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Example 3.8.12 Enneper's surface can be given by one single parametrisation:\n\n\[ F : {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{3} \]\n\n\[ F\left( {{u}^{1},{u}^{2}}\right) = \left( \begin{matrix} {u}^{1} - \frac{{\left( {u}^{1}\right) }^{3}}{3} + {u}^{1}{\left( {u}^{2}\right) }^{2} \\ {u}^{2} - \frac{{\left( {u}^{2}... | Plate 3 shows that Enneper's surface intersects itself. To obtain a regular surface, we need to restrict the domain of \( F \) in a suitable way. | No |
Theorem 3.8.15 For every regular surface we have\n\n\[ K \leq {H}^{2} \]\n\nIn particular, the Gauss curvature of minimal surfaces satisfies\n\n\[ K \leq 0. \] | Proof Expressing the mean curvature and the Gauss curvature in terms of the principal curvatures, \( H = \left( {{\kappa }_{1} + {\kappa }_{2}}\right) /2, K = {\kappa }_{1}{\kappa }_{2} \), we observe that\n\n\[ 4 \cdot \left( {{H}^{2} - K}\right) = {\left( {\kappa }_{1} + {\kappa }_{2}\right) }^{2} - 4 \cdot {\kappa }... | Yes |
Corollary 3.8.16 Compact minimal surfaces do not exist. | Proof By theorem 3.6.17 every compact surface \( S \subset {\mathbb{R}}^{3} \) has a point with positive Gauss curvature. By theorem 3.8.15 it follows that \( S \) cannot be a minimal surface. | Yes |
Example 3.8.17 The hyperboloid of revolution \( S = \left\{ {{\left( x, y, z\right) }^{\top } \in {\mathbb{R}}^{3} \mid z = }\right. \) \( \left. {{x}^{2} + {y}^{2}}\right\} \) is a surface of revolution with function \( r\left( t\right) = \sqrt{t}, t > 0 \) . | Application of the formulae derived above gives:\n\n\[ K = \frac{4}{{\left( 1 + 4t\right) }^{2}},\;H = \frac{2 + {4t}}{{\left( 1 + 4t\right) }^{3/2}}. \] | Yes |
Example 4.2.2 Let \( f : S \rightarrow \mathbb{R} \) be a smooth function. Since the first fundamental form is non-degenerate, there exists, for a fixed point \( p \), exactly one vector \( v\left( p\right) \in {T}_{p}S \) with the property\n\n\[ \n{d}_{p}f\left( X\right) = I\left( {v\left( p\right), X}\right) \n\] \n\... | The differentiability of a vector field is best verified using a local parametri-sation. Let \( \left( {U, F, V}\right) \) be a local parametrisation of the regular surface \( S \) . Then for every point \( p \in V \) the vectors \( \left( {\partial F/\partial {u}^{1}}\right) \left( {{F}^{-1}\left( p\right) }\right) \)... | Yes |
Example 4.2.3 Let us check that the gradient vector field of a smooth function \( f : S \rightarrow \mathbb{R} \) is smooth. For this purpose let \( \left( {U, F, V}\right) \) be a local parametrisation. Then \( \widetilde{f} \mathrel{\text{:=}} f \circ F : U \rightarrow \mathbb{R} \) is also a smooth function. We need... | We calculate\n\n\[ \frac{\partial \widetilde{f}}{\partial {u}^{k}}\left( {{F}^{-1}\left( p\right) }\right) = {d}_{p}f\left( {\frac{\partial F}{\partial {u}^{k}}\left( {{F}^{-1}\left( p\right) }\right) }\right) \]\n\n\[ = I\left( {\operatorname{grad}f\left( p\right) ,\frac{\partial F}{\partial {u}^{k}}\left( {{F}^{-1}\l... | Yes |
Example 4.2.10 Let \( S = {\mathbb{R}}^{2} \times \{ 0\} \) be the \( x - y \) plane and \( c \) a parametrised plane curve, \( c\left( t\right) = {\left( {c}_{1}\left( t\right) ,{c}_{2}\left( t\right) ,0\right) }^{\top } \) . A vector field \( v \) on \( S \) along \( c \) is then of the form \( v\left( t\right) = {\l... | \[\n\frac{\nabla }{dt}v\left( t\right) = {\Pi }_{c\left( t\right) }\left( {\dot{v}\left( t\right) }\right)\]\n\[= {\Pi }_{c\left( t\right) }\left( {\left( {\dot{v}}_{1}\left( t\right) ,{\dot{v}}_{2}\left( t\right) ,0\right) }^{\top }\right)\]\n\[= {\left( {\dot{v}}_{1}\left( t\right) ,{\dot{v}}_{2}\left( t\right) ,0\ri... | Yes |
Lemma 4.2.14 The Christoffel symbols satisfy the following:\n\n\[ \n{\Gamma }_{ij}^{k} = \frac{1}{2}\mathop{\sum }\limits_{{m = 1}}^{2}\left( {\frac{\partial {g}_{jm}}{\partial {u}^{i}} + \frac{\partial {g}_{im}}{\partial {u}^{j}} - \frac{\partial {g}_{ij}}{\partial {u}^{m}}}\right) {g}^{mk} \n\] | Proof We calculate\n\n\[ \n\frac{\partial {g}_{jm}}{\partial {u}^{i}} = \frac{\partial }{\partial {u}^{i}}\left\langle {\frac{\partial F}{\partial {u}^{j}},\frac{\partial F}{\partial {u}^{m}}}\right\rangle \n\]\n\n\[ \n= \left\langle {\frac{{\partial }^{2}F}{\partial {u}^{i}\partial {u}^{j}},\frac{\partial F}{\partial ... | Yes |
Lemma 4.2.17 Let \( S \) be a regular surface, let \( {c}_{1},{c}_{2} \in \mathbb{R}, v,{v}_{1},{v}_{2}, w,{w}_{1} \) and \( {w}_{2} \) be differentiable vector fields on \( S \), and let \( f : S \rightarrow \mathbb{R} \) be a differentiable function.\n\nThen the following hold:\n\n(a) linearity in the vector field th... | Proof Properties (a), (b) and (c) follow directly from the corresponding properties of the covariant derivatives of vector fields along curves in lemma 4.2.12. Points (d) and (e) are most easily deduced from the formula in terms of a local parametrisation, given in the above exercise. | No |
Lemma 4.3.2 Let \( S \) be a regular surface and \( v, w \) and \( z \) be vector fields on S. Let \( \left( {U, F, V}\right) \) be a local parametrisation of \( S \) . As usual we express \( v \) in the basis given by the parametrisation, \( v = \mathop{\sum }\limits_{{i = 1}}^{2}{v}^{i}\left( {\partial F/\partial {u}... | \[ \left( {\mathop{\sum }\limits_{{i, j}}\frac{{\partial }^{2}{z}^{m}}{\partial {u}^{i}\partial {u}^{j}}{v}^{i}{w}^{j} + \mathop{\sum }\limits_{{i, j, k}}{\Gamma }_{ij}^{m}\frac{\partial {z}^{i}}{\partial {u}^{k}}\left( {{v}^{j}{w}^{k} + {v}^{k}{w}^{j}}\right) - \mathop{\sum }\limits_{{i, j, k}}{\Gamma }_{ij}^{k}\frac{... | Yes |
Corollary 4.3.3 The value of the second covariant derivative \( {\nabla }_{v, w}^{2}z \) at a point \( p \in S \) depends only on \( v\left( p\right), w\left( p\right) \) and the derivatives of \( z \) at \( p \) up to order 2 . | Proof of lemma 4.3.2 The proof consists of thorough calculations, in which it is most important to keep a level head. The vector field \( {\nabla }_{w}z \) has the components\n\n\[ \n{\left( \mathop{\sum }\limits_{\ell }\frac{\partial {z}^{k}}{\partial {u}^{\ell }}{w}^{\ell } + \mathop{\sum }\limits_{{ij}}{\Gamma }_{ij... | Yes |
Lemma 4.3.5 The Riemann curvature tensor w.r.t. a local parametrisation has the form\n\n\[ R\left( {{v}_{p},{w}_{p}}\right) z = \mathop{\sum }\limits_{{{ijk}\ell = 1}}^{2}{R}_{ijk}^{\ell }\left( {u}_{0}\right) {v}^{i}{w}^{j}{z}^{k}\frac{\partial F}{\partial {u}^{\ell }}\left( {u}_{0}\right) , \]\n\nwhere\n\n\[ {R}_{ijk... | Proof All terms from lemma 4.3.2 that involve derivatives of \( z \) are symmetric in \( {v}_{p} \) and \( {w}_{p} \) and therefore cancel. Considering further that the Christoffel symbols \( {\Gamma }_{ij}^{k} \) are symmetric in the lower indices \( i \) and \( j \) proves the claim. | No |
Theorem 4.3.7 (Gauss’s equation) Let \( S \subset {\mathbb{R}}^{3} \) be an oriented regular surface, \( p \in S \) . Then any \( v, w, z \in {T}_{p}S \) satisfy the following:\n\n\[ R\left( {v, w}\right) z = H\left( {w, z}\right) \cdot W\left( v\right) - H\left( {v, z}\right) \cdot W\left( w\right) . \]\n\nWith respec... | Proof We prove the local version. Let \( \left( {U, F, V}\right) \) be a local parametrisation. We recall (4.1):\n\n\[ \frac{{\partial }^{2}F}{\partial {u}^{i}\partial {u}^{j}} = \mathop{\sum }\limits_{k}{\Gamma }_{ij}^{k}\frac{\partial F}{\partial {u}^{k}} + {h}_{ij} \cdot \left( {N \circ F}\right) . \]\n\nDifferentia... | Yes |
Theorem 4.3.8 (Theorema Egregium) Gauss curvature can be calculated form the Riemann curvature tensor as follows: let \( p \in S \) be a point. Choose an orthonormal basis \( v, w \) of \( {T}_{p}S \) . Then\n\n\[ K\left( p\right) = I\left( {{R}_{p}\left( {v, w}\right) w, v}\right) . \]\n\nIn particular, Gauss curvatur... | Proof According to the Gauss equation we have\n\n\[ I\left( {R\left( {v, w}\right) w, v}\right) = I\left( {{II}\left( {w, w}\right) \cdot W\left( v\right) - {II}\left( {v, w}\right) \cdot W\left( w\right), v}\right) \]\n\n\[ = {II}\left( {w, w}\right) {II}\left( {v, v}\right) - {II}\left( {v, w}\right) {II}\left( {w, v... | Yes |
The two regular surfaces \( {S}_{1} = \{ {\left( x, y,0\right) }^{\top } \mid {x}^{2} + {y}^{2} < 1\} \) (circular disc) and \( {S}_{2} = \left\{ {{\left( x, y, z\right) }^{\top } \mid {x}^{2} + {y}^{2} < 1, z = \sqrt{1 - {x}^{2} - {y}^{2}}}\right\} \) (hemisphere) are diffeomorphic. | The projection \( {\left( x, y, z\right) }^{\top } \mapsto {\left( x, y,0\right) }^{\top } \) gives a diffeomorphism from \( {S}_{2} \) to \( {S}_{1} \). | No |
Lemma 4.3.10 Let \( S \) be a regular surface, let \( p \in S \), let \( v, w, x, y \in {T}_{p}S \) . The curvature tensor has the following symmetries:\n\n\( R\left( {v, w}\right) x = - R\left( {w, v}\right) x \)\n\n(b) \( I\left( {R\left( {v, w}\right) x, y}\right) = - I\left( {R\left( {v, w}\right) y, x}\right) ;\n\... | Proof Part (a) is trivial. Statement (c) follows from Gauss's equation:\n\n\[ I\left( {R\left( {v, w}\right) x, y}\right) = I\left( {{II}\left( {w, x}\right) \cdot W\left( v\right) - {II}\left( {v, x}\right) \cdot W\left( w\right), y}\right) \]\n\n\[ = H\left( {w, x}\right) \cdot H\left( {v, y}\right) - H\left( {v, x}\... | Yes |
Lemma 4.3.11 Let \( S \) be a regular surface, let \( p \in S \) . For all \( v, w, x \in {T}_{p}S \)\n\nwe have\n\n\[ R\left( {v, w}\right) x = K\left( p\right) \cdot \left( {I\left( {w, x}\right) v - I\left( {v, x}\right) w}\right) . \]\n\nIn local coordinates,\n\n\[ {R}_{ijk}^{\ell } = K \cdot \left( {{g}_{jk}{\delt... | Proof (a) Let \( V \) be a two-dimensional real vector space. We first show that the vector space of all multi-linear maps\n\n\[ \mathcal{R} : V \times V \times V \times V \rightarrow \mathbb{R} \]\n\nthat satisfy the symmetries\n\n\[ \mathcal{R}\left( {v, w, x, y}\right) = - \mathcal{R}\left( {w, v, x, y}\right) \;\te... | Yes |
What is the Gauss curvature for this Riemannian metric? | Well, the definition is formulated such that with the parametrisation from above the Riemannian metric has components\n\n\[ \n{\left( {g}_{ij}\right) }_{ij} = \left( \begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right) \n\]\n\nThe fact that all the \( {g}_{ij} \) are constant causes the Christoffel symbols to vanish, an... | Yes |
Lemma 4.5.3 Let \( S \) be a regular surface with Riemannian metric \( g \) . Let \( c \) : \( \left\lbrack {a, b}\right\rbrack \rightarrow S \) be a parametrised curve. Then\n\n\[ L{\left\lbrack c\right\rbrack }^{2} \leq 2\left( {b - a}\right) E\left\lbrack c\right\rbrack \]\n\nwhere the equality applies if and only i... | Proof We set \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R}, f\left( t\right) \mathrel{\text{:=}} \sqrt{{g}_{c\left( t\right) }\left( {\dot{c}\left( t\right) ,\dot{c}\left( t\right) }\right) } \) . By the Cauchy-Schwarz inequality we have\n\n\[ L{\left\lbrack c\right\rbrack }^{2} = {\left( {\int }_{a}^{... | Yes |
Lemma 4.5.4 Let \( S \) be a regular surface with Riemannian metric \( g \) . Let \( c \) : \( I \times J \rightarrow S,\left( {s, t}\right) \mapsto c\left( {s, t}\right) \), be a smooth map. Then\n\n\[ \frac{\nabla }{\partial s}\frac{\partial c}{\partial t} = \frac{\nabla }{\partial t}\frac{\partial c}{\partial s} \] | Proof From the formula\n\n\[ {\Gamma }_{ij}^{k} = \frac{1}{2}\mathop{\sum }\limits_{{m = 1}}^{2}\left( {\frac{\partial {g}_{jm}}{\partial {u}_{i}} + \frac{\partial {g}_{im}}{\partial {u}_{j}} - \frac{\partial {g}_{ij}}{\partial {u}_{m}}}\right) {g}^{mk} \]\n\nwe see that in the case of general Riemannian metrics the Ch... | Yes |
Theorem 4.5.5 (Variation of energy) Let \( S \) be a regular surface with Riemannian metric \( g \) . Let \( p, q \in S \) . Let \( c : \left( {-\varepsilon ,\varepsilon }\right) \times \left\lbrack {a, b}\right\rbrack \rightarrow S \) be a smooth map such that for \( {c}_{s} : \left\lbrack {a, b}\right\rbrack \rightar... | Proof We differentiate inside the integral and use lemma 4.5.4:\n\n\[ \n{\left. \frac{d}{ds}E\left\lbrack {c}_{s}\right\rbrack \right| }_{s = 0}\n\]\n\n\[ \n= {\left. \frac{1}{2}{\int }_{a}^{b}\frac{d}{ds}{g}_{{c}_{s}\left( t\right) }\left( {\dot{c}}_{s}\left( t\right) ,{\dot{c}}_{s}\left( t\right) \right) \right| }_{s... | Yes |
Example 4.5.8 Let \( S \subset {\mathbb{R}}^{3} \) be the \( x - y \) plane with the first fundamental form as Riemannian metric. As we already know, the covariant derivative agrees with the usual derivative in this case: | \[ \frac{\nabla }{dt}\dot{c}\left( t\right) = \ddot{c}\left( t\right) \] The geodesics are therefore precisely those straight lines that have a constant speed: \[ c\left( t\right) = p + t \cdot v. \] | Yes |
Lemma 4.5.10 Geodesics are parametrised proportional to arc-length. | Proof Let \( c \) be a geodesic. We differentiate and use the product rule II from lemma 4.2.17:\n\n\[ \frac{d}{dt}{g}_{c\left( t\right) }\left( {\dot{c}\left( t\right) ,\dot{c}\left( t\right) }\right) = {g}_{c\left( t\right) }\left( {\frac{\nabla }{dt}\dot{c}\left( t\right) ,\dot{c}\left( t\right) }\right) + {g}_{c\le... | Yes |
Theorem 4.5.12 (Uniqueness of geodesics) Let \( S \subset {\mathbb{R}}^{3} \) be a regular surface with Riemannian metric \( g \) . Let \( I \) be an interval, \( {t}_{0} \in I \) . Let \( c : I \rightarrow S \) be a geodesic. Then \( c \) is uniquely determined by \( c\left( {t}_{0}\right) \in S \) and \( \dot{c}\left... | Proof If we knew that the trace of \( c \) lay entirely in one open neighbourhood of the local parametrisation, then we could argue in a way similar to that used for theorem 4.5.11. We would simply cite the uniqueness statement instead of the existence statement from the theory of ordinary differential equations.\n\nBu... | Yes |
Theorem 4.5.13 (Clairaut's theorem) Let \( S \) be a surface of revolution, given by the parametrisation \( F\left( {t,\varphi }\right) = {\left( r\left( t\right) \cos \left( \varphi \right), r\left( t\right) \sin \left( \varphi \right), t\right) }^{\top } \) . We take the first fundamental form as the Riemannian metri... | Proof The first fundamental form was calculated in section 3.8.3. The result is \[ {\left( {g}_{ij}\left( t,\varphi \right) \right) }_{ij} = \left( \begin{matrix} 1 + \dot{r}{\left( t\right) }^{2} & 0 \\ 0 & r{\left( t\right) }^{2} \end{matrix}\right) . \] We set \[ v \mathrel{\text{:=}} \frac{\partial F}{\partial t}\;... | Yes |
Example 4.6.2 Let \( S = {\mathbb{R}}^{2} \times \{ 0\} \) be the \( x - y \) plane with the first fundamental form as Riemannian metric. Let \( p \in S \) and \( v \in {T}_{p}S = {\mathbb{R}}^{2} \times \{ 0\} \). The geodesic \( c \) in \( S \) with \( c\left( 0\right) = p \) and \( \dot{c}\left( 0\right) = v \) is t... | \[ {\exp }_{p}\left( v\right) = p + v \] | Yes |
Example 4.6.3 Now let \( S = \left\{ {{\left( x, y,0\right) }^{\top } \in {\mathbb{R}}^{3} \mid {x}^{2} + {y}^{2} < 1}\right\} \) be the unit disc in the \( x - y \) plane with the first fundamental form as a Riemannian metric, as above. The geodesics are again segments of straight lines, but as they leave the disc aft... | \[ {\exp }_{p}\left( v\right) = p + v \] but the domain of the exponential map is now \[ {\mathcal{D}}_{p} = \left\{ {v \in {\mathbb{R}}^{3} \mid p + v \in S}\right\} = S - p. \] | Yes |
Example 4.6.4 Let \( S = {S}^{2} \) be the sphere, again with the first fundamental form as a Riemannian metric. Let \( p \in S \) and \( v \in {T}_{p}S = {p}^{ \bot } \) . We write \( v = {\delta w} \) , where \( w \in {T}_{p}S \) is a unit vector, \( \parallel w\parallel = 1 \) and \( \delta = \parallel v\parallel \g... | \[ {\exp }_{p}\left( v\right) = \left\{ \begin{matrix} \cos \left( {\parallel v\parallel }\right) \cdot p + \sin \left( {\parallel v\parallel }\right) \cdot v/\parallel v\parallel , & v \neq 0, \\ p, & v = 0. \end{matrix}\right. \] | Yes |
Lemma 4.6.5 The differential of the exponential map at the point 0 is the identity, \[ {d}_{0}{\exp }_{p} = \operatorname{Id} : {T}_{p}S \rightarrow {T}_{p}S. \] | Proof Let \( v \in {T}_{p}S \) . We know from (4.12) that the geodesic \( c \) with initial conditions \( c\left( 0\right) = p \) and \( \dot{c}\left( 0\right) = v \) is given by \[ c\left( t\right) = {\exp }_{p}\left( {tv}\right) \] Further, \( \widetilde{c}\left( t\right) = {tv} \) is a curve in \( {T}_{p}S \) with \... | Yes |
Example 4.6.6 Let \( S \) be an arbitrary regular surface with a Riemannian metric. Let \( p \in S \) and let \( {X}_{1},{X}_{2} \) be an orthonormal basis of the tangent plane \( {T}_{p}S \) . We take the parametrisation in Cartesian coordinates for \( {T}_{p}S \), i.e. \( {U}_{1} = {\mathbb{R}}^{2} \) and \( {F}_{1}\... | \[ F\left( {{u}^{1},{u}^{2}}\right) = {\exp }_{p}\left( {\mathop{\sum }\limits_{i}{u}^{i}{X}_{i}}\right) \] is called parametrisation in Riemann normal coordinates (at the point \( p \) ). | Yes |
Theorem 4.6.7 Let \( S \) be a regular surface, let \( p \in S \) and let \( F \) be a local parametrisation in Riemann normal coordinates at the point \( p \) . Then the corresponding component functions of the metric and the Christoffel symbols satisfy the following:\n\n(i) \( \;F\left( {0,0}\right) = p \) .\n\n(ii) ... | Proof Statement (i) is clear and statement (ii) means precisely that \( {d}_{0}{\exp }_{p} \) is a linear isometry. To prove statement (iii), we recall that the exponential map maps the straight lines through the origin to the geodesics through \( p \) . The map \( t \mapsto {tx} \) describes a geodesic in Riemann norm... | Yes |
Lemma 4.6.10 Keep the notation from theorem 4.6.9. Then the Gauss curvature satisfies\n\n\\[ K\\left( {F\\left( {r,\\varphi }\\right) }\\right) = - \\frac{1}{f\\left( {r,\\varphi }\\right) }\\frac{{\\partial }^{2}f}{\\partial {r}^{2}}\\left( {r,\\varphi }\\right) . \\] | Proof Theorem 4.6.9 tells us that in geodesic coordinates the Riemannian metric has the form\n\n\\[ {\\left( {g}_{ij}\\left( r,\\varphi \\right) \\right) }_{ij} = \\left( \\begin{matrix} 1 & 0 \\\\ 0 & f{\\left( r,\\varphi \\right) }^{2} \\end{matrix}\\right) . \\]\n\nThe inverse matrix is then\n\n\\[ {\\left( {g}^{ij}... | Yes |
Lemma 4.6.12 Keep the notation from theorem 4.6.9. Let \( r > 0 \) be such that \( \bar{D}\left( {p, r}\right) \) is (up to a zero set) covered by the geodesic polar coordinate system. Then \[ {\int }_{\bar{D}\left( {p, r}\right) }{KdA} = {2\pi } - {\int }_{{\varphi }_{0}}^{{\varphi }_{0} + {2\pi }}\frac{\partial f}{\p... | Proof By the formula for the Riemannian metric from Gauss's lemma the surface element is given in geodesic polar coordinates by \[ {dA} = \sqrt{1 \cdot {f}^{2}}{drd\varphi } = {fdrd\varphi }. \] For the geodesic coordinate system to cover \( \bar{D}\left( {p, r}\right) \) up to a zero set, the domain of the local param... | Yes |
Lemma 4.6.13 Let \( S \) be a regular surface with Riemannian metric \( g \) . Let \( c : I \rightarrow S \) be a curve parametrised by arc-length, defined on an open interval \( I \) . Let \( n : I \rightarrow {\mathbb{R}}^{3} \) be a vector field on \( S \) along \( c \) that has constant length 1 and \( \langle \dot... | Proof The partial derivatives of \( F \) at the point \( \left( {t,0}\right) \) are given by\n\n\[ \frac{\partial F}{\partial t}\left( {t,0}\right) = \frac{d}{dt}{\exp }_{c\left( t\right) }\left( 0\right) = \dot{c}\left( t\right) \]\n\nand\n\n\[ \frac{\partial F}{\partial s}\left( {t,0}\right) = n\left( t\right) \]\n\n... | Yes |
Proposition 4.7.4 Let \( S \) be a regular surface with Riemannian metric \( g \) . Let \( c : \left\lbrack {{t}_{0},{t}_{1}}\right\rbrack \rightarrow S \) be a smooth curve. Then the following hold:\n\n(i) If \( {v}_{0} \in {T}_{c\left( {t}_{0}\right) }S \), then the parallel vector field \( v \) along \( c \) with \(... | Proof Statement (i) follows directly from the uniqueness of parallel transport. We have linearity of parallel transport because of the following: let \( {v}_{0} \) and \( {w}_{0} \) be from \( {T}_{c\left( {t}_{0}\right) }S \), and let \( v \) and \( w \) be the parallel vector fields along \( c \) with \( v\left( {t}_... | Yes |
Example 4.7.5 Let \( S \) be the conical surface,\n\n\[ S = \\left\\{ {{\\left( \\xi ,\\eta ,\\zeta \\right) }^{\\top }\\left| {\\;{\\xi }^{2} + {\\eta }^{2} = \\frac{1}{3}{\\zeta }^{2}}\\right. ,\\zeta > 0}\\right\\} .\n\nTo investigate parallel vector fields along the curve \( c : \\left\\lbrack {0,\\pi }\\right\\rbr... | Using\n\n\[ \\frac{\\partial f}{\\partial x} = \\frac{1}{\\sqrt{{x}^{2} + {y}^{2}}}\\left( \\begin{matrix} x - \\frac{1}{2}\\frac{\\left( {{x}^{2} - {y}^{2}}\\right) x}{{x}^{2} + {y}^{2}} \\\\\ny - \\frac{{x}^{2}y}{{x}^{2} + {y}^{2}} \\\\\n\\frac{\\sqrt{3}}{2}x \\end{matrix}\\right) ,\n\n\]\n\nwe calculate\n\n\[ v\\lef... | Yes |
Proposition 4.8.3 If \( c \) is a geodesic variation on \( S \), then the corresponding variation field \( J\left( t\right) \mathrel{\text{:=}} \left( {\partial c/\partial s}\right) \left( {0, t}\right) \) is a Jacobi field. | Proof (a) Let \( J \) be a variation field to a geodesic variation \( c \) . That every curve \( {c}_{s} \) is a geodesic means that\n\n\[ \frac{\nabla }{\partial t}\frac{\partial c}{\partial t} = 0 \]\n\nWe differentiate this equation with respect to \( s \) and obtain, using lemma 4.5.4, that\n\n\[ 0 = \frac{\nabla }... | Yes |
If the Gauss curvature of a surface is constant, \( K \equiv \kappa \), then (4.16) can be solved explicitly. | Setting\n\n\[ \n{\mathfrak{s}}_{\kappa }\left( t\right) \mathrel{\text{:=}} \left\{ {\begin{matrix} \sin \left( {\sqrt{\kappa }t}\right) /\sqrt{\kappa }, & \kappa > 0, \\ t, & \kappa = 0, \\ \sinh \left( {\sqrt{\left| \kappa \right| }t}\right) /\sqrt{\left| \kappa \right| }, & \kappa < 0, \end{matrix}\;{\mathfrak{c}}_{... | Yes |
Theorem 4.9.2 The traces of geodesics of \( {\mathbb{M}}_{\kappa } \) with the Riemannian metric defined by the restriction of \( \langle \cdot , \cdot {\rangle }_{\kappa } \) are exactly the (non-empty) intersections of \( {\mathbb{M}}_{\kappa } \) with the two-dimensional subspaces of \( {\mathbb{R}}^{3} \) . | In the case \( \kappa = 1 \) these are the great circles on \( {S}^{2} \), in the case \( \kappa = 0 \) they are the straight lines in the plane and in the case \( \kappa = - 1 \) we obtain hyperbolas. | No |
Lemma 4.10.1 Let \( g \) and \( {g}^{\prime } \) be two Euclidean metrics on \( {\mathbb{R}}^{n}, n \geq 2 \) . Then \( g \) and \( {g}^{\prime } \) define the same angles if and only if there exists a number \( c > 0 \) such that\n\n\[ \n{g}^{\prime } = c \cdot g \n\] | Proof If \( {g}^{\prime } = c \cdot g \), then for any non-zero vectors \( X, Y \in {\mathbb{R}}^{n} \) we have\n\n\[ \n\frac{g\left( {X, Y}\right) }{\sqrt{g\left( {X, X}\right) }\sqrt{g\left( {Y, Y}\right) }} = \frac{{g}^{\prime }\left( {X, Y}\right) }{\sqrt{{g}^{\prime }\left( {X, X}\right) }\sqrt{{g}^{\prime }\left(... | Yes |
Example 5.1.2 The closed circular disc \( S = \left\{ {{\left( x, y,0\right) }^{\top } \mid {x}^{2} + {y}^{2} \leq 1}\right\} \) is a surface with boundary. | We can take the \( x - y \) plane as the regular surface \( {S}_{\text{reg }} = \) \( {\mathbb{R}}^{2} \times \{ 0\} \) . The points \( {\left( x, y,0\right) }^{\top } \) with \( {x}^{2} + {y}^{2} < 1 \) are interior points, since for \( r \mathrel{\text{:=}} \sqrt{1 - \left( {{x}^{2} + {y}^{2}}\right) } > 0 \) the set... | Yes |
Lemma 5.1.5 If we write the vector field \( X \) with respect to a local parametri-sation \( F \) as \( X = \mathop{\sum }\limits_{i}{\xi }^{i}\partial F/\partial {u}^{i} \), then the divergence is given by\n\n\[ \operatorname{div}X = \mathop{\sum }\limits_{j}\left( {\frac{\partial {\xi }^{j}}{\partial {u}^{j}} + \math... | Proof We find the matrix representation of the endomorphism \( {Y}_{p} \mapsto \) \( {\nabla }_{{Y}_{p}}X \) with respect to the basis \( \partial F/\partial {u}^{1},\partial F/\partial {u}^{2} \) .\n\n\[ {\nabla }_{\frac{\partial F}{\partial {u}^{j}}}X = {\nabla }_{\frac{\partial F}{\partial {u}^{j}}}\left( {\mathop{\... | Yes |
Lemma 5.1.6 (Derivative of the determinant) Let \( t \mapsto g\left( t\right) \) be a differentiable curve of invertible real \( n \times n \) matrices. Then\n\n\[ \frac{d}{dt}\ln \det g = \operatorname{Trace}\left( {{g}^{-1}\frac{d}{dt}g}\right) . \] | Proof We first prove the equation for \( t = {t}_{0} \) if \( g\left( {t}_{0}\right) = \mathrm{{Id}} \) . Then the claim is\n\nsimply\n\n\[ \frac{d}{dt}\det g\left( {t}_{0}\right) = \operatorname{Trace}\left( {\frac{dg}{dt}\left( {t}_{0}\right) }\right) . \]\n\n(5.4)\n\nAs commonly known (see [17, p. 171, theorem 7.2])... | Yes |
Let \( S \subset {\mathbb{R}}^{3} \) be a minimal surface with the first fundamental form as Riemannian metric. Let \( \ell : {\mathbb{R}}^{3} \rightarrow \mathbb{R} \) be a linear function, e.g. one of the three Cartesian coordinate functions. We then argue that \( f \mathrel{\text{:=}} {\left. \ell \right| }_{S} : S ... | As \( \ell \) is linear, there exists a vector \( Z \in {\mathbb{R}}^{3} \) such that \( \ell \left( X\right) = \langle X, Z\rangle \) for all \( X \in {\mathbb{R}}^{3} \) . The gradient of \( f \) at the point \( p \) is given by the projection of \( Z \) on \( {T}_{p}S \) : \( \operatorname{grad}f\left( p\right) = Z ... | Yes |
If there are two metrics \( {g}_{0} \) and \( {g}_{1} \) defined on a regular surface \( S \), then | \[ {g}_{t, p} \mathrel{\text{:=}} \left( {1 - t}\right) {g}_{0, p} + t{g}_{1, p},\;t \in \left\lbrack {0,1}\right\rbrack \] defines a one-parameter family of Riemannian metrics, which represents a transition from \( {g}_{0} \) to \( {g}_{1} \). | Yes |
If \( g \) is a Riemannian metric on \( S \), then\n\n\[ \n{g}_{t, p} \mathrel{\text{:=}} t{g}_{p},\;t \in \left( {0,\infty }\right) \n\]\n\ndefines a one-parameter family of Riemannian metrics, which are simply scalings of the original metric. | Given a one-parameter family of Riemannian metrics with \( {t}_{0} \in I \) we can find its Taylor expansion w.r.t. \( t \) about \( t = {t}_{0} \) and write\n\n\[ \n{g}_{ij}\left( {t,{u}^{1},{u}^{2}}\right) = {g}_{ij}\left( {{u}^{1},{u}^{2}}\right) + \left( {t - {t}_{0}}\right) \cdot {\dot{g}}_{ij}\left( {{u}^{1},{u}^... | Yes |
Lemma 5.2.4 The following are implications of the definitions made above:\n\n(a)\n\n\[ \n{\dot{g}}^{jk} = - \mathop{\sum }\limits_{{i\ell }}{g}^{ij}{\dot{g}}_{i\ell }{g}^{\ell k} \n\] | Proof In the proof we will use the abbreviation \( {\partial }_{i} \mathrel{\text{:=}} \partial /\partial {u}^{i} \) and the Einstein summation, which allows a more compact notation and is popular in literature in physics. The convention consists of omitting the sigma and regarding an expression as a sum if an index ap... | Yes |
Lemma 5.2.5 The variation of the surface element is given by\n\n\\[ \n\\dot{dA} = \\frac{1}{2}\\operatorname{Trace}\\left( \\dot{g}\\right) {dA}.\n\\] | Proof The claim essentially follows from lemma 5.1.6:\n\n\\[ \n\\sqrt{\\det \\left( {g}_{ij}\\right) } = \\frac{\\det {\\left( {g}_{ij}\\right) }^{ \\cdot }}{2\\sqrt{\\det \\left( {g}_{ij}\\right) }}\n\\]\n\n\\[ \n= \\frac{{g}^{k\\ell }{\\dot{g}}_{k\\ell }\\det \\left( {g}_{ij}\\right) }{2\\sqrt{\\det \\left( {g}_{ij}\... | Yes |
Theorem 5.2.7 Let \( S \) be a compact regular surface. Then the number \[ {\int }_{S}{KdA} \] is independent of the Riemannian metric. | Proof We see from lemma 5.2.5 and lemma 5.2.6 that every one-parameter family of Riemannian metrics \( {g}_{t} \) with Gauss curvature \( {K}_{t} \) and surface element \( d{A}_{t} \) satisfies \[ \frac{d}{dt}{\int }_{S}{K}_{t}d{A}_{t} = {\int }_{S}\left( {{\dot{K}}_{t}d{A}_{t} + {K}_{t}\dot{d{A}_{t}}}\right) \] \[ = {... | Yes |
Corollary 5.2.8 Let \( {S}_{1} \) and \( {S}_{2} \) be compact regular surfaces with Riemannian metrics. If \( {S}_{1} \) and \( {S}_{2} \) are diffeomorphic, then\n\n\[ {\int }_{{S}_{1}}{KdA} = {\int }_{{S}_{2}}{KdA} \] | Proof The equation certainly holds if the metric on \( {S}_{1} \) is the pulled-back metric of \( {S}_{2} \), since the two metrics would then be isomorphic. Otherwise theorem 5.2.7 tells us that \( {\int }_{{S}_{1}}{KdA} \) w.r.t. the pulled-back metric and w.r.t. given metric agree. | No |
Theorem 6.1.5 (the Gauss-Bonnet theorem for polyhedra) Let \( X \) be a polyhedron such that exactly two triangles meet at every edge. Then\n\n\[ \mathop{\sum }\limits_{v}\operatorname{def}\left( v\right) = {2\pi \chi }\left( X\right) \]\n\nwhere the sum is taken over all vertices \( v \) of \( X \) . | Proof Since exactly two triangles meet at every edge by assumption and every triangle has exactly three edges, we have\n\n\[ {2k}\left( X\right) = {3f}\left( X\right) \]\n\nIt follows that\n\n\[ \chi \left( X\right) = f\left( X\right) - k\left( X\right) + e\left( X\right) \]\n\n\[ = f\left( X\right) - \frac{3}{2}f\left... | Yes |
Lemma 6.2.6 If \( S \) is contained in a ball of radius \( R \), then every \( \varepsilon \) -separated subset of \( A \subset S \) has at most\n\n\[ \n{\left( \frac{{2R} + \varepsilon }{\varepsilon }\right) }^{3} \n\]\n\nmany elements. In particular, \( S \) has an \( \varepsilon \) -configuration with finitely many ... | Proof Let \( S \subset B\left( {q, R}\right) \) . As the balls of radius \( \varepsilon /2 \) around the points from \( A \) are pairwise disjoint and since\n\n\[ \n\mathop{\bigcup }\limits_{{p \in A}}B\left( {p,\varepsilon /2}\right) \subset {U}_{\varepsilon /2}\left( S\right) \subset B\left( {q, R + \varepsilon /2}\r... | Yes |
Lemma 6.2.7 Let \( S \subset {\mathbb{R}}^{3} \), let \( \varepsilon > 0 \), let \( A \) be an \( \varepsilon \) -configuration on \( S \) . Then the \( \varepsilon \) -balls with centres in \( A \) cover \( S \) entirely: | Proof Suppose that there is a \( q \in S \) with \( q \notin \mathop{\bigcup }\limits_{{p \in A}}B\left( {p,\varepsilon }\right) \) . Then \( \parallel q - p\parallel \geq \) \( \varepsilon \) for all \( p \in A \) and \( {A}^{\prime } \mathrel{\text{:=}} A \cup \{ q\} \) is \( \varepsilon \) -separated as well. This c... | Yes |
Lemma 6.2.9 Let \( S \subset {\mathbb{R}}^{3} \), let \( \varepsilon > 0 \) and let \( A \) be an \( \varepsilon \) -configuration on \( S \) . Then all \( p \in A \) satisfy\n\n\[ \operatorname{St}\left( p\right) \cap S \subset B\left( {p,\varepsilon }\right) . \]\n\nFor any two distinct neighbouring points \( p,{p}^{... | Proof Let \( q \in S \) with \( \parallel q - p\parallel \geq \varepsilon \) . As the \( \varepsilon \) -balls with centres in \( A \) cover \( S \) entirely by lemma 6.2.7, there is a \( {p}^{\prime } \in A \) with \( \begin{Vmatrix}{q - {p}^{\prime }}\end{Vmatrix} < \varepsilon \) . Hence \( \begin{Vmatrix}{q - {p}^{... | Yes |
Lemma 6.2.11 Let \( S \subset {\mathbb{R}}^{3} \) be a compact orientable regular surface. Then there exists a positive \( {\rho }_{1} \), smaller than the \( \rho \) from lemma 6.2.3, such that for every \( \varepsilon \) -configuration \( A \) on \( S \) with \( \varepsilon \in \left( {0,{\rho }_{1}}\right\rbrack \) ... | \[ {\left. \mathcal{P}\right| }_{U\left( \Delta \right) } : U\left( \Delta \right) \rightarrow \mathcal{P}\left( {U\left( \Delta \right) }\right) \subset S. \] Proof (a) We first show that for a sufficiently small \( \varepsilon \) the restriction of \( \mathcal{P} \) to a neighbourhood of \( \Delta \) in \( E \mathrel... | Yes |
Lemma 6.2.12 Let \( S \subset {\mathbb{R}}^{3} \) be a compact orientable regular surface. Then there exists a positive \( {\rho }_{2} \) such that for every \( \varepsilon \) -configuration \( A \) on \( S \) with \( \varepsilon \in \) \( \left( {0,{\rho }_{2}}\right\rbrack \) and for any three common neighbours \( {p... | Proof The condition that \( {p}_{1},{p}_{2} \) and \( {p}_{3} \) are common neighbours says precisely that the set \( \operatorname{St}\left( {p}_{1}\right) \cap \operatorname{St}\left( {p}_{2}\right) \cap \operatorname{St}\left( {p}_{3}\right) \cap S \) is not empty. We therefore need to show that if \( \varepsilon \)... | Yes |
Corollary 6.2.13 Let \( S \subset {\mathbb{R}}^{3} \) be a compact orientable regular surface with an \( \varepsilon \) -configuration \( A \) with \( \varepsilon \in \left( {0,{\rho }_{2}}\right\rbrack \) and \( {\rho }_{2} \) as in lemma 6.2.12. Let \( {M}_{1},{M}_{2} \subset A \) be two sets, each consisting of comm... | Proof Let \( {p}_{1},{p}_{2},{p}_{3} \) be three common points of \( {M}_{1} \) and \( {M}_{2} \) . Because of\n\n\[ 1 = \left| {\operatorname{St}\left( {p}_{1}\right) \cap \operatorname{St}\left( {p}_{2}\right) \cap \operatorname{St}\left( {p}_{3}\right) \cap S}\right| \geq \left| {\mathop{\bigcap }\limits_{{p \in {M}... | Yes |
Lemma 6.2.14 Let \( D \subset {\mathbb{R}}^{2} \) be an open disc. Let \( f : D \rightarrow \mathbb{R} \) be a smooth function. Let \( \eta > 0 \). Further suppose that \( \parallel \operatorname{grad}f\parallel \leq \eta \) on the whole of \( D \). Then \( p = \left( {\bar{p}, f\left( \bar{p}\right) }\right) \) and \(... | Proof The inequality \[ \parallel \bar{p} - \bar{q}\parallel \leq \parallel p - q\parallel \] is clear. From the estimate \[ \left| {f\left( \bar{p}\right) - f\left( \bar{q}\right) }\right| = \left| {{\int }_{0}^{1}\frac{d}{dt}f\left( {t\bar{p} + \left( {1 - t}\right) \bar{q}}\right) {dt}}\right| \] \[ = \left| {{\int ... | Yes |
Lemma 6.2.15 Let \( S \subset {\mathbb{R}}^{3} \) be a compact orientable regular surface. Then there exists a positive \( {\rho }_{3} \) such that for every \( \varepsilon \) -configuration \( A \) on \( S \) with \( \varepsilon \in \left( {0,{\rho }_{3}}\right\rbrack \) the resulting set of triangles forms a polyhedr... | Proof Choose \( {\rho }_{3} > 0 \) so small that all previous lemmas apply. Let \( \Delta \) and \( {\Delta }^{\prime } \) be two triangles obtained by the above construction. We have to show that they intersect at exactly one edge, or at exactly one vertex, or not at all. If all their vertices belong to one maximal su... | Yes |
Lemma 6.2.16 Let \( S \) be a compact orientable regular surface. Then there exists a \( {\rho }_{4} > 0 \) such that for every \( \varepsilon \) -configuration \( A \) with \( \varepsilon \in \left( {0,{\rho }_{4}}\right\rbrack \) there is a polyhedron \( X \) whose vertices are points in \( A \), such that \[ \Phi \m... | Proof (a) We first show injectivity. If there were a sequence of \( {\varepsilon }_{i} \) - configurations \( {A}_{i},{\varepsilon }_{i} \searrow 0 \), and points \( {x}_{i} \neq {y}_{i} \) in the corresponding polyhedra with \( \Phi \left( {x}_{i}\right) = \Phi \left( {y}_{i}\right) = q \), then we would have \( \begi... | Yes |
Corollary 6.2.18 Let \( S \) be a compact orientable regular surface. Then there exist three local parametrisations \( \left( {{U}_{i},{F}_{i},{V}_{i}}\right) \) of \( S \), which cover \( S \) entirely: | Proof Let \( \Phi : \left| X\right| \rightarrow S \) be a triangulation, \( X = \left\{ {{\Delta }_{1},\ldots ,{\Delta }_{k}}\right\} \) . For every triangle we choose a congruence to a triangle in the plane:\n\n\[ {L}_{j} : {\Delta }_{j}\overset{ \cong }{ \rightarrow }{\Delta }_{j}^{\prime } \subset {\mathbb{R}}^{2} \... | No |
Lemma 1.2.15 Let \( N \) be a fixed integer. Then, for every \( {a}_{\epsilon }^{i} \geq 0 \) , \[ \mathop{\limsup }\limits_{{\epsilon \rightarrow 0}}\epsilon \log \left( {\mathop{\sum }\limits_{{i = 1}}^{N}{a}_{\epsilon }^{i}}\right) = \mathop{\max }\limits_{{i = 1}}^{N}\mathop{\limsup }\limits_{{\epsilon \rightarrow ... | Proof: First note that for all \( \epsilon \) , \[ 0 \leq \epsilon \log \left( {\mathop{\sum }\limits_{{i = 1}}^{N}{a}_{\epsilon }^{i}}\right) - \mathop{\max }\limits_{{i = 1}}^{N}\epsilon \log {a}_{\epsilon }^{i} \leq \epsilon \log N. \] Since \( N \) is fixed, \( \epsilon \log N \rightarrow 0 \) as \( \epsilon \right... | Yes |
Lemma 1.2.18 Let \( \left\{ {\mu }_{\epsilon }\right\} \) be an exponentially tight family.\n\n(a) If the upper bound (1.2.7) holds for some \( \alpha < \infty \) and all compact subsets of \( {\Psi }_{I}{\left( \alpha \right) }^{c} \), then it also holds for all measurable sets \( \Gamma \) with \( \bar{\Gamma } \subs... | Proof: We consider the general situation, the case where \( {\mathcal{B}}_{\mathcal{X}} \subseteq \mathcal{B} \) being included in it.\n\n(a) To establish (1.2.7), fix a set \( \Gamma \in \mathcal{B} \) and \( \alpha < \infty \) such that \( \bar{\Gamma } \subset {\Psi }_{I}{\left( \alpha \right) }^{c} \) . Let \( {K}_... | Yes |
Lemma 2.1.2 (a) \( \\left| {\\mathcal{L}}_{n}\\right| \\leq {\\left( n + 1\\right) }^{\\left| \\sum \\right| } \) . | Proof: Note that every component of the vector \( {L}_{n}^{\\mathbf{y}} \) belongs to the set \( \\left\{ {\\frac{0}{n},\\frac{1}{n},\\ldots ,\\frac{n}{n}}\\right\} \), whose cardinality is \( \\left( {n + 1}\\right) \). Part (a) of the lemma follows, since the vector \( {L}_{n}^{\\mathbf{y}} \) is specified by at most... | Yes |
Lemma 2.1.6 If \( \mathbf{y} \in {T}_{n}\left( \nu \right) \) for \( \nu \in {\mathcal{L}}_{n} \), then\n\n\[ \n{\mathrm{P}}_{\mu }\left( {\left( {{Y}_{1},\ldots ,{Y}_{n}}\right) = \mathbf{y}}\right) = {e}^{-n\left\lbrack {H\left( \nu \right) + H\left( {\nu \mid \mu }\right) }\right\rbrack }.\n\] | Proof: The random empirical measure \( {L}_{n}^{\mathbf{Y}} \) concentrates on types \( \nu \in {\mathcal{L}}_{n} \) for which \( {\sum }_{\nu } \subseteq {\sum }_{\mu } \) i.e., \( H\left( {\nu \mid \mu }\right) < \infty \) . Therefore, assume without loss of generality that \( {L}_{n}^{\mathbf{y}} = \nu \) and \( {\s... | Yes |
Lemma 2.1.9 (Large deviations probabilities) For any \( \nu \in {\mathcal{L}}_{n} \) ,\n\n\[ \n{\left( n + 1\right) }^{-\left| \sum \right| }{e}^{-{nH}\left( {\nu \mid \mu }\right) } \leq {\mathrm{P}}_{\mu }\left( {{L}_{n}^{\mathbf{Y}} = \nu }\right) \leq {e}^{-{nH}\left( {\nu \mid \mu }\right) }.\n\] | Proof: By Lemma 2.1.6,\n\n\[ \n{\mathrm{P}}_{\mu }\left( {{L}_{n}^{\mathbf{Y}} = \nu }\right) \; = \;\left| {{T}_{n}\left( \nu \right) }\right| {\mathrm{P}}_{\mu }\left( {\left( {{Y}_{1},\ldots ,{Y}_{n}}\right) = \mathbf{y},{L}_{n}^{\mathbf{y}} = \nu }\right)\n\]\n\n\[ \n= \left| {{T}_{n}\left( \nu \right) }\right| {e}... | No |
Theorem 2.1.10 (Sanov) For every set \( \Gamma \) of probability vectors in \( {M}_{1}\left( \sum \right) \), \[ - \mathop{\inf }\limits_{{\nu \in {\Gamma }^{o}}}H\left( {\nu \mid \mu }\right) \leq \mathop{\liminf }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log {\mathrm{P}}_{\mu }\left( {{L}_{n}^{\mathbf{Y}} \in \Gam... | Proof: First, from Lemma 2.1.9, upper and lower bounds for all finite \( n \) are deduced. By the upper bound of Lemma 2.1.9, \[ {\mathrm{P}}_{\mu }\left( {{L}_{n}^{\mathbf{Y}} \in \Gamma }\right) = \mathop{\sum }\limits_{{\nu \in \Gamma \cap {\mathcal{L}}_{n}}}{\mathrm{P}}_{\mu }\left( {{L}_{n}^{\mathbf{Y}} = \nu }\ri... | Yes |
Theorem 2.1.24 (Cramér’s theorem for finite subsets of \( \mathbb{R} \) ) For any set \( A \subset \mathbb{R} \) , \n\n\[ \n- \mathop{\inf }\limits_{{x \in {A}^{o}}}I\left( x\right) \leq \mathop{\liminf }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log {\mathrm{P}}_{\mu }\left( {{\widehat{S}}_{n} \in A}\right) \n\] \n\... | Proof: When the set \( A \) is open, so is the set \( \Gamma \) of (2.1.23), and the bounds of (2.1.25) are simply the bounds of (2.1.11) for \( \Gamma \) . By Jensen’s inequality, for every \( \nu \in {M}_{1}\left( \sum \right) \) and every \( \lambda \in \mathbb{R} \) , \n\n\[ \n\Lambda \left( \lambda \right) = \log ... | Yes |
Lemma 2.1.33 For every probability vector \( \nu \in {\mathcal{L}}_{n} \): (a) If \( I\left( {\nu \mid \frac{n}{m},{L}_{m}^{\mathbf{y}}}\right) < \infty \), then \[ \left| {\frac{1}{n}\log \mathrm{P}\left( {{L}_{n}^{\mathbf{Y}} = \nu }\right) + I\left( {\nu \left| {\;\frac{n}{m}}\right. ,{L}_{m}^{\mathbf{y}}}\right) }\... | Proof: (a) Under sampling without replacement, the probability of the event \( \left\{ {{L}_{n}^{\mathbf{Y}} = \nu }\right\} \) for \( \nu \in {\mathcal{L}}_{n} \) is exactly the number of \( n \)-tuples \( {i}_{1} \neq {i}_{2} \neq \cdots \neq {i}_{n} \) resulting in type \( \nu \), compared to the overall number of \... | Yes |
Theorem 2.1.41 Suppose \( {L}_{m}^{\mathbf{y}} \) converges to \( \mu \) and \( n/m \rightarrow \beta \in \left( {0,1}\right) \) as \( n \rightarrow \infty \) . Then the random empirical measures \( {L}_{n}^{\mathbf{Y}} \) satisfy the LDP with the good rate function \( I\left( {\nu \mid \beta ,\mu }\right) \) . Explici... | \[ - \mathop{\inf }\limits_{{\nu \in {\Gamma }^{o}}}I\left( {\nu \mid \beta ,\mu }\right) \leq \mathop{\liminf }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log \mathrm{P}\left( {{L}_{n}^{\mathbf{Y}} \in \Gamma }\right) \] \[ \leq \mathop{\limsup }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log \mathrm{P}\left( {{L}_{... | Yes |
Lemma 2.2.5 (a) \( \Lambda \) is a convex function and \( {\Lambda }^{ * } \) is a convex rate function. | Proof: (a) The convexity of \( \Lambda \) follows by Hölder’s inequality, since\n\n\[ \Lambda \left( {\theta {\lambda }_{1} + \left( {1 - \theta }\right) {\lambda }_{2}}\right) = \log E\left\lbrack {{\left( {e}^{{\lambda }_{1}{X}_{1}}\right) }^{\theta }{\left( {e}^{{\lambda }_{2}{X}_{1}}\right) }^{\left( 1 - \theta \ri... | Yes |
Corollary 2.2.19 For any \( y \in \mathbb{R} \) ,\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log {\mu }_{n}\left( {\lbrack y,\infty }\right) ) = - \mathop{\inf }\limits_{{x \geq y}}{\Lambda }^{ * }\left( x\right) . \] | Proof: Since \( \lbrack x, x + \delta ) \subset \lbrack y,\infty ) \) for all \( x \geq y \) and all \( \delta > 0 \), it follows that\n\n\[ \mathop{\liminf }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log {\mu }_{n}\left( {\lbrack y,\infty }\right) ) \geq \mathop{\sup }\limits_{{x \geq y}}\mathop{\liminf }\limits_{{n... | Yes |
Lemma 2.2.20 If \( 0 \in {\mathcal{D}}_{\Lambda }^{o} \) then \( {\Lambda }^{ * } \) is a good rate function. Moreover, if \( {\mathcal{D}}_{\Lambda } = \mathbb{R} \), then\n\n\[\n\mathop{\lim }\limits_{{\left| x\right| \rightarrow \infty }}{\Lambda }^{ * }\left( x\right) /\left| x\right| = \infty\n\] | Proof: As \( 0 \in {\mathcal{D}}_{\Lambda }^{o} \), there exist \( {\lambda }_{ - } < 0 \) and \( {\lambda }_{ + } > 0 \) that are both in \( {\mathcal{D}}_{\Lambda } \) . Since for any \( \lambda \in \mathbb{R} \), \n\n\[ \n\frac{{\Lambda }^{ * }\left( x\right) }{\left| x\right| } \geq \lambda \operatorname{sign}\left... | Yes |
Lemma 2.2.31 (a) \( \Lambda \left( \cdot \right) \) is convex and differentiable everywhere, and \( {\Lambda }^{ * }\left( \cdot \right) \) is a good convex rate function.\n\n(b) \( \;y = \nabla \Lambda \left( \eta \right) \Rightarrow \;{\Lambda }^{ * }\left( y\right) = \langle \eta, y\rangle - \Lambda \left( \eta \rig... | Proof: (a) The convexity of \( \Lambda \) follows by Hölder’s inequality. Its differentiability follows by dominated convergence. (See the proof of Lemma 2.2.5.) Convexity and lower semicontinuity of \( {\Lambda }^{ * } \) follow from Definition 2.2.2 by an argument similar to the proof of part (a) of Lemma 2.2.5. Sinc... | Yes |
Lemma 2.3.12 (Rockafellar) If \( \Lambda : {\mathbb{R}}^{d} \rightarrow ( - \infty ,\infty \rbrack \) is an essentially smooth, lower semicontinuous, convex function, then \( {ri}{\mathcal{D}}_{{\Lambda }^{ * }} \subseteq \mathcal{F} \) . | Proof: The proof is based on the results of Appendix A. Note first that there is nothing to prove if \( {\mathcal{D}}_{{\Lambda }^{ * }} \) is empty. Hence, it is assumed hereafter that \( {\mathcal{D}}_{{\Lambda }^{ * }} \) is non-empty. Fix a point \( x \in \operatorname{ri}{\mathcal{D}}_{{\Lambda }^{ * }} \) and def... | Yes |
Corollary 2.4.5 Fix \( a < b \) . Suppose that \( a \leq X \leq b \) is a real-valued random variable with \( \bar{x} = E\left( X\right) \) . Then, for any \( \lambda \in \mathbb{R} \) ,\n\n\[ E\left( {e}^{\lambda X}\right) \leq \frac{\bar{x} - a}{b - a}{e}^{\lambda b} + \frac{b - \bar{x}}{b - a}{e}^{\lambda a}. \] | Proof: Set \( {\sigma }^{2} = \left( {b - \bar{x}}\right) \left( {\bar{x} - a}\right) \) . If \( \sigma = 0 \) then either \( X = \bar{x} = a \) almost surely or \( X = \bar{x} = b \) almost surely, with (2.4.6) trivially holding with equality in both cases. Assuming hereafter that \( \sigma > 0 \), since \( {x}^{2} - ... | Yes |
Corollary 2.4.7 Suppose \( v > 0 \) and the real valued random variables \( \left\{ {Y}_{n}\right. \) : \( n = 1,2,\ldots \} \) are such that both \( {Y}_{n} \leq 1 \) almost surely, and \( E\left\lbrack {{Y}_{n} \mid {S}_{n - 1}}\right\rbrack = 0 \) , \( E\left\lbrack {{Y}_{n}^{2} \mid {S}_{n - 1}}\right\rbrack \leq v... | Proof: Applying Lemma 2.4.1 for the conditional law of \( {Y}_{k} \) given \( {S}_{k - 1} \) , \( k = 1,2,\ldots \), for which \( \bar{x} = 0, b = 1 \) and \( {\sigma }^{2} = v \), it follows that almost surely\n\n\[ E\left( {{e}^{\lambda {Y}_{k}} \mid {S}_{k - 1}}\right) \leq \frac{{e}^{-{v\lambda }} + v{e}^{\lambda }... | Yes |
Corollary 2.4.14 Let \( {Z}_{n} = {g}_{n}\left( {{X}_{1},\ldots ,{X}_{n}}\right) \) for independent \( \sum \) -valued random variables \( \left\{ {X}_{i}\right\} \) and real-valued, measurable \( {g}_{n}\left( \cdot \right) \) . Let \( \left\{ {\widehat{X}}_{i}\right\} \) be an independent copy of \( \left\{ {X}_{i}\r... | Proof: For \( k = 1,\ldots, n \) let \( {S}_{k} \triangleq E\left\lbrack {{Z}_{n} \mid {X}_{1},\ldots ,{X}_{k}}\right\rbrack - E{Z}_{n} \) with \( {S}_{0} \triangleq 0 \) . In particular, \( {S}_{n} = {Z}_{n} - E{Z}_{n} \) and \( {Y}_{k} \triangleq {S}_{k} - {S}_{k - 1} \) is such that\n\n\[ E\left\lbrack {{Y}_{k} \mid... | Yes |
Corollary 2.4.19 For any \( n, t > 0 \) , \[ P\left( {\left| {{B}_{n}\left( \mathbf{X}\right) - {E}_{n}}\right| \geq t}\right) \leq 2\exp \left( {-{t}^{2}/{2n}}\right) . \] | Proof: Clearly, \( \left| {{B}_{n}\left( \mathbf{x}\right) - {B}_{n}\left( {\mathbf{x}}^{\prime }\right) }\right| \leq 1 \) when \( \mathbf{x} \) and \( {\mathbf{x}}^{\prime } \) differ only in one coordinate. It follows that \( {B}_{n}\left( \mathbf{X}\right) \) satisfies (2.4.15). The claim follows by an application ... | Yes |
For any product probability measure \( \mathbf{R} \) on \( {\sum }^{n} \), any \( \alpha > \) \( 0, t > 0 \) and \( A \in {\mathcal{B}}_{{\sum }^{n}} \) ,\n\n\[ \mathbf{R}\left( \left\{ {\mathbf{x} : {f}_{\alpha }\left( {A,\mathbf{x}}\right) \geq t}\right\} \right) {e}^{t} \leq {\int }_{{\sum }^{n}}{e}^{{f}_{\alpha }\l... | Proof: Fix a product probability measure \( \mathbf{R},\alpha > 0 \) and \( A \in {\mathcal{B}}_{{\sum }^{n}} \) . The right inequality in (2.4.27) trivially holds when \( \mathbf{R}\left( A\right) = 0 \) . When \( \mathbf{R}\left( A\right) > 0 \) , set \( \mathbf{Q}\left( \cdot \right) = \mathbf{R}\left( {\cdot \cap A... | Yes |
For any product probability measure \( \mathbf{R} \) on \( {\sum }^{n}, A \in {\mathcal{B}}_{{\sum }^{n}} \) , and any \( u > 0 \) , \[ \mathbf{R}\left( {\{ \mathbf{x} : g\left( {A,\mathbf{x}}\right) \geq u\} }\right) \mathbf{R}\left( A\right) \leq {e}^{-{u}^{2}/4}. \] | Proof: Note first that \[ \mathop{\inf }\limits_{\left\{ \nu \in {M}_{1}\left( {\sum }^{n}\right) : \nu \left( A\right) = 1\right\} }{\int }_{{\sum }^{n}}\left( {\mathop{\sum }\limits_{{k = 1}}^{n}{\beta }_{k}{1}_{{y}_{k} \neq {x}_{k}}}\right) \nu \left( {d\mathbf{y}}\right) = \mathop{\inf }\limits_{{\mathbf{y} \in A}}... | Yes |
Corollary 2.4.36 For any \( v \in {\mathbb{Z}}_{ + } \) , \[ P\left( {{Z}_{n}\left( \mathbf{X}\right) \geq {M}_{n} + v}\right) \leq 2\exp \left( {-\frac{{v}^{2}}{4\left( {{M}_{n} + v}\right) }}\right) , \] (2.4.37) \[ P\left( {{Z}_{n}\left( \mathbf{X}\right) \leq {M}_{n} - v}\right) \leq 2\exp \left( {-\frac{{v}^{2}}{4... | Proof: Fix \( v \in {\mathbb{Z}}_{ + } \), and let \( \sum = \left\lbrack {0,1}\right\rbrack \), with \( A\left( j\right) \triangleq \left\{ {\mathbf{y} : {Z}_{n}\left( \mathbf{y}\right) \leq j}\right\} \subset \) \( {\sum }^{n} \) for \( j \in \{ 1,2,\ldots, n\} \) . Suppose \( \mathbf{x} \in {\sum }^{n} \) is such th... | Yes |
For any \( \alpha > 0 \) and probability measures \( P, Q \) on \( \sum \), let\n\n\[ \n{\Delta }_{\alpha }\left( {Q, P}\right) \triangleq {\int }_{\widetilde{\sum }}{\phi }_{\alpha }\left( {\min \left\{ {\frac{dQ}{dP},1}\right\} }\right) {dP} \n\]\n\nwhere \( \widetilde{\sum } \) is such that \( P\left( {\widetilde{\s... | Proof: (a) Let \( {\left( P - Q\right) }_{ + } \) denote the positive part of the finite (signed) measure \( P - Q \) while \( Q \land P \) denotes the positive measure \( P - {\left( P - Q\right) }_{ + } = Q - \) \( {\left( Q - P\right) }_{ + } \) . Let \( q = {\left( P - Q\right) }_{ + }\left( \sum \right) = {\left( ... | Yes |
Theorem 3.1.1 (Perron-Frobenius) Let \( \mathbf{B} = \{ B\left( {i, j}\right) {\} }_{i, j = 1}^{\left| \sum \right| } \) be an irreducible matrix. Then \( \mathbf{B} \) possesses an eigenvalue \( \rho \) (called the Perron-Frobenius eigenvalue) such that:\n\n(a) \( \rho > 0 \) is real.\n\n(b) For any eigenvalue \( \lam... | Proof: The proofs of parts (a)-(d) can be found in [Sen81], Theorem 1.5. To prove part (e), let \( \alpha \triangleq \mathop{\sup }\limits_{i}{\vartheta }_{i},\beta \triangleq \mathop{\inf }\limits_{i}{\vartheta }_{i} > 0,\gamma \triangleq \mathop{\sup }\limits_{j}{\phi }_{j} \), and \( \delta \triangleq \mathop{\inf }... | Yes |
Theorem 3.1.2 Let \( \\left\\{ {Y}_{k}\\right\\} \) be a finite state Markov chain possessing an irreducible transition matrix \( \\mathbf{\\Pi } \) . For every \( z \\in {\\mathbb{R}}^{d} \\), define\n\n\[ I\\left( z\\right) \\triangleq \\mathop{\\sup }\\limits_{{\\lambda \\in {\\mathbb{R}}^{d}}}\\left\\{ {\\langle \\... | Proof: Define\n\n\[ {\\Lambda }_{n}\\left( \\lambda \\right) \\triangleq \\log {E}_{\\sigma }^{\\pi }\\left\\lbrack {e}^{\\left\\langle \\lambda ,{Z}_{n}\\right\\rangle }\\right\\rbrack .\n\]\n\nIn view of the Gärtner-Ellis theorem (Theorem 2.3.6), it is enough to check that the limit\n\n\[ \\Lambda \\left( \\lambda \\... | Yes |
Theorem 3.1.13 Assume that \( \Pi \) is strictly positive. Then for every probability measure \( q \in {M}_{1}\left( {\sum }^{2}\right) \) , \[ {I}_{2}\left( q\right) = \left\{ \begin{matrix} \mathop{\sum }\limits_{i}{q}_{1}\left( i\right) H\left( {{q}_{f}\left( {\cdot \mid i}\right) \mid \pi \left( {i, \cdot }\right) ... | Proof: By Theorem 3.1.6, \[ {I}_{2}\left( q\right) = \mathop{\sup }\limits_{{\mathbf{u} \gg 0}}\mathop{\sum }\limits_{{j = 1}}^{\left| \sum \right| }\mathop{\sum }\limits_{{i = 1}}^{\left| \sum \right| }q\left( {i, j}\right) \log \frac{u\left( {i, j}\right) }{\left( {\mathbf{u}{\mathbf{\Pi }}^{\left( 2\right) }}\right)... | Yes |
Theorem 3.3.3 (Gibbs's principle) Let\n\n\\[ \n\\mathcal{M} \\triangleq \\left\\{ {\\nu \\in \\bar{\\Gamma } : H\\left( {\\nu \\mid \\mu }\\right) = {I}_{\\Gamma }}\\right\\} \n\\]\n\n(3.3.4)\n\n(a) All the limit points of \\( \\left\\{ {\\mu }_{n}^{ * }\\right\\} \\) belong to \\( \\overline{co}\\left( \\mathcal{M}\\r... | Proof: Since \\( \\left| \\sum \\right| < \\infty ,\\bar{\\Gamma } \\) is a compact set and thus \\( \\mathcal{M} \\) is non-empty. Moreover, part (b) of the theorem follows from part (a) by Exercise 2.1.19 and the compactness of \\( {M}_{1}\\left( \\sum \\right) \\) . Next, for every \\( U \\subset {M}_{1}\\left( \\su... | No |
Theorem 3.4.3 The Neyman-Pearson test with the constant threshold \( \gamma \in \) \( \left( {{\bar{x}}_{0},{\bar{x}}_{1}}\right) \) satisfies\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log {\alpha }_{n} = - {\Lambda }_{0}^{ * }\left( \gamma \right) < 0 \]\n\nand\n\n\[ \mathop{\lim }\limits_{{n \r... | Proof: Note that\n\n\[ {\alpha }_{n} = {\mathrm{P}}_{{\mu }_{0}}\left( {{\widehat{S}}_{n} \in \left( {\gamma ,\infty }\right) }\right) .\n\]\n\nMoreover, by dominated and monotone convergence,\n\n\[ {\bar{x}}_{0} = \mathop{\lim }\limits_{{\lambda \searrow 0}}{\Lambda }_{0}^{\prime }\left( \lambda \right) ,\;{\bar{x}}_{... | No |
Corollary 3.4.6 (Chernoff’s bound) If \( 0 < \mathrm{P}\left( {H}_{0}\right) < 1 \), then\n\n\[ \mathop{\inf }\limits_{\mathcal{S}}\mathop{\liminf }\limits_{{n \rightarrow \infty }}\left\{ {\frac{1}{n}\log {P}_{n}^{\left( e\right) }}\right\} = - {\Lambda }_{0}^{ * }\left( 0\right) ,\]\n\nwhere the infimum is over all d... | Proof: It suffices to consider only Neyman-Pearson tests. Let \( {\alpha }_{n}^{ * } \) and \( {\beta }_{n}^{ * } \) be the error probabilities for the zero threshold Neyman-Pearson test. For any other Neyman-Pearson test, either \( {\alpha }_{n} \geq {\alpha }_{n}^{ * } \) (when \( {\gamma }_{n} \leq 0 \) ) or \( {\be... | Yes |
Lemma 3.4.7 (Stein’s lemma) Let \( {\beta }_{n}^{\epsilon } \) be the infimum of \( {\beta }_{n} \) among all tests with \( {\alpha }_{n} < \epsilon \) . Then, for any \( \epsilon < 1 \) , \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log {\beta }_{n}^{\epsilon } = {\bar{x}}_{0} \] | Proof: It suffices to consider only Neyman-Pearson tests. Then \[ {\alpha }_{n} = {\mathrm{P}}_{{\mu }_{0}}\left( {{\widehat{S}}_{n} > {\gamma }_{n}}\right) \] and \[ {\beta }_{n} = {\mathrm{P}}_{{\mu }_{1}}\left( {{\widehat{S}}_{n} \leq {\gamma }_{n}}\right) = {E}_{{\mu }_{1}}\left\lbrack {1}_{{\widehat{S}}_{n} \leq {... | Yes |
Lemma 3.5.3 For every test \( \mathcal{S} \) with error probabilities \( {\left\{ {\alpha }_{n},{\beta }_{n}\right\} }_{n = 1}^{\infty } \), there exists a test \( \widetilde{\mathcal{S}} \) of the form \( {\widetilde{\mathcal{S}}}^{n}\left( \mathbf{y}\right) = \widetilde{\mathcal{S}}\left( {{L}_{n}^{\mathbf{y}}, n}\ri... | Proof: Let \( {\mathcal{S}}_{0}^{n} \triangleq {\left( {\mathcal{S}}^{n}\right) }^{-1}\left( 0\right) \) and \( {\mathcal{S}}_{1}^{n} \triangleq {\left( {\mathcal{S}}^{n}\right) }^{-1}\left( 1\right) \) denote the subsets of \( {\sum }^{n} \) that the maps \( {\mathcal{S}}^{n} \) assign to \( {H}_{0} \) and \( {H}_{1} ... | Yes |
Theorem 3.5.4 (Hoeffding) Let the test \( {\mathcal{S}}^{ * } \) consist of the maps\n\n\[ \n{\mathcal{S}}^{*n}\left( \mathbf{y}\right) = \left\{ \begin{array}{ll} 0 & \text{ if }H\left( {{L}_{n}^{\mathbf{y}} \mid {\mu }_{0}}\right) < \eta \\ 1 & \text{ otherwise. } \end{array}\right.\n\]\n\n(3.5.5)\n\nThen \( {\mathca... | Proof: By the upper bound of Sanov's theorem (Theorem 2.1.10),\n\n\[ \n\mathop{\limsup }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log {\mathrm{P}}_{{\mu }_{0}}\left( {{H}_{0}\text{ rejected by }{\mathcal{S}}^{*n}}\right)\n\]\n\n\[ \n= \mathop{\limsup }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log {\mathrm{P}}_{{\... | Yes |
Theorem 3.6.2 (Shannon's weak source coding theorem) For any \( D \geq 0 \) such that \( {R}_{1}\left( D\right) < \infty \) and for any \( \delta > 0 \), there exists a sequence of codes \( {\left\{ {C}_{n}\right\} }_{n = 1}^{\infty } \) with distortion at most \( D \), and rates \( {R}_{{C}_{n}} \leq {R}_{1}\left( D\r... | The proof of this theorem is based on a random coding argument, where instead of explicitly constructing the codes \( {C}_{n} \), the classes \( {\mathcal{C}}_{n} \) of all codes of some fixed size are considered. A probability measure on \( {\mathcal{C}}_{n} \) is constructed using a law that is independent of the seq... | Yes |
Lemma 3.6.5 Suppose \( Q \) is a probability measure on \( \sum \times \sum \) for which \( H\left( {Q \mid {Q}_{X} \times {Q}_{Y}}\right) < \infty \) and \( {Q}_{X} = {\mathcal{P}}_{1} \) . Fix \( \delta > 0 \) arbitrarily small and let \( {\mathcal{C}}_{n} \) be the class of all codes \( {C}_{n} \) of size \( {k}_{n}... | Proof: For any \( \mathbf{x} = \left( {{x}_{1},\ldots ,{x}_{n},\ldots }\right) \), define the set\n\n\[ \n{S}_{n}\left( \mathbf{x}\right) \triangleq \left\{ {\left( {{y}_{1},\ldots ,{y}_{n}}\right) : \frac{1}{n}\mathop{\sum }\limits_{{j = 1}}^{n}\rho \left( {{x}_{j},{y}_{j}}\right) < {\rho }_{Q} + \delta }\right\} , \n... | Yes |
Theorem 3.7.4 (Bahadur and Rao) Let \( {\mu }_{n} \) denote the law of \( {\widehat{S}}_{n} = \) \( \frac{1}{n}\mathop{\sum }\limits_{{i = 1}}^{n}{X}_{i} \), where \( {X}_{i} \) are i.i.d. real valued random variables with logarithmic moment generating function \( \Lambda \left( \lambda \right) = \log E\left\lbrack {e}... | Proof: Consider the probability measure \( \widetilde{\mu } \) defined by \( d\widetilde{\mu }/{d\mu }\left( x\right) = \) \( {e}^{{\eta x} - \Lambda \left( \eta \right) } \), and let \( {Y}_{i} \triangleq \left( {{X}_{i} - q}\right) /\sqrt{{\Lambda }^{\prime \prime }\left( \eta \right) } \), for \( i = 1,2,\ldots, n \... | No |
Lemma 4.1.4 A family of probability measures \( \left\{ {\mu }_{\epsilon }\right\} \) on a regular topological space can have at most one rate function associated with its LDP. | Proof: Suppose there exist two rate functions \( {I}_{1}\left( \cdot \right) \) and \( {I}_{2}\left( \cdot \right) \), both associated with the LDP for \( \left\{ {\mu }_{\epsilon }\right\} \) . Without loss of generality, assume that for some \( {x}_{0} \in \mathcal{X},{I}_{1}\left( {x}_{0}\right) > {I}_{2}\left( {x}_... | Yes |
Lemma 4.1.5 Let \( \mathcal{E} \) be a measurable subset of \( \mathcal{X} \) such that \( {\mu }_{\epsilon }\left( \mathcal{E}\right) = 1 \) for all \( \epsilon > 0 \) . Suppose that \( \mathcal{E} \) is equipped with the topology induced by \( \mathcal{X} \) . (a) If \( \mathcal{E} \) is a closed subset of \( \mathca... | Proof: In the topology induced on \( \mathcal{E} \) by \( \mathcal{X} \), the open sets are the sets of the form \( G \cap \mathcal{E} \) with \( G \subseteq \mathcal{X} \) open. Similarly, the closed sets in this topology are the sets of the form \( F \cap \mathcal{E} \) with \( F \subseteq \mathcal{X} \) closed. Furt... | Yes |
Lemma 4.1.6 Let \( I \) be a good rate function.\n\n(a) Let \( {\left\{ {F}_{\delta }\right\} }_{\delta > 0} \) be a nested family of closed sets, i.e., \( {F}_{\delta } \subseteq {F}_{{\delta }^{\prime }} \) if \( \delta < {\delta }^{\prime } \) . Define \( {F}_{0} = { \cap }_{\delta > 0}{F}_{\delta } \) . Then\n\n\[ ... | Proof: (a) Since \( {F}_{0} \subseteq {F}_{\delta } \) for all \( \delta > 0 \), it suffices to prove that for all \( \eta > 0 \),\n\n\[ \gamma \triangleq \mathop{\lim }\limits_{{\delta \rightarrow 0}}\mathop{\inf }\limits_{{y \in {F}_{\delta }}}I\left( y\right) \geq \mathop{\inf }\limits_{{y \in {F}_{0}}}I\left( y\rig... | Yes |
Theorem 4.1.11 Let \( \mathcal{A} \) be a base of the topology of \( \mathcal{X} \) . For every \( A \in \mathcal{A} \) , define\n\n\[ \n{\mathcal{L}}_{A} \triangleq - \mathop{\liminf }\limits_{{\epsilon \rightarrow 0}}\epsilon \log {\mu }_{\epsilon }\left( A\right)\n\]\n\n(4.1.12)\n\nand\n\n\[ \nI\left( x\right) \tria... | Proof: Since \( \mathcal{A} \) is a base for the topology of \( \mathcal{X} \), for any open set \( G \) and any point \( x \in G \) there exists an \( A \in \mathcal{A} \) such that \( x \in A \subset G \) . Therefore, by definition,\n\n\[ \n\mathop{\liminf }\limits_{{\epsilon \rightarrow 0}}\epsilon \log {\mu }_{\eps... | Yes |
Lemma 4.1.15 Let \( {\mu }_{\epsilon ,\sigma } \) be a family of probability measures on \( \mathcal{X} \), indexed by \( \sigma \), whose range is the set \( \sum \) . Let \( \mathcal{A} \) be a base for the topology of \( \mathcal{X} \). For each \( A \in \mathcal{A} \), define\n\n\[ \n{\mathcal{L}}_{A} \triangleq - ... | ## Proof: The proof parallels that of Theorem 4.1.11. (See Exercise 4.1.29.) | No |
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