }^{d} \) to \( \mathbb{R} \) such that the axis \( \left( {0,{t}_{2},\ldots ,{t}_{d}}\right) ,\left( {{t}_{1},0,\ldots ,{t}_{d}}\right) ,\ldots ,\left( {{t}_{1},\ldots ,{t}_{d - 1},0}\right) \) are mapped to zero, and equip... | Proof of Lemma 5.3.3: Applying the projective limit argument as in the proof of Lemma 5.1.6, one concludes that \( {\widetilde{\mu }}_{n} \) satisfies the LDP in \( \mathcal{X} \) with the good rate function\n\n\[ \begin{array}{l} {I}_{\mathcal{X}}\left( \phi \right) = \mathop{\sup }\limits_{{k < \infty }}\mathop{\sup ... | Yes |
Theorem 5.4.6\n\n\[ \n- \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}\epsilon \log {P}_{e}^{\epsilon } = \mathop{\inf }\limits_{{\phi \in \Phi }}{I}_{w}\left( \phi \right) \triangleq {I}_{0} < \infty , \n\] \n\nwhere \n\n\[ \n{I}_{w}\left( \phi \right) = \left\{ \begin{matrix} \frac{1}{2}{\int }_{0}^{2T}{\dot{\phi }... | Proof: Let \( {\widehat{\phi }}_{t} = - {\pi t}/{MT} \) . Since \( F\left( \widehat{\phi }\right) = 0 \), it follows that \n\n\[ \n{I}_{0} \leq {I}_{w}\left( \widehat{\phi }\right) = \frac{{\pi }^{2}}{{M}^{2}T} < \infty . \n\] \n\n(5.4.7) \n\nNow, note that \n\n\[ \n\mathrm{P}\left( {F\left( {\sqrt{\epsilon }{w}_{ \cdo... | Yes |
Theorem 5.4.9\n\[ \frac{{\pi }^{2}}{2{M}^{2}T} \leq {I}_{0} \leq \frac{{\pi }^{2}}{{M}^{2}T}. \] | Proof: The upper bound on \( {I}_{0} \) is merely (5.4.7), and it implies that\n\[ {I}_{0} = \mathop{\inf }\limits_{{\phi \in \widehat{\Phi }}}{I}_{w}\left( \phi \right) \]\nwhere\n\[ \widehat{\Phi } = \Phi \cap \left\{ {\phi : {I}_{w}\left( \phi \right) \leq \frac{{\pi }^{2}}{{M}^{2}T}}\right\} .\n\nFix \( \phi \in \w... | No |
Lemma 5.4.15 If \( \left| \alpha \right| \leq {2\pi }/\sqrt{3}M \), then\n\n\[ \widetilde{I}\left( \alpha \right) = \frac{3{\alpha }^{2}}{4T}. \] | Proof: Let \( {\psi }_{s} = \min \{ {2T}, T + s\} - \max \{ T, s\} \) (see Fig. 5.4.2), and note that for any \( \phi \in {H}_{1}\left( \left\lbrack {0,{2T}}\right\rbrack \right) \) ,\n\n\[ {\int }_{T}^{2T}\left( {{\phi }_{t} - {\phi }_{t - T}}\right) {dt} = {\int }_{T}^{2T}{\int }_{t - T}^{t}{\dot{\phi }}_{s}{dsdt} = ... | Yes |
Lemma 5.5.3 For all \( x \in {\mathbb{R}}^{d}, t > 0 \) ,\n\n\[ V\left( {x, t}\right) = \mathop{\sup }\limits_{{\lambda \in {\mathbb{R}}^{d}}}\{ \langle \lambda, x\rangle + t\left\lbrack {\langle \lambda, b\rangle - \Lambda \left( \lambda \right) }\right\rbrack \} = t{\Lambda }^{ * }\left( {\frac{x}{t} + b}\right) ,\]\... | Proof: Recall that \( {\Lambda }^{ * }\left( \cdot \right) \) is convex. Hence, for all \( t > 0 \) and any \( \phi \in {\mathcal{{AC}}}^{t} \) with \( {\phi }_{0} = 0 \), by Jensen’s inequality,\n\n\[ {I}_{t}\left( \phi \right) = t{\int }_{0}^{t}{\Lambda }^{ * }\left( {{\dot{\phi }}_{s} + b}\right) \frac{ds}{t} \geq t... | Yes |
Theorem 5.5.6 Let Assumption 5.5.5 hold. Then\n\n\[ \n{V}_{A} = \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}\epsilon \log {T}_{\epsilon }\text{ in probability } \n\]\n\n(5.5.7)\n\nand\n\n\[ \n\bar{t} = \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}{L}_{\epsilon }\text{ in probability. } \n\]\n\n(5.5.8) | Proof: The following properties of the cost function, whose proofs are deferred to the end of the section, are needed.\n\nLemma 5.5.10\n\n\[ \n\matho | No |
Lemma 5.5.19 For any \( \delta > 0 \) ,\n\n\[ \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}\mathrm{P}\left( {{T}_{\epsilon } > {e}^{\left( {{V}_{A} + \delta }\right) /\epsilon }}\right) = 0. \] | Proof: Let \( \Delta = \left\lbrack {\left( {\bar{t} + 2}\right) /\epsilon }\right\rbrack \epsilon \) and split the time interval \( \left\lbrack {0,{e}^{\left( {{V}_{A} + \delta }\right) /\epsilon }}\right\rbrack \) into disjoint intervals of length \( \Delta \) each. Let \( {N}_{\epsilon } \) be the (integer part of ... | Yes |
\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\lim }\limits_{{\epsilon \rightarrow 0}}\mathrm{P}\left( {{T}_{\epsilon, n} \neq {T}_{\epsilon }}\right) = 0 \] | Proof: Divide \( \left\lbrack {{C}_{\epsilon },\infty }\right) \) into the disjoint intervals \( {I}_{\ell } \triangleq \left\lbrack {\left( {2\ell - 1}\right) {C}_{\epsilon },\left( {2\ell + 1}\right) {C}_{\epsilon }}\right) \) , \( \ell = 1,\ldots \) Define the events\n\n\[ {J}_{\ell } \triangleq \left\{ {{Y}_{t}^{\e... | Yes |
Theorem 5.6.3 \( \left\{ {x}_{t}^{\epsilon }\right\} \) satisfies the LDP in \( {C}_{0}\left( \left\lbrack {0,1}\right\rbrack \right) \) with the good rate function | Proof: Theorem 4.2.1 is applicable here, as \( F \) is continuous on \( {C}_{0}\left( \left\lbrack {0,1}\right\rbrack \right) \) . Indeed, if \( {f}_{1} = F\left( {g}_{1}\right) ,{f}_{2} = F\left( {g}_{2}\right) \), then by (5.6.2), \[ {f}_{1}\left( t\right) - {f}_{2}\left( t\right) = {\int }_{0}^{t}\left\lbrack {b\lef... | Yes |
Theorem 5.6.7 If all the entries of \( b \) and \( \sigma \) are bounded, uniformly Lipschitz continuous functions, then \( \left\{ {x}_{t}^{\epsilon }\right\} \), the solution of (5.6.5), satisfies the LDP in \( C\left( \left\lbrack {0,1}\right\rbrack \right) \) with the good rate function \( {I}_{x}\left( \cdot \righ... | Proof: It suffices to prove the theorem for \( x = 0 \) (as \( x \) may always be moved to the origin by a translation of the coordinates). Then the measure \( {\widetilde{\mu }}_{\epsilon } \) of \( {x}^{\epsilon } \) is supported on \( {C}_{0}\left( \left\lbrack {0,1}\right\rbrack \right) \) . The proof here is based... | Yes |
Corollary 5.6.15 Assume the conditions of Theorem 5.6.7. Then for any compact \( K \subset {\mathbb{R}}^{d} \) and any closed \( F \subset C\left( \left\lbrack {0,1}\right\rbrack \right) \) , \[ \mathop{\limsup }\limits_{{\epsilon \rightarrow 0}}\epsilon \log \mathop{\sup }\limits_{{y \in K}}\mathrm{P}\left( {{X}^{\eps... | Proof: Let \( - {I}_{K} \) denote the right side of (5.6.16). Fix \( \delta > 0 \) and let \( {I}_{K}^{\delta } \triangleq \min \left\{ {{I}_{K} - \delta ,1/\delta }\right\} \) . Then from (5.6.13), it follows that for any \( x \in K \) , there exists an \( {\epsilon }_{x} > 0 \) such that for all \( \epsilon \leq {\ep... | Yes |
Lemma 5.6.18 Let \( {b}_{t},{\sigma }_{t} \) be progressively measurable processes, and let\n\n\[ d{z}_{t} = {b}_{t}{dt} + \sqrt{\epsilon }{\sigma }_{t}d{w}_{t} \]\n\nwhere \( {z}_{0} \) is deterministic. Let \( {\tau }_{1} \in \left\lbrack {0,1}\right\rbrack \) be a stopping time with respect to the filtration of \( \... | Proof: Let \( {u}_{t} = \phi \left( {z}_{t}\right) \), where \( \phi \left( y\right) = {\left( {\rho }^{2} + {\left| y\right| }^{2}\right) }^{1/\epsilon } \). By Itô’s formula, \( {u}_{t} \) is the strong solution of the stochastic differential equation\n\n\[ d{u}_{t} = \nabla \phi {\left( {z}_{t}\right) }^{\prime }d{z... | Yes |
Assume the conditions of Theorem 5.7.3. Let \( {u}_{1}\left( x\right) = \) \( {E}_{x}\left( {\tau }^{\epsilon }\right) \) . Then \( {u}_{1} \) is the unique solution of\n\n\[ \n{L}^{\epsilon }{u}_{1} = - 1,\;\text{ in }\;G\;;\;{u}_{1} = 0,\;\text{ on }\;\partial G.\n\] | Proof: To establish (5.7.5), specialize Theorem 5.7.3 to \( f \equiv 0 \) and \( g \equiv 1 \) . | No |
Lemma 5.7.8 Assume (A-4). For any \( \delta > 0 \), there exists \( \rho > 0 \) small enough such that\n\n\[ \mathop{\sup }\limits_{{x, y \in {B}_{\rho }}}\mathop{\inf }\limits_{{t \in \left\lbrack {0,1}\right\rbrack }}V\left( {x, y, t}\right) < \delta \]\n\n\( \left( {5.7.9}\right) \)\n\nand\n\n\[ \mathop{\sup }\limit... | Proof: Observe that the function \( {\phi }_{s} \) described in Assumption (A-4) results in the upper bound \( V\left( {x, y, t}\right) \leq {M}^{2}t/2 \), where \( t = T\left( \left| {x - y}\right| \right) \rightarrow 0 \) as \( \left| {x - y}\right| \rightarrow 0 \) . Equations (5.7.9) and (5.7.10) follow from this b... | Yes |
Corollary 5.7.16 Part (a) of Theorem 5.7.11 holds true under Assumptions (A-1), (A-3), and (A-4). Moreover, it remains true for the exit time of any processes \( \\left\\{ {\\widetilde{x}}^{\\epsilon }\\right\\} \) (not necessarily Markov) that satisfy for any \( {T}^{\\prime },\\delta \) fixed, and any stopping times ... | \[ \n\\mathop{\\limsup }\\limits_{{\\epsilon \\rightarrow 0}}\\epsilon \\log \\mathrm{P}\\left( {\\mathop{\\sup }\\limits_{{t \\in \\left\\lbrack {0,{T}^{\\prime }}\\right\\rbrack }}\\left| {{x}_{t}^{\\epsilon } - {\\widetilde{x}}_{t + {T}_{\\epsilon }}^{\\epsilon }}\\right| > \\delta \\left| {\\;{\\mathcal{F}}_{{T}_{\... | Yes |
Lemma 5.7.23 For every \( \rho > 0 \) and every \( c > 0 \), there exists a constant \( T\left( {c,\rho }\right) < \infty \) such that | \[ \mathop{\limsup }\limits_{{\epsilon \rightarrow 0}}\epsilon \log \mathop{\sup }\limits_{{x \in G}}{P}_{x}\left( {\mathop{\sup }\limits_{{t \in \left\lbrack {0, T\left( {c,\rho }\right) }\right\rbrack }}\left| {{x}_{t}^{\epsilon } - x}\right| \geq \rho }\right) < - c. \] | No |
\[ \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}\epsilon \log E\left( \tau \right) = \frac{{\theta }_{cr}^{2}}{2} \] | Proof: Let \( {z}_{t} = {\theta }_{t} - {u}_{t} \) denote the angular tracking error. Then \( {z}_{t} \) satisfies the equation\n\n\[ d{z}_{t} = \left( {m\left( {\theta }_{t}\right) - m\left( {u}_{t}\right) }\right) {dt} - \frac{1}{\epsilon }{z}_{t}{dt} - d{v}_{t} + d{w}_{t},\;{z}_{0} = 0, \]\n\nwhich under the time ch... | Yes |
Lemma 5.8.8\n\n\[ \mathop{\sup }\limits_{{{t}^{\prime } \geq 0}}\left| {{z}_{{t}^{\prime }} - {\widehat{z}}_{{t}^{\prime }}}\right| \leq {2\epsilon }\mathop{\sup }\limits_{{x \in {\mathbb{R}}^{2}}}\left| {m\left( x\right) }\right| \text{ almost surely }.\] | Proof: Let \( {e}_{{t}^{\prime }} = {z}_{{t}^{\prime }} - {\widehat{z}}_{{t}^{\prime }} \) . Then\n\n\[ {\dot{e}}_{{t}^{\prime }} = \epsilon \left( {m\left( {\theta }_{{t}^{\prime }}\right) - m\left( {u}_{{t}^{\prime }}\right) }\right) - {e}_{{t}^{\prime }}\;,\;{e}_{0} = 0.\]\n\nHence,\n\n\[ {e}_{{t}^{\prime }} = \epsi... | Yes |
Theorem 6.1.3 Let Assumption 6.1.2 hold. Then \( \left\{ {\mu }_{n}\right\} \) satisfies in \( \mathcal{X} \) (and \( \mathcal{E} \) ) a weak LDP with rate function \( {\Lambda }^{ * } \) . Moreover, for every open, convex subset \( A \subset \mathcal{X} \) , \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n... | (6.1.4) | No |
Corollary 6.1.6 The sequence \( \left\{ {\mu }_{n}\right\} \) of the laws of empirical means of \( {\mathbb{R}}^{d} \) - valued i.i.d. random variables satisfies a weak LDP with the convex rate function \( {\Lambda }^{ * } \) . Moreover, if \( 0 \in {\mathcal{D}}_{\Lambda }^{o} \), then \( \left\{ {\mu }_{n}\right\} \)... | Proof: The weak LDP is merely a specialization of Theorem 6.1.3. If \( 0 \in {\mathcal{D}}_{\Lambda }^{o} \), the full LDP follows, since \( \left\{ {\mu }_{n}\right\} \subset {M}_{1}\left( {\mathbb{R}}^{d}\right) \) is then exponentially tight. (See the inequality (2.2.33) and the discussion there.) | Yes |
Lemma 6.1.11 (Sub-additivity) If \( f : {\mathbb{Z}}_{ + } \rightarrow \left\lbrack {0,\infty }\right\rbrack \) is a sub-additive function such that \( f\left( n\right) < \infty \) for all \( n \geq N \) and some \( N < \infty \), then\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{f\left( n\right) }{n} = \... | Proof: Fix \( m \geq N \) and let \( {M}_{m} \triangleq \max \{ f\left( r\right) : m \leq r \leq {2m}\} \) . By assumption, \( {M}_{m} < \infty \) . For each \( n \geq m \geq N \), let \( s = \lfloor n/m\rfloor \geq 1 \) and \( r = n - m\left( {s - 1}\right) \in \{ m,\ldots ,{2m}\} \) . Since \( f \) is sub-additive,\n... | Yes |
Lemma 6.1.12 Let part (a) of Assumption 6.1.2 hold true. Then, for every convex \( A \in {\mathcal{B}}_{\mathcal{X}} \), the function \( f\left( n\right) \triangleq - \log {\mu }_{n}\left( A\right) \) is sub-additive. | Proof: Without loss of generality, it may be assumed that \( A \subset \mathcal{E} \) . Now,\n\n\[ \n{\widehat{S}}_{m + n} = \frac{m}{m + n}{\widehat{S}}_{m} + \frac{n}{m + n}{\widehat{S}}_{m + n}^{m}.\n\]\n\nTherefore, \( {\widehat{S}}_{m + n} \) is a convex combination (with deterministic coefficients) of the indepen... | Yes |
Corollary 6.2.3 The empirical measures \( {L}_{n}^{\mathbf{Y}} \) satisfy a weak LDP in \( {M}_{1}\left( \sum \right) \) (equipped with the weak topology and \( \mathcal{B} = {\mathcal{B}}^{w} \) ) with the convex rate function | \[ {\Lambda }^{ * }\left( \nu \right) = \mathop{\sup }\limits_{{\phi \in {C}_{b}\left( \sum \right) }}\{ \langle \phi ,\nu \rangle - \Lambda \left( \phi \right) \} ,\;\nu \in {M}_{1}\left( \sum \right) ,\] where for \( \phi \in {C}_{b}\left( \sum \right) \), \[ \Lambda \left( \phi \right) \triangleq \log E\left\lbrack ... | Yes |
Lemma 6.2.6 The laws of \( {L}_{n}^{\mathbf{Y}} \) of (6.2.1) are exponentially tight. | Proof: Note that, by Theorem D.7, \( \mu \in {M}_{1}\left( \sum \right) \) is tight, and in particular, there exist compact sets \( {\Gamma }_{\ell } \subset \sum ,\ell = 1,2,\ldots \) such that\n\n\[ \mu \left( {\Gamma }_{\ell }^{c}\right) \leq {e}^{-2{\ell }^{2}}\left( {{e}^{\ell } - 1}\right) . \]\n\n(6.2.7)\n\nThe ... | Yes |
Lemma 6.2.13 The identity \( H\left( {\cdot \mid \mu }\right) = {\Lambda }^{ * }\left( \cdot \right) \) holds over \( \mathcal{X} \) . Moreover, the definitions (6.2.11) and (6.2.4) yield the same function over \( {M}_{1}\left( \sum \right) \) . | Proof: Observe that \( \mathcal{X} \) is, by Theorem B.8, a locally convex Hausdorff topological vector space whose (topological) dual \( {\mathcal{X}}^{ * } \) is \( B\left( \sum \right) \) . By combining Lemma 6.2.12 with the duality lemma (Lemma 4.5.8) (for \( f = H\left( {\cdot \mid \mu }\right) \) ), the identity ... | Yes |
Lemma 6.2.16 Suppose \( \gamma \) is a convex, good rate function on \( \mathbb{R} \) such that \( \gamma \left( x\right) /\left| x\right| \rightarrow \infty \) for \( \left| x\right| \rightarrow \infty \) . Then, \[ {I}_{\gamma }\left( \nu \right) \overset{\bigtriangleup }{ = }\left\{ \begin{matrix} {\int }_{\sum }\ga... | Proof: Since \( \gamma \left( \cdot \right) \geq 0 \), also \( {I}_{\gamma }\left( \cdot \right) \geq 0 \) . Fix an \( \alpha < \infty \) and consider the set \[ {\Psi }_{I}\left( \alpha \right) \triangleq \left\{ {f \in {L}_{1}\left( \mu \right) : {\int }_{\sum }\gamma \left( f\right) {d\mu } \leq \alpha }\right\} . \... | Yes |
Lemma 6.3.1 Let \( A \in {\mathcal{B}}_{M\left( \sum \right) } \) be convex. Define\n\n\[ \n{\widetilde{\mu }}_{n}\left( A\right) \triangleq \mathop{\inf }\limits_{{\sigma \in \sum }}{\mu }_{n,\sigma }\left( A\right)\n\]\n\nThen \( {\widetilde{\mu }}_{n}\left( A\right) \) is super multiplicative, i.e.,\n\n\[ \n{\wideti... | Proof: Note that\n\n\[ \n{\mu }_{n + m,\sigma }\left( A\right) = {P}_{\sigma }\left( {{L}_{n + m}^{\mathbf{Y}} \in A}\right) \geq {P}_{\sigma }\left( {{L}_{n}^{\mathbf{Y}} \in A,{L}_{n + m}^{\mathbf{Y}, n} \in A}\right)\n\]\n\n\[ \n= {\int }_{{\sum }^{n}}{1}_{\left\{ {L}_{n}^{\mathbf{Y}} \in A\right\} }{\mu }_{m,{Y}_{n... | Yes |
Lemma 6.3.3 For any \( {A}_{\delta } \in \Theta \), either\n\n\[ \n{\widetilde{\mu }}_{n}\left( {A}_{\delta /2}\right) = 0,\;\forall n \in {\mathbb{Z}}_{ + },\n\]\n\nor\n\n\[ \n{\widetilde{\mu }}_{n}\left( {A}_{\delta }\right) > 0,\;\forall n \geq {n}_{0}\left( {A}_{\delta }\right) .\n\] | Proof: Let \( {A}_{\delta } = { \cap }_{i = 1}^{N}{U}_{{f}_{i},{x}_{i},{\delta }_{i}} \) . Assume that \( {\widetilde{\mu }}_{m}\left( {A}_{\delta /2}\right) > 0 \) for some \( m \) . For any \( n \geq m \), let \( {q}_{n} = \left\lbrack {n/m}\right\rbrack ,{r}_{n} = n - {q}_{n}m \) . Then\n\n\[ \n{\mu }_{n,\sigma }\le... | Yes |
Lemma 6.3.5 Assume (U). For any \( {A}_{\delta } \in \Theta \), there exists an \( {n}_{0}\left( {A}_{\delta }\right) \) such that for all \( n > {n}_{0}\left( {A}_{\delta }\right) \), \[ \mathop{\sup }\limits_{{\sigma \in \sum }}{\mu }_{n,\sigma }\left( {A}_{\delta /2}\right) \leq M{\widetilde{\mu }}_{n}\left( {A}_{\d... | Proof: Let \( \ell \) be as in (U), and \( n = {q}_{n}\ell + {r}_{n} \) where \( 0 \leq {r}_{n} \leq \ell - 1 \) . Then with \( {A}_{\delta } = { \cap }_{i = 1}^{{N}^{\prime }}{U}_{{f}_{i},{x}_{i},{\delta }_{i}} \), \[ {\mu }_{n,\sigma }\left( {A}_{\delta /2}\right) = {P}_{\sigma }\left( {\frac{1}{n}\mathop{\sum }\limi... | Yes |
Theorem 6.3.8 Assume (U). Then the following limits exist:\n\n\[ \Lambda \left( f\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log {E}_{\sigma }\left( {\exp \left( {\mathop{\sum }\limits_{{i = 1}}^{n}f\left( {Y}_{i}\right) }\right) }\right) \]\n\nwhere \( f \in {C}_{b}\left( \sum \right) \). More... | Proof: Note that, since \( f \) is bounded,\n\n\[ \mathop{\limsup }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log {E}_{\sigma }\left( {\exp \left( {\mathop{\sum }\limits_{{i = 1}}^{n}f\left( {Y}_{i}\right) }\right) }\right) \leq \parallel f\parallel < \infty . \]\n\nThe reader will prove the exponential tightness of ... | No |
Theorem 6.4.4 Let Assumption 6.4.1 hold. Then \( \left\{ {\mu }_{n}\right\} \) satisfies the LDP in \( {\mathbb{R}}^{d} \) with the good convex rate function \( {\Lambda }^{ * }\left( \cdot \right) \), which is the Fenchel-Legendre transform of \[ \Lambda \left( \lambda \right) = \mathop{\lim }\limits_{{n \rightarrow \... | Proof: Note that \( {\widehat{S}}_{n} \in K \) for all \( n \in {\mathbb{Z}}_{ + } \), and hence the sequence \( \left\{ {\mu }_{n}\right\} \) is exponentially tight. Consequently, when combined with Lemma 4.4.8 and Theorem 4.4.10, the following lemma, whose proof is deferred, implies that \( {\mu }_{n} \) satisfies th... | No |
Lemma 6.4.7 Let assumption 6.4.1 hold. Suppose \( \delta > 0 \) and \( {x}_{1},{x}_{2} \) are such that for \( i = 1,2 \) , \[ \mathop{\liminf }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log {\mu }_{n}\left( {B}_{{x}_{i},\delta /2}\right) > - \infty . \] Then for all \( n \) large enough, \[ {\mu }_{2n}\left( {B}_{\l... | Fix \( {x}_{1},{x}_{2} \) such that \( I\left( {x}_{1}\right) < \infty \) and \( I\left( {x}_{2}\right) < \infty \) . Then due to (6.4.9), Lemma 6.4.7 holds for \( {x}_{1},{x}_{2} \) and all \( \delta > 0 \) . Note that \( y \in {B}_{\left( {{x}_{1} + {x}_{2}}\right) /2,\delta } \) implies the inclusion \( {B}_{\left( ... | Yes |
Lemma 6.4.10 (Approximate sub-additivity) Assume \( f : {\mathbb{Z}}_{ + } \rightarrow \mathbb{R} \) is such that for all \( n, m \geq 1 \), \[ f\left( {n + m}\right) \leq f\left( n\right) + f\left( m\right) + \epsilon \left( {n + m}\right) ,\] where for some \( \delta > 0 \), \[ \mathop{\limsup }\limits_{{n \rightarro... | Proof of Lemma 6.4.10: Fix \( s \in {\mathbb{Z}}_{ + }, s \geq 2 \). Observe that for all \( m \geq 1 \), \( 1 \leq r \leq \left( {s - 1}\right) \[ f\left( {{ms} + r}\right) \leq f\left( {ms}\right) + f\left( r\right) + \epsilon \left( {{ms} + r}\right) . \] Hence, \[ \frac{f\left( {{ms} + r}\right) }{{ms} + r} \leq \f... | Yes |
Theorem 6.4.14 (a) Let \( {\left\{ {g}_{j}\right\} }_{j = 1}^{d} \in B\left( \sum \right) \) . Define the \( {\mathbb{R}}^{d} \) -valued stationary process \( {X}_{1},\ldots ,{X}_{n},\ldots \) by \( {X}_{i} = \left( {{g}_{1}\left( {Y}_{i}\right) ,\ldots ,{g}_{d}\left( {Y}_{i}\right) }\right) \) . Suppose that, for any ... | Proof: The proof relies on part (b) of Corollary 4.6.11. Indeed, the triplet \( \left( {B{\left( \sum \right) }^{\prime },{\mathcal{B}}^{cy},{\mu }_{n}}\right) \) satisfies Assumption 4.6.8 with \( \mathcal{W} = B\left( \sum \right) \) . Moreover, by Theorem 6.4.4, for any \( {g}_{1},\ldots ,{g}_{d} \in B\left( \sum \r... | Yes |
Lemma 6.4.18 Let \( {Y}_{1},\ldots ,{Y}_{n},\ldots \) be the stationary process defined before. (a) Assume that (H-1) holds and \( \Lambda \left( f\right) \) exists for all \( f \in B\left( \sum \right) \) . Then the inequality (6.4.16) holds true. | Proof: (a) Since \( \Lambda \left( f\right) \) exists, it is enough to consider limits along the sequence \( n = m\ell \) . By Jensen’s inequality,\n\n\[ \n{E}_{P}\left\lbrack {e}^{\mathop{\sum }\limits_{{i = 1}}^{{m\ell }}f\left( {Y}_{i}\right) }\right\rbrack = {E}_{P}\left\lbrack {e}^{{\ell }^{-1}\mathop{\sum }\limit... | Yes |
Theorem 6.5.2 Let \( {P}_{1} \) be an invariant measure for \( \pi \left( {\cdot , \cdot }\right) \) . Then for all \( \nu \in {M}_{1}\left( \sum \right) \)\n\n\[ \mathop{\sup }\limits_{{f \in B\left( \sum \right) }}\{ \langle \nu, f\rangle - \Lambda \left( f\right) \}\]\n\n\[ = \left\{ \begin{matrix} \mathop{\sup }\li... | Proof: (a) Suppose that \( {d\nu }/d{P}_{1} \) does not exist, i.e., there exists a set \( A \in {\mathcal{B}}_{\sum } \) such that \( {P}_{1}\left( A\right) = 0 \) while \( \nu \left( A\right) > 0 \) . For any \( \alpha > 0 \), the function \( \alpha {1}_{A} \) belongs to \( B\left( \sum \right) \), and by the union o... | Yes |
Theorem 6.5.4 Assume that (U) of Section 6.3 holds.\n\nThen, for all \( \nu \in {M}_{1}\left( \sum \right) \),\n\n\[ {I}^{U}\left( \nu \right) = \mathop{\sup }\limits_{{u \in {C}_{b}\left( \sum \right), u \geq 1}}\left\{ {-{\int }_{\sum }\log \left( \frac{\pi u}{u}\right) {d\nu }}\right\} .\n\] | Proof: Recall that Assumption (U) implies that for any \( \mu \in {M}_{1}\left( \sum \right) \) and any \( \sigma \in \sum \),\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log {E}_{\sigma }\left\lbrack {e}^{\mathop{\sum }\limits_{{i = 1}}^{n}f\left( {Y}_{i}\right) }\right\rbrack = \mathop{\lim }\lim... | No |
Corollary 6.5.10 (a) Assume \( \pi \) satisfies Assumption (U) of Section 6.3. Then \( {L}_{n, k}^{\mathbf{Y}} \) satisfies (in the weak topology of \( {M}_{1}\left( {\sum }^{k}\right) \) ) the LDP with the good rate function | \[ {I}_{k}^{U}\left( \nu \right) \triangleq \mathop{\sup }\limits_{{u \in B\left( {\sum }^{k}\right), u \geq 1}}\left\{ {-{\int }_{{\sum }^{k}}\log \left( \frac{{\pi }_{k}u}{u}\right) {d\nu }}\right\} . \] | Yes |
Lemma 6.5.16 (Pinsker) Let \( \sum \) be Polish and \( \nu ,\mu \in {M}_{1}\left( {\sum }^{{\mathbb{Z}}_{1}}\right) \) . Then\n\n\[ H\left( {{\bar{p}}_{k}\nu \mid {\bar{p}}_{k}\mu }\right) \nearrow H\left( {\nu \mid \mu }\right) \text{ as }k \rightarrow \infty . \] | Proof of Lemma 6.5.16: Recall that\n\n\[ H\left( {{\bar{p}}_{k}\nu \mid {\bar{p}}_{k}\mu }\right) = \mathop{\sup }\limits_{{\phi \in {C}_{b}\left( {\sum }^{k}\right) }}\left\{ {{\int }_{{\sum }^{k}}{\phi d}{\bar{p}}_{k}\nu - \log \left( {{\int }_{{\sum }^{k}}{e}^{\phi }d{\bar{p}}_{k}\mu }\right) }\right\} ,\]\n\nwherea... | Yes |
Theorem 6.6.1 (Dupuis-Ellis) Suppose \( I\left( \cdot \right) \) is a good rate function on a Polish space \( \mathcal{X} \) such that for a family of probability measures \( \left\{ {\widetilde{\mu }}_{\epsilon }\right\} \) on \( \mathcal{X} \) and any \( f \in {C}_{b}\left( \mathcal{X}\right) \) , \[ {\Lambda }_{f} \... | Proof: The duality in Lemma 6.2.13 (see in particular (6.2.14)) implies that for all \( f \in {C}_{b}\left( \mathcal{X}\right) \) , \[ \log {\int }_{\mathcal{X}}{e}^{f\left( x\right) /\epsilon }{\widetilde{\mu }}_{\epsilon }\left( {dx}\right) = \mathop{\sup }\limits_{{\widetilde{\nu } \in {M}_{1}\left( \mathcal{X}\righ... | Yes |
Lemma 6.6.3 For any \( f \in {C}_{b}\left( \mathcal{X}\right) \) and \( {\mu }^{\left( n\right) } \in {M}_{1}\left( {\sum }^{n}\right) \) , \[ {\Lambda }_{f, n}\overset{\bigtriangleup }{ \triangleq }\mathop{\sup }\limits_{{\widetilde{\nu } \in {M}_{1}\left( \mathcal{X}\right) }}\left\{ {\langle f,\widetilde{\nu }\rangl... | Proof: Fix \( f \in {C}_{b}\left( \mathcal{X}\right), n \in {\mathbb{Z}}_{ + } \) and \( {\mu }^{\left( n\right) } \in {M}_{1}\left( {\sum }^{n}\right) \) . For computing \( {\Lambda }_{f, n} \) it suffices to consider \( \widetilde{\nu } = {\widetilde{\nu }}_{n} \) such that \( {\rho }_{n} = d{\widetilde{\nu }}_{n}/d{... | Yes |
Theorem 7.1.3 Let Assumption 7.1.2 hold. For any \( \delta > 0 \) and any \( \eta \geq 0 \) , let \( {\mathcal{S}}_{\delta }^{ * } \) denote the test\n\n\[ \n{\mathcal{S}}_{\delta }^{ * }\left( z\right) = \left\{ \begin{array}{ll} 0 & \text{ if }{J}_{\delta }\left( z\right) < \eta \\ 1 & \text{ otherwise,} \end{array}\... | The proof is based on the following lemmas: | Yes |
Lemma 7.1.8 For any test satisfying (7.1.5), and all \( n > {n}_{0}\left( \delta \right) \) , \n\n\[ \n{\mathcal{S}}_{0}^{*,\delta } \subseteq {\mathcal{S}}_{0}^{n,\delta } \n\] | Proof of Lemma 7.1.8: Assume that \( \left\{ {\mathcal{S}}^{n}\right\} \) is a test for which (7.1.9) does not hold. Then for some infinite subsequence \( \left\{ {n}_{m}\right\} \), there exist \( {z}_{m} \in {\mathcal{S}}_{0}^{*,\delta } \) such that \( {z}_{m} \in {\mathcal{S}}_{1}^{{n}_{m},\delta } \) . Therefore, ... | Yes |
Theorem 7.1.11 Let Assumption 7.1.10 hold.\n\n(a) For all \( \theta \in \Theta \) ,\n\n\[ \mathop{\limsup }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log {\mu }_{\theta, n}\left( {\mathcal{S}}_{1}^{*,\delta }\right) \leq - \eta \]\n\n(7.1.12) | (a) By a repeat of the proof of Lemma 7.1.7, it is concluded that for all \( \delta > 0 \) and all \( \theta \in \Theta \) ,\n\n\[ {c}_{\theta }^{\delta } \triangleq \mathop{\inf }\limits_{{z \in \overline{{\mathcal{S}}_{1}^{*,\delta }}}}{I}_{\theta }\left( z\right) \geq \eta \]\n\nTherefore, (7.1.12) follows by the la... | Yes |
Theorem 7.1.19 If \( \bar{A} \) is a compact subset of \( {M}_{1}\left( \sum \right) \), then (7.1.14) holds, and the test \( {\mathcal{S}}^{ * } \) is optimal. | Proof: Assume that the set of measures \( \bar{A} \) is compact. Then by Prohorov’s theorem (Theorem D.9), for each \( \delta > 0 \) there exists a compact set \( {K}_{\delta } \subseteq \) \( \sum \) such that \( \mu \left( {K}_{\delta }^{c}\right) < \delta \) for all \( \mu \in A \) . Fix \( \eta < \infty \) and defi... | Yes |
Theorem 7.2.3 Let \( {X}_{i} \) be real-valued i.i.d. random variables with \( {\Lambda }_{X}\left( \lambda \right) = \) \( \log E\left\lbrack {e}^{\lambda {X}_{1}}\right\rbrack \) finite everywhere and \( {y}_{i} \in \sum \) non-random such that \( {L}_{m}^{\mathbf{y}} \rightarrow \mu \) weakly in \( {M}_{1}\left( \su... | Proof: Note that \( {\Lambda }_{X}^{ * }\left( \cdot \right) \) is a convex good rate function and by Lemma 2.2.20, \( {\Lambda }_{X}^{ * }\left( x\right) /\left| x\right| \rightarrow \infty \) as \( \left| x\right| \rightarrow \infty \) . Hence, by Lemma 6.2.16, \( {I}_{X}\left( \cdot \right) \) is a convex, good rate... | Yes |
Theorem 7.3.3 Assume (A-1). Then \( \mathcal{M} \triangleq \left\{ {\nu \in {F}_{0} : H\left( {\nu \mid \mu }\right) = {I}_{F}}\right\} \) is a non-empty, compact set. Further, for any \( \Gamma \in {\mathcal{B}}^{cy} \) with \( \mathcal{M} \subset {\Gamma }^{o} \), \[ \mathop{\limsup }\limits_{{\delta \rightarrow 0}}\... | Proof: Note that \( {A}_{0} \subseteq {F}_{0} \), so \( {\nu }_{ * } \in \mathcal{M} \) by Assumption (A-1). Moreover, \( {I}_{F} < \infty \) implies that \( \mathcal{M} \) being the intersection of the closed set \( {F}_{0} \) and the compact set \( \left\{ {\nu : H\left( {\nu \mid \mu }\right) \leq {I}_{F}}\right\} \... | Yes |
Lemma 7.3.4 Assume (A-1). Then, for all \( \delta > 0 \) ,\n\n\[ \mathop{\liminf }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log {Q}_{n}\left( {A}_{\delta }\right) \geq - {I}_{F} \] | Proof of Lemma 7.3.4: Since \( {A}_{\delta } \) in general may contain no neighborhood of points from \( \mathcal{M} \), the lower bound of Sanov’s theorem cannot be used directly. Instead, a direct computation of the lower bound via the change of measure argument will be used in conjunction with (7.3.2).\n\nLet \( {\n... | Yes |
Corollary 7.3.5 If \( \mathcal{M} = \left\{ {\nu }_{ * }\right\} \) then \( {\mu }_{{\mathbf{Y}}^{k} \mid {A}_{\delta }}^{n} \rightarrow {\left( {\nu }_{ * }\right) }^{k} \) weakly in \( {M}_{1}\left( {\sum }^{k}\right) \) for \( n \rightarrow \infty \) followed by \( \delta \rightarrow 0 \) . | Proof: Assume \( \mathcal{M} = \left\{ {\nu }_{ * }\right\} \) and fix \( {\phi }_{j} \in {C}_{b}\left( \sum \right), j = 1,\ldots, k \) . By the invariance of \( {\mu }_{{\mathbf{Y}}^{n} \mid {A}_{\delta }}^{n} \) with respect to permutations of \( \left\{ {{Y}_{1},\ldots ,{Y}_{n}}\right\} \) ,\n\n\[ \left\langle {\ma... | Yes |
Theorem 7.3.8 Let \( U,\mu \) and \( {\beta }^{ * } \) be as in the preceding lemma. If either \( U \) is bounded or \( {\beta }^{ * } \geq 0 \), then Theorem 7.3.3 applies, with \( \mathcal{M} \) consisting of a unique Gibbs measure \( {\gamma }_{{\beta }^{ * }} \) . | Proof: Note that by the monotone convergence theorem, \( \langle U, \cdot \rangle = \mathop{\sup }\limits_{n}\langle U \land \) \( n, \cdot \rangle \) . Since \( U \land n \in B\left( \sum \right) \), it follows that \( \Phi \left( \cdot \right) = \langle U, \cdot \rangle - 1 \) is a \( \tau \) -lower semicontinuous fu... | Yes |
Lemma 7.3.12 The functional \( \nu \mapsto \langle {U\nu },\nu \rangle \) is continuous with respect to the \( \tau \) -topology on \( {M}_{1}\left( \sum \right) \) . | Proof: Clearly, it suffices to prove the continuity of the functional \( \nu \mapsto \) \( \langle {U\nu },\nu \rangle \) with respect to the weak topology on \( {M}_{1}\left( \sum \right) \) . For \( U\left( {x, y}\right) = \) \( f\left( x\right) g\left( y\right) \) with \( f, g \in {C}_{b}\left( \sum \right) \), the ... | Yes |
Theorem 7.3.16 Assume (A-2)-(A-4). Then Theorem 7.3.3 applies, with \( \mathcal{M} \) consisting of a unique Gibbs measure \( {\gamma }_{{\beta }^{ * }} \), where \( {\beta }^{ * } \) is as defined in (7.3.15). | Proof: Since \( \Phi \left( \cdot \right) \) is a continuous functional (see Lemma 7.3.12), \( {F}_{\delta } = {A}_{\delta } \) may be taken here, yielding \( {F}_{0} = {A}_{0} = \{ \nu : \langle {U\nu },\nu \rangle = 1\} \) . Recall that \( \mu \) and \( {\gamma }_{{\beta }^{ * }} \) are equivalent with a globally bou... | Yes |
Lemma 7.3.25 For \( \mathcal{X} \) a Polish space, any \( P \in {M}_{1}\left( {\mathcal{X}}^{m}\right), m \in {\mathbb{Z}}_{ + } \) and any \( Q \in {M}_{1}\left( \mathcal{X}\right) \) , \[ H\left( {P \mid {Q}^{m}}\right) = H\left( {P \mid {P}_{1} \times \cdots \times {P}_{m}}\right) + \mathop{\sum }\limits_{{i = 1}}^{... | Proof: Suppose \( {P}_{i}\left( B\right) > Q\left( B\right) = 0 \) for some \( B \subset \mathcal{X} \) and \( i = 1,\ldots, m \) . Then, \( P\left( \widetilde{B}\right) > {Q}^{m}\left( \widetilde{B}\right) = 0 \) for \( \widetilde{B} = {\mathcal{X}}^{i - 1} \times B \times {\mathcal{X}}^{m - i} \) in which case both s... | Yes |
Lemma 7.3.27 (Csiszàr) Suppose that \( H\left( {{\nu }_{0} \mid \mu }\right) = \mathop{\inf }\limits_{{\nu \in A}}H\left( {\nu \mid \mu }\right) \) for a convex set \( A \subset {M}_{1}\left( \sum \right) \) and some \( {\nu }_{0} \in A \) . Then, for any \( \nu \in A \) , \[ H\left( {\nu \mid \mu }\right) \geq H\left(... | Proof: Fix \( \nu \triangleq {\nu }_{1} \in A \) and let \( {\nu }_{\alpha } = {\alpha \nu } + \left( {1 - \alpha }\right) {\nu }_{0} \in A \) for \( \alpha \in \lbrack 0,1) \) . There is nothing to prove unless \( H\left( {{\nu }_{0} \mid \mu }\right) \leq H\left( {\nu \mid \mu }\right) < \infty \), in which case \( {... | Yes |
Corollary 7.3.34 Let \( A = \left\{ {\nu \in {M}_{1}\left\lbrack {0,1}\right\rbrack : \langle U,\nu \rangle \leq 1}\right\} \) for a bounded non-negative Borel function \( U\left( \cdot \right) \), such that \( \mu \circ {U}^{-1} \) is a non-lattice law, \( {\int }_{0}^{1}U\left( x\right) {d\mu }\left( x\right) > 1 \) ... | Proof: It is shown in the course of proving Theorem 7.3.8 that \( {\nu }_{ * } = {\gamma }_{{\beta }^{ * }} \) is such that \( \left\langle {U,{\nu }_{ * }}\right\rangle = 1 \) and\n\n\[ H\left( {{\nu }_{ * } \mid \mu }\right) = \mathop{\inf }\limits_{{\nu \in A}}H\left( {\nu \mid \mu }\right) < \infty . \]\n\nNote tha... | Yes |
Corollary 1.2 For any \( j, l \geq 2 \) ,\n\n\[ P\left( {\partial {\Lambda }_{{2}^{j}} \leftrightarrow \partial {\Lambda }_{{2}^{j + l}}}\right) \leq {\left( 1 - {a}_{4}^{6}\right) }^{l}. \] | This corollary follows immediately from the previous lemma because of the independence of the realization of percolation in each of the disjoint annuli \( {A}_{j},{A}_{j + 1},\ldots ,{A}_{j + l - 1} \) and the fact that if \( \partial {\Lambda }_{{2}^{j}} \leftrightarrow \partial {\Lambda }_{{2}^{j + l}} \) then none o... | Yes |
Corollary 1.3 There exist constants \( K \) and \( c \) such that for all \( n \geq 2 \) and \( m \geq 1 \), the probability that there exist \( K \) disjoint open paths joining \( \partial {\Lambda }_{n} \) to \( \partial {\Lambda }_{mn} \) is smaller than \( c \times {m}^{-3} \) . | This follows readily from our estimates and the fact (called the Van den Berg-Kesten (BK) inequality, see [15]) that the probability that there exist \( K \) disjoint open paths joining \( \partial {\Lambda }_{n} \) to \( \partial {\Lambda }_{nm} \) is bounded from above by\n\n\[ P{\left( \partial {\Lambda }_{n} \leftr... | Yes |
Theorem 3.1 The law of \( {\gamma }^{\delta } \) converges to the law of chordal SLE(6) from \( x \) to \( c \) in the domain \( D \) . | When one states a convergence in law, one has to be precise about the topology. Here, we use the distance between two continuous paths \( \gamma \) and \( {\gamma }^{\prime } \) (with endpoints) defined by\n\n\[ \mathop{\inf }\limits_{\varphi }\mathop{\sup }\limits_{{u \in \left\lbrack {0,1}\right\rbrack }}\left| {\gam... | No |
Lemma 3.1 For each fixed \( z \in D \) with rational coordinates, \( {\sigma }_{z}^{{\delta }_{n}} \rightarrow {\sigma }_{z} \) almost surely as \( n \rightarrow \infty \) . | Note that the convergence of \( {\gamma }^{{\delta }_{n}} \) to \( \gamma \) implies immediately that almost surely, \[ {\sigma }_{z} \geq \mathop{\limsup }\limits_{{n \rightarrow \infty }}{\sigma }_{z}^{{\delta }_{n}} \] What one has to prove here is that when \( {\gamma }^{{\delta }_{n}} \) comes close to disconnect ... | No |
Lemma 3.2 For all \( u < {u}^{\prime } \), there almost surely exists \( v \in \left( {u,{u}^{\prime }}\right) \) such that \( {\gamma }_{v} \notin \gamma \left\lbrack {0, u}\right\rbrack \cap \partial D \) . | Proof. Define a countable dense set of points \( \left( {x}_{j}\right) \) on the union of the boundaries of the connected components of the complement of \( \gamma \left\lbrack {0, u}\right\rbrack \) . Because of the Russo-Seymour-Welsh estimates, almost surely, none of the points \( {x}_{j} \) are visited by \( \gamma... | No |
Lemma 3.3 For all \( t,{X}_{t} = P\left( {\mathcal{A} \mid \gamma \left\lbrack {0, t}\right\rbrack }\right) \) . | Proof. Let us consider any continuous bounded function \( f \) on the space of curves. Using three times dominated convergence, and our previously collected results, we obtain that\n\n\[ E\left( {{1}_{\mathcal{A}}f\left( {\gamma \left\lbrack {0, t}\right\rbrack }\right) }\right) = \mathop{\lim }\limits_{{n \rightarrow ... | Yes |
Proposition 4.1 Consider a chordal SLE(6) process \( {\gamma }^{1} \) from 1 to -1 in the unit disc \( \mathbb{U} \), and a radial SLE(6) process \( {\gamma }^{2} \) from 1 to 0 in \( \mathbb{U} \) . Define\n\n\[ \n{T}^{l} = \inf \left\{ {t > 0 : {\gamma }^{l}\left\lbrack {0, t}\right\rbrack \text{ disconnect }0\text{ ... | Outline of the proof of Proposition 4.1. Consider the chordal \( \mathrm{{SLE}}\left( 6\right) \) \( {\gamma }^{1} \) and define, for each \( t \leq {T}^{1} \), the conformal map \( {f}_{t} \) from \( {U}_{t} \) onto \( \mathbb{U} \) such that \( {f}_{t}\left( 0\right) = 0 \) and \( {f}_{t}\left( {\gamma }_{t}^{1}\righ... | Yes |
Theorem 4.1 When the mesh of the lattice goes to zero, then the law of the radial discrete exploration process converges to that of radial SLE(6). | Outline of the proof. Basically, one can note that the fact that up to the first disconnection time \( {T}_{1} \), the law of the discrete exploration process converges to that of the radial \( \operatorname{SLE}\left( 6\right) \) up to its first disconnection time, is a combination of the fact that chordal exploration... | No |
Proposition 4.3 There exist two constants \( {c}_{1} \) and \( {c}_{2} \) such that for all \( t \geq 2 \) ,\n\n\[ \n{c}_{1}{e}^{-{5t}/{48}} \leq {J}_{t} \leq {c}_{2}{e}^{-{5t}/{48}}. \n\] | Note that at time \( t \), the boundary of \( {U}_{t} \) can be decomposed into three parts. The arc on \( \partial \mathbb{U} \) that the path has not yet disconnected, an arc \( {\partial }_{t}^{1} \) that corresponds to \ | No |
Proposition 4.4 There is a constant \( c > 0 \) such that for all \( t \geq 1 \), for all \( x \in \left( {0,{2\pi }}\right) \)\n\n\[{e}^{-{\lambda t}}{\left( \sin \left( x/2\right) \right) }^{q} \leq f\left( {x, t}\right) \leq c{e}^{-{\lambda t}}{\left( \sin \left( x/2\right) \right) }^{q}\] | Proof. We can assume that \( b > 0 \) since the case \( b = 0 \) was treated in the previous section. Let \( {Y}_{t}^{x} \) be as before and define for all \( t < {\tau }^{x} \)\n\n\[{\Phi }_{t}^{x} \mathrel{\text{:=}} \left| {{g}_{t}^{\prime }\left( {\exp \left( {ix}\right) }\right) }\right| .\n\]\nOn \( t \geq {\tau ... | Yes |
Lemma 5.1 There exists a constant \( c = c\left( \delta \right) > 0 \) such that for all large enough \( {r}_{1} < {r}_{2}/4 < {r}_{3}/{32} \)\n\n\[{\pi }_{4}^{\delta }\left( {{r}_{1},{r}_{3}}\right) \geq c\left( \delta \right) {\pi }_{4}^{\delta }\left( {{r}_{1},{r}_{2}}\right) {\pi }_{4}^{\delta }\left( {2{r}_{2},{r}... | The proof is again an application of the \ | No |
Lemma 5.2 If one chooses \( \delta \) small enough, then for all large enough \( R \), and for all large enough \( n \geq N\left( {R,\delta }\right) \), \[ {\pi }_{4}^{\delta }\left( {n,{nR}}\right) \geq \frac{1}{2}{\widehat{\pi }}_{4}\left( {n,{nR}}\right) \] | The proof is based on the one hand on the fact that that for \( {r}_{1} < {r}_{2}/8 \), \[ {\widehat{\pi }}_{4}\left( {{r}_{1},{r}_{2}}\right) - {\pi }_{4}^{\delta }\left( {{r}_{1},{r}_{2}}\right) \leq {\widehat{\pi }}_{4}\left( {2{r}_{1},{r}_{2}/2}\right) \times a\left( \delta \right) , \] where the probability \( a\l... | Yes |
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