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Theorem 1.6. Let the structure conditions (F1), (F2) be in force. Suppose that \( u \in {C}^{2}\left( \Omega \right) \cap C\left( \overline{\Omega }\right) \) is a solution of (1.1), \( \Omega \) satisfies a uniform exterior sphere condition, \( {\left| u\right| }_{0;\Omega } \leq {M}_{0} \), and \( {\left\lbrack u\rig... | The proof is similar to that of Theorem 3.2 in Chapter 6. | No |
Theorem 2.1. Let the structure conditions (F1), (F2) be in force. Suppose that \( u \in {C}^{2}\left( \Omega \right) \cap C\left( \overline{\Omega }\right) \) is a solution of (1.1), \( \Omega \) satisfies a uniform exterior sphere condition, \( \partial \Omega \in {C}^{1} \), and \( u = 0 \) on \( \partial \Omega ,{\l... | Proof. Take \( r = \rho \) in Lemma 1.5, where \( {x}_{0} \) is an arbitrary point on \( \partial \Omega \) . Since \( {\left. u\right| }_{\partial \Omega } = 0 \)\n\n\[ \left| {u\left( x\right) }\right| \leq \left| {\varphi \left( d\right) }\right| ,\;0 < d < \delta . \]\n\nIt follows that\n\n\[ {\left| \frac{\partial... | Yes |
Lemma 3.1. Suppose that \( {\omega }_{i}\left( R\right) \left( {i = 1,2,\cdots, N}\right) \) are nonnegative, nondecreasing functions defined on \( \left( {0,{R}_{0}}\right\rbrack \) . For each \( R \in \left( {0,{R}_{0}/2}\right\rbrack \), we assume that there exists an index set \( A\left( R\right) \subset \{ 1,2,\cd... | Proof. For \( 0 < \beta < 1 \), we write\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{N}{\omega }_{i}\left( R\right) = \mathop{\sum }\limits_{{i \in A\left( R\right) }}{\omega }_{i}\left( R\right) + \beta \mathop{\sum }\limits_{{i \notin A\left( R\right) }}{\omega }_{i}\left( R\right) + \left( {1 - \beta }\right) \mathop{\su... | Yes |
Lemma 3.4 (Krylov). Suppose that the coefficient matrix \( \left( {a}^{ij}\right) \) satisfies (3.16) on \( {B}_{1}^{ + }, f/\lambda \in {L}^{\infty }\left( {B}_{1}^{ + }\right) \), and \( u \in C\left( {\bar{B}}_{1}^{ + }\right) \cap {W}_{loc}^{2, n}\left( {B}_{1}^{ + }\right) \) is a solution of (3.15). Then there ex... | Proof. Set\n\n\[ M\left( R\right) = \mathop{\sup }\limits_{{B}_{R,\delta }}v, m\left( R\right) = \mathop{\inf }\limits_{{B}_{R,\delta }}v,\omega \left( R\right) = M\left( R\right) - m\left( R\right) ,\]\n\nwhere \( v = u/{x}_{n} \) and \( \delta \) is a constant determined in Lemma 3.3. We now apply Lemma 3.3 to the fu... | Yes |
Theorem 3.5. Let the structure conditions (F1) and (F3) be in force. Suppose that \( \partial \Omega \in {C}^{2},\varphi \in {C}^{2}\left( \bar{\Omega }\right) \), and that \( u \in {C}^{3}\left( \Omega \right) \cap {C}^{1}\left( \bar{\Omega }\right) \) is a solution of (1.1)-(1.2), with \( {\left| u\right| }_{0;\Omega... | Proof. The function \( v = u - \varphi \) satisfies the equation\n\n\[ - {a}^{ij}{D}_{ij}v = - {a}^{ij}{D}_{ij}\varphi + F\left( {x, u,{Du},0}\right) ,\]\n\nwhere\n\n\[ {a}^{ij} = {\int }_{0}^{1}\frac{\partial F}{\partial {r}_{ij}}\left( {x, u,{Du},\tau {D}^{2}u}\right) {d\tau }.\]\n\nLet \( {x}_{0} \in \partial \Omega... | Yes |
Theorem 4.1. Let the structure conditions \( {\left( \mathrm{F}1\right) }^{\prime },{\left( \mathrm{F}2\right) }^{\prime },{\left( \mathrm{F}3\right) }^{\prime } \) and \( {\left( \mathrm{F}5\right) }^{\prime } \) be in force. Suppose that \( 0 < \alpha < 1,\partial \Omega \in {C}^{2,\alpha },{a}^{ij}, b \in {C}^{1}\le... | Proof. First we assume that \( {a}^{ij}, b \in {C}^{1,\delta }\left( {\bar{\Omega } \times \mathbb{R} \times {\mathbb{R}}^{n}}\right) \) for some \( 0 < \delta < 1 \) . Take the Banach space \( X = {C}^{1,\alpha }\left( \bar{\Omega }\right) \) . For any \( v \in X \), we solve the Dirichlet problem\n\n(4.2)\n\n\[ \left... | Yes |
If \( F\left\lbrack u\right\rbrack = F\left( {{D}^{2}u}\right) ,{X}_{1} = {C}^{2,\alpha }\left( \bar{\Omega }\right) ,{X}_{2} = {C}^{\alpha }\left( \bar{\Omega }\right) \left( {0 < \alpha < 1}\right) \), and the function \( F \in {C}^{1}\left( {\mathcal{S}}^{n}\right) \), then\n\n(5.2)\n\n\[ \n{F}_{u} = \frac{\partial ... | In fact,\n\n\[ \n{\begin{Vmatrix}F\left\lbrack u + h\right\rbrack - F\left\lbrack u\right\rbrack - \frac{\partial F}{\partial {r}_{ij}}\left( {D}^{2}u\right) {D}_{ij}h\end{Vmatrix}}_{{C}^{\alpha }} \n\]\n\n\[ \n= {\begin{Vmatrix}F\left( {D}^{2}u + {D}^{2}h\right) - F\left( {D}^{2}u\right) - \frac{\partial F}{\partial {... | Yes |
Theorem 5.1. Suppose that \( F \in {C}^{1}\left( \Gamma \right) \) and satisfies the structure condition (F1), \( \partial \Omega \in {C}^{2,\beta } \) for some \( 0 < \beta < 1 \), and \( \varphi \in {C}^{2,\beta }\left( \bar{\Omega }\right) \) . We assume, under some additional structure conditions (may depend on \( ... | Proof. Take \( {X}_{1} = \left\{ {v \in {C}^{2,\beta }\left( \bar{\Omega }\right) \mid v = 0}\right. \) on \( \left. {\partial \Omega }\right\} ,{X}_{2} = {C}^{\beta }\left( \bar{\Omega }\right) ,\sum = \mathbb{R} \) . Define the map \( G : {X}_{1} \times \left\lbrack {0,1}\right\rbrack \rightarrow {X}_{2} \) :\n\n(5.4... | Yes |
Theorem 6.1. Suppose that \( F \) satisfies the structure conditions (F1) and (F4), \( F \in {C}^{2}\left( \Gamma \right) ,\partial \Omega \in {C}^{3},\varphi \in {C}^{3}\left( \bar{\Omega }\right) \) . Then there exist \( C \geq 1 \) and \( 0 < \beta < 1 \) such that a solution \( u \in {C}^{2,\beta }\left( \bar{\Omeg... | First, we notice that for any \( 1 < p < \infty, u \in {W}_{loc}^{4, p}\left( \Omega \right) \cap {C}^{2,\beta }\left( \bar{\Omega }\right) \) . In fact, using finite differences, we can prove that \( {Du} \in {C}^{2,\beta }\left( \Omega \right) \), and\n\n\[ \frac{\partial F}{\partial {r}_{ij}}\left( {{D}^{2}u}\right)... | No |
Theorem 6.3. Let the structure conditions (F1), (F4) be in force. Suppose that \( \partial \Omega \in {C}^{3} \) and \( \varphi \in {C}^{3}\left( \bar{\Omega }\right) \) . Then there exists \( \beta = \beta \left( {n,{\mu }_{1}}\right) \in \left( {0,1}\right) \) such that the Dirichlet problem (6.1),(1.2) admits a solu... | If \( F\left( r\right) \) is twice continuously differentiable, then this theorem is a direct corollary of Theorems 5.1 and 6.1. If \( F\left( r\right) \) is not twice differentiable, then it can be approximated by twice continuously differentiable concave functions. The details of the proof are left to the reader. | No |
Theorem 7.2. Let the structure conditions (F1) and (F4) be in force. Set\n\n\[ \n{M}_{0} = {\left| \varphi \right| }_{0;\Omega } + \frac{d}{n\sqrt[n]{{\omega }_{n}}}\parallel F\left( {\cdot ,0}\right) /\lambda {\parallel }_{{L}^{n}\left( \Omega \right) },\;{\mu }_{1} = {\mu }_{1}\left( {M}_{0}\right) .\n\]\n\nUnder the... | Proof. First we assume that \( \partial \Omega \in {C}^{3,\delta },\varphi \in {C}^{3,\delta }\left( \bar{\Omega }\right) \), where \( 0 < \delta < 1 \) . Suppose that \( u \in {C}^{2,\beta }\left( \bar{\Omega }\right) \) is a solution of (7.1) and (1.2). Then by using the technique of locally flattening the boundary a... | Yes |
Theorem 7.3. Let the structure conditions (F1), (F3) and (F4) be in force. Suppose that \( u \) is a solution of the Dirichlet problem (7.1), (1.2), satisfying \( {\left| u\right| }_{0;\Omega } \leq \) \( {M}_{0},{\left\lbrack u\right\rbrack }_{1;\Omega } \leq {M}_{1},{\left\lbrack u\right\rbrack }_{1,\bar{\alpha };\Om... | Proof. Set\n\n(7.17)\n\n\[ \n\widetilde{F}\left( {x, r}\right) = F\left( {x, u\left( x\right) ,{Du}\left( x\right), r}\right) .\n\]\n\nUsing the structure condition (F3) and the assumption \( {\left\lbrack u\right\rbrack }_{1;\Omega } \leq {M}_{1} \), we get\n\n\[ \n\left| {\widetilde{F}\left( {x, r}\right) - \widetild... | Yes |
Theorem 7.4. Let the structure conditions (F1)-(F6) be in force. Suppose that \( \Lambda \) is bounded on \( \Omega \times \mathbb{R} \times {\mathbb{R}}^{n} \) . Under these assumptions, there exists \( 0 < \alpha < 1 \) , such that for any \( 0 < \beta < \alpha \), if \( \partial \Omega \in {C}^{2,\beta },\varphi \in... | Proof. This is a direct corollary of Theorems 5.1 and 7.3. | No |
Theorem 7.5. Under assumptions (B1) and (B3), there exists \( \alpha = \alpha \left( {n,{\mu }_{1}}\right) \) , such that for any \( 0 < \beta < \alpha \), if (B2) is valid, \( \partial \Omega \in {C}^{2,\beta },\varphi \in {C}^{2,\beta }\left( \bar{\Omega }\right) \), then the Dirichlet problem (7.19),(7.20) admits a ... | Proof. Let \( {h}_{\tau }\left( y\right) \) be the mollified function of \( h\left( y\right) \) . Consider the fully nonlinear equation\n\n\( \left( {7.22}\right) \)\n\n\[ \n{F}_{\tau }\left( {x, u,{Du},{D}^{2}u}\right) = {h}_{\tau }\left( {{L}_{1}u - {f}_{1},\cdots .{L}_{N}u - {f}_{N}}\right) = 0.\n\]\n\nIt is easy to... | Yes |
Theorem 1.1. Let the assumptions (1.2),(1.3) be in force. Suppose that \( {f}_{i}^{\alpha } \in \) \( {L}^{2}\left( \Omega \right) \left( {\alpha = 1,\cdots, n;i = 1,\cdots, N}\right) \) . Then the problem (1.1),(1.5) admits a unique weak solution. | Proof. Set\n\n\[ a\left( {u, v}\right) = {\int }_{\Omega }{A}_{ij}^{\alpha \beta }\left( x\right) {D}_{\beta }{u}^{j}{D}_{\alpha }{v}^{i}{dx} \]\n\n\[ \langle T, v\rangle = {\int }_{\Omega }{f}_{i}^{\alpha }{D}_{\alpha }{v}^{i}{dx} \]\n\nIt is easy to verify that \( a\left( {u, v}\right) \) is a bounded bilinear form o... | Yes |
Example 1.1. Let \( N = n = 2,{A}_{ij}^{\alpha \beta } = \varepsilon {\delta }_{\alpha \beta }{\delta }_{ij} + {a}_{ij}^{\alpha \beta }\left( {\alpha ,\beta = 1,2;i, j = 1,2}\right) \) , where \( 0 < \varepsilon < 1/2 \) , \[ {\delta }_{\alpha \beta } = \left\{ \begin{array}{ll} 1, & \alpha = \beta \\ 0, & \alpha \neq ... | Obviously, \[ {A}_{ij}^{\alpha \beta }{\xi }_{\alpha }{\xi }_{\beta }{\eta }^{i}{\eta }^{j} = \varepsilon {\left| \xi \right| }^{2}{\left| \eta \right| }^{2} + \left( {{\xi }_{2}{\xi }_{1}{\eta }^{2}{\eta }^{1} - {\xi }_{2}{\xi }_{1}{\eta }^{1}{\eta }^{2}}\right) \] \[ = \varepsilon {\left| \xi \right| }^{2}{\left| \et... | Yes |
Theorem 2.1 (Caccioppoli’s inequality). Under the assumptions of Theorem 1.1, if \( u \in {H}_{loc}^{1}\left( {\Omega ,{\mathbb{R}}^{N}}\right) \) is a weak solution of (1.1), then for all \( {x}^{0} \in \Omega \) and all \( \rho, R \) with \( 0 < \rho < R < \operatorname{dist}\left( {{x}^{0},\partial \Omega }\right) \... | Proof. In the integral identity (1.4) we choose \( \varphi = {\eta }^{2}\left( {u - \nu }\right) \), where \( \eta \in \) \( {C}_{0}^{\infty }\left( {{B}_{R}\left( {x}^{0}\right) }\right) \) is a cutoff function:\n\n\[0 \leq \eta \leq 1\;\text{ on }{B}_{R}\left( {x}^{0}\right) ,\;\eta \equiv 1\;\text{ on }{B}_{\rho }\l... | Yes |
Theorem 2.2. Suppose that \( {A}_{ij}^{\alpha \beta }\left( x\right) \) satisfy (1.2), \( {A}_{ij}^{\alpha \beta } \in {W}^{1,\infty }\left( \Omega \right) (\alpha ,\beta = \) \( 1,\cdots, n;i, j = 1,\cdots, N),{f}_{i}^{\alpha } \in \check{{H}^{1}}\left( \Omega \right) \left( {\alpha ,\beta = 1,\cdots, n;i, j = 1,\cdot... | Proof. Denote by \( {e}_{s} \) the unit vector in the \( {x}_{s} \) direction \( \left( {s = 1,\cdots, n}\right) \) . Let \( {\Delta }_{h, s}u\left( x\right) = \frac{1}{h}\left\lbrack {u\left( {x + h{e}_{s}}\right) - u\left( x\right) }\right\rbrack \) be the difference quotient in the \( {x}_{s} \) direction. If \( \va... | Yes |
Corollary 2.5. Suppose that \( {A}_{ij}^{\alpha \beta } \) are constants satisfying (1.2),(1.3) \( (\alpha ,\beta = \) \( 1,\cdots, n;i, j = 1,\cdots, N),{f}_{i}^{\alpha } \equiv 0\left( {\alpha = 1,\cdots, n;i = 1,\cdots, N}\right) \) . If \( u \in {H}_{loc}^{1}\left( {\Omega ,{\mathbb{R}}^{N}}\right) \) is a weak sol... | Proof. Under our assumptions, Corollary 2.4 implies that \( u \in {C}^{\infty }\left( {\Omega ,{\mathbb{R}}^{N}}\right) \) . It\nsatisfies\n\[ \n{\int }_{\Omega }{A}_{ij}^{\alpha \beta }{D}_{\beta }{u}^{j}{D}_{\alpha }{\varphi }^{i}{dx} = 0,\;\forall \varphi \in {H}_{0}^{1}\left( {\Omega ,{\mathbb{R}}^{N}}\right) .\n\]... | Yes |
Lemma 1.2. \( {L}^{p,\mu }\left( \Omega \right) \) has the following properties:\n\n(i) \( \;{L}^{p,0}\left( \Omega \right) \simeq {L}^{p}\left( \Omega \right) \) . | Proof. (i) is obvious. | No |
Lemma 1.3. Let \( \Omega \) be a domain of type (A). If \( u \in {\mathcal{L}}^{p,\mu }\left( \Omega \right) \), where \( p \geq 1 \) , \( \mu \geq 0 \), then for all \( x \in \bar{\Omega } \) and all \( \widetilde{r},\widetilde{R} \) with \( 0 < \widetilde{r} < \widetilde{R} \), we have\n\n(1.2)\n\n\[ \left| {{u}_{x,\... | Proof. For simplicity, we use \( R \) and \( r \) in place of \( \widetilde{R} \) and \( \widetilde{r} \) in our proof. Clearly,\n\n\[ {\left| {u}_{x, R} - {u}_{x, r}\right| }^{p} \leq {2}^{p - 1}\left( {{\left| u\left( z\right) - {u}_{x, R}\right| }^{p} + {\left| u\left( z\right) - {u}_{x, r}\right| }^{p}}\right) . \]... | Yes |
Theorem 1.4. Let \( \Omega \) be a domain of type \( \left( A\right) \) . Then\n\n\[{\mathcal{L}}^{p,\mu }\left( \Omega \right) \simeq {L}^{p,\mu }\left( \Omega \right) \;\text{ for }0 \leq \mu < n.\] | Proof. First we show that for \( \mu \geq 0 \), if \( u \in {L}^{p,\mu }\left( \Omega \right) \), then \( u \in {\mathcal{L}}^{p,\mu }\left( \Omega \right) \), and\n\n\[ \parallel u{\parallel }_{{\mathcal{L}}^{p,\mu }\left( \Omega \right) } \leq C\parallel u{\parallel }_{{L}^{p,\mu }\left( \Omega \right) }.\]\n\nSuppos... | Yes |
Theorem 1.5 (Integral characterization of Hölder continuity). Let \( \Omega \) be a domain of type (A). If \( n < \mu \leq n + p \), then \( {\mathcal{L}}^{p,\mu }\left( \Omega \right) \simeq {C}^{0,\delta }\left( \bar{\Omega }\right) ,\delta = \left( {\mu - n}\right) /p \) ; if \( \mu > n + p \), then \( {\mathcal{L}}... | Proof. We first show that: if \( u \in {C}^{0,\delta }\left( \bar{\Omega }\right) ,0 < \delta \leq 1 \), then \( u \in {\mathcal{L}}^{p,\mu }\left( \Omega \right) \) , \( \mu = n + {p\delta } \), and \( \parallel u{\parallel }_{{\mathcal{L}}^{p,\mu }\left( \Omega \right) } \leq C\parallel u{\parallel }_{{C}^{0,\delta }... | Yes |
Lemma 2.2. Let \( u \) be a weak solution of (2.1), where we assume that \( {A}_{ij}^{\alpha \beta } \) are constants satisfying (2.2) and \( \left| {A}_{ij}^{\alpha \beta }\right| \leq \Lambda \left( {i, j = 1,\cdots, N;\alpha ,\beta = 1,\cdots, n}\right) \) , \( {f}_{i}^{\alpha } \equiv 0\left( {i = 1,\cdots, N;\alph... | Proof. First we prove (2.4).\n\nCorollary 2.5 in Chapter 8 implies that \( u \in {H}^{k}\left( {{B}_{R/2}\left( {x}^{0}\right) ,{\mathbb{R}}^{N}}\right) \) and that\n\n(2.6)\n\n\[ \parallel u{\parallel }_{{H}^{k}\left( {{B}_{R/2}\left( {x}^{0}\right) ,{\mathbb{R}}^{N}}\right) } \leq C\parallel u{\parallel }_{{L}^{2}\le... | Yes |
Theorem 2.4. Suppose that \( {A}_{ij}^{\alpha \beta } \) are constants satisfying (2.2), \( \left| {A}_{ij}^{\alpha \beta }\right| \leq \Lambda \) \( \left( {i, j = 1,\cdots, N;\alpha ,\beta = 1,\cdots, n}\right) ,{f}_{i}^{\alpha } \in {\mathcal{L}}^{2,\mu }\left( \Omega \right) ,0 \leq \mu < n + 2\left( {i = 1,\cdots,... | Remark 2.1. Since \( {L}^{2,\mu }\left( \Omega \right) \subset {\mathcal{L}}^{2,\mu }\left( \Omega \right) \) for \( 0 \leq \mu < n + 2 \) and \( \parallel \cdot {\parallel }_{{\mathcal{L}}^{2,\mu }\left( \Omega \right) } \leq \) \( C\parallel \cdot {\parallel }_{{L}^{2,\mu }\left( \Omega \right) } \), the assumption o... | No |
Theorem 2.5. Suppose that \( {A}_{ij}^{\alpha \beta }\left( x\right) \) satisfy (2.2), \( {A}_{ij}^{\alpha \beta } \in {C}^{0}\left( \bar{\Omega }\right) ,\left| {A}_{ij}^{\alpha \beta }\right| \leq \) \( \Lambda \left( {i, j = 1,\cdots, N;\alpha ,\beta = 1,\cdots, n}\right) \), and \( {f}_{i}^{\alpha } \in {L}^{2,\mu ... | Proof. We use the technique of freezing coefficients.\n\nFor all \( \widetilde{\Omega } \subset \subset \Omega \), all \( {x}^{0} \in \widetilde{\Omega } \), and all \( R \) with \( 0 < R < d, d = \operatorname{dist}\left( {\widetilde{\Omega },\partial \Omega }\right) \), we rewrite (2.1) as\n\n\[ - {D}_{\alpha }\left(... | Yes |
Theorem 2.7. Suppose that \( {A}_{ij}^{\alpha \beta }\left( x\right) \) satisfy (2.2), \( {A}_{ij}^{\alpha \beta } \in {C}^{0,\delta }\left( \bar{\Omega }\right) \;(i, j = \) \( 1,\cdots, N;\alpha ,\beta = 1,\cdots, n),{f}_{i}^{\alpha } \in {C}^{0,\delta }\left( \bar{\Omega }\right) \left( {i = 1,\cdots, N;\alpha = 1,\... | \[ {Du} \in {C}^{0,\delta }\left( {\bar{\Omega },{\mathbb{R}}^{nN}}\right) . \] | Yes |
Theorem 2.8. Suppose that \( {A}_{ij}^{\alpha \beta }\left( x\right) \) satisfy (2.2), \( {A}_{ij}^{\alpha \beta } \in {C}^{k,\delta }\left( \bar{\Omega }\right) \;(i, j = \) \( 1,\cdots, N;\alpha ,\beta = 1,\cdots, n),{f}_{i}^{\alpha } \in {C}^{k,\delta }\left( \bar{\Omega }\right) \left( {i = 1,\cdots, N;\alpha = 1,\... | \[ {Du} \in {C}^{k,\delta }\left( {\bar{\Omega },{\mathbb{R}}^{nN}}\right) . \] | Yes |
Theorem 1.1 (John and Nirenberg). There exist two constants \( {C}_{1} \) and \( {C}_{2} \) , depending only on \( n \), such that for all \( Q \subset {Q}_{0} \), \[ \operatorname{meas}\left\{ {x \in {Q}_{0}\left| \right| u\left( x\right) - {u}_{Q} \mid > t}\right\} \leq {C}_{1}\left| Q\right| \exp \left\{ {-\frac{{C}... | The proof of this theorem is given in Appendix 3 (or see [JN]). | No |
Theorem 1.2 (Stampacchia interpolation theorem). Let \( 1 < q < + \infty \) . Suppose that \( T \) is both a bounded linear operator: \( {L}^{q}\left( {Q}_{0}\right) \rightarrow {L}^{q}\left( {Q}_{0}\right) \), and a bounded linear operator: \( {L}^{\infty }\left( {Q}_{0}\right) \rightarrow \operatorname{BMO}\left( {Q}... | For a proof of this theorem, see Appendix 4. | No |
Theorem 2.1. Suppose that \( u \in {H}_{0}^{1}\left( {{B}_{R},{\mathbb{R}}^{N}}\right) \) satisfies the equation\n\n(2.1)\n\n\[ \n{\int }_{{B}_{R}}{A}_{ij}^{\alpha \beta }{D}_{\beta }{u}^{j}{D}_{\alpha }{\varphi }^{i}{dx} = {\int }_{{B}_{R}}{f}_{i}^{\alpha }{D}_{\alpha }{\varphi }^{i}{dx},\;\forall \varphi \in {H}_{0}^... | Proof. Let \( u \in {H}_{0}^{1}\left( {{B}_{R},{\mathbb{R}}^{N}}\right) \) be the unique solution of (2.1) corresponding to \( f \) . We define the operator \( T \) as follows:\n\n\[ \n{Tf} \triangleq {Du}\n\]\n\nBy choosing \( \varphi = u \) in (2.1), we obtain the estimate\n\n\[ \n\parallel {Du}{\parallel }_{{L}^{2}\... | Yes |
Theorem 2.2. Suppose that \( u \in {H}^{1}\left( {\Omega ,{\mathbb{R}}^{N}}\right) \) satisfies the equation \[ {\int }_{\Omega }{A}_{ij}^{\alpha \beta }\left( x\right) {D}_{\beta }{u}^{j}{D}_{\alpha }{\varphi }^{i}{dx} = {\int }_{\Omega }{f}_{i}^{\alpha }{D}_{\alpha }{\varphi }^{i}{dx},\;\forall \varphi \in {H}_{0}^{1... | Proof. It suffices to prove, \( \forall \widetilde{\Omega } \subset \subset \Omega ,\forall {x}^{0} \in \widetilde{\Omega },\forall R : 0 < R < \operatorname{dist}\left( {\widetilde{\Omega },\partial \Omega }\right) \), that \( {Du} \in {L}^{p}\left( {{B}_{R/2}\left( {x}^{0}\right) ,{\mathbb{R}}^{nN}}\right) . \n\nBy o... | Yes |
Theorem 2.2 (E. Acerbi, N Fusco). Suppose that for any \( \\left( {u, p}\\right) \\in {\\mathbb{R}}^{N} \\times {\\mathbb{R}}^{nN} \) the function \( F\\left( {x, u, p}\\right) : \\Omega \\times {\\mathbb{R}}^{N} \\times {\\mathbb{R}}^{nN} \\rightarrow \\mathbb{R} \) is measurable in \( x \), for almost every \( x \\in... | We refer readers to \\( \\left\\lbrack {\\mathrm{{AF}}1}\\right\\rbrack \\) for a proof. | No |
Theorem 2.4. Suppose that\n\n\\( {1}^{ \\circ }\\;F : \\Omega \\times {\\mathbb{R}}^{N} \\times {\\mathbb{R}}^{nN} \\rightarrow \\mathbb{R}\\;\\) is measurable in \\(\\;x,\\;\\) and for a.e. \\(\\;x \\in \\Omega ,\\;\\) it is \\(\\;{C}^{1}\\) in \\(\\left( {u, p}\\right)\\) .\n\n\\( {2}^{ \\circ }\\;\\) For \\(\\left| ... | We refer readers to \\(\\left\\lbrack {\\mathrm{{MR}}4}\\right\\rbrack \\) for a proof. | No |
Theorem 1.1. Let \( A, B \in {C}^{1} \) satisfy the controllable structure conditions (1.2),(1.3). If \( u \in {H}^{1}\left( {\Omega ,{\mathbb{R}}^{N}}\right) \) satisfies\n\n\[{\int }_{\Omega }\left\lbrack {{A}_{i}^{\alpha }\left( {x, u,{Du}}\right) {D}_{\alpha }{\varphi }^{i} + {B}_{i}\left( {x, u,{Du}}\right) {\varp... | Proof. We first consider the simple case \( {A}_{i}^{\alpha }\left( {x, u, p}\right) = {A}_{i}^{\alpha }\left( p\right) ,{B}_{i}\left( {x, u, p}\right) \equiv 0 \) . In this case, (1.6) reduces to\n\n\[{\int }_{\Omega }{A}_{i}^{\alpha }\left( {Du}\right) {D}_{\alpha }{\varphi }^{i}{dx} = 0,\;\forall \varphi \in {H}_{0}... | Yes |
Example 1 (De Giorgi [DG2],1968). Let \( \Omega = {B}_{1}\left( 0\right) \subset {\mathbb{R}}^{n}, N = n \geq 3 \) . Consider\n\n(2.3)\n\n\[ \n{\int }_{{B}_{1}\left( 0\right) }{A}_{ij}^{\alpha \beta }\left( x\right) {D}_{\beta }{u}^{j}{D}_{\alpha }{\varphi }^{i}{dx} = 0,\;\forall \varphi \in {H}_{0}^{1}\left( {{B}_{1}\... | However, the vector valued function \( {u}_{\gamma }\left( x\right) = \frac{x}{{\left| x\right| }^{\gamma }} \) is an unbounded weak solution of (2.3), where\n\n(2.4)\n\n\[ \n\gamma = \frac{n}{2}\left\{ {1 - {\left\lbrack {\left( 2n - 2\right) }^{2} + 1\right\rbrack }^{-1/2}}\right\} .\n\]\n\nFirst, \( n - {2\gamma } >... | Yes |
Let \( \Omega = {B}_{1}\left( 0\right) \subset {\mathbb{R}}^{n}, N = \) \( n \geq 3 \) . Consider\n\n\[ \n{\int }_{{B}_{1}\left( 0\right) }{A}_{ij}^{\alpha \beta }\left( u\right) {D}_{\beta }{u}^{j}{D}_{\alpha }{\varphi }^{i}{dx} = 0,\;\forall \varphi \in {H}_{0}^{1}\left( {{B}_{1}\left( 0\right) ,{\mathbb{R}}^{n}}\rig... | It is not difficult to verify that \( {A}_{ij}^{\alpha \beta }\left( u\right) {\xi }_{\alpha }^{i}{\xi }_{\beta }^{j} \geq {\left| \xi \right| }^{2} \) and \( {A}_{ij}^{\alpha \beta }\left( u\right) \) is real analytic in \( u \) . However, the vector valued function \( {u}_{1}\left( x\right) = \frac{x}{\left| x\right|... | Yes |
Let \( \Omega = {B}_{1}\left( 0\right) \subset {\mathbb{R}}^{n} \) and \( N = n \geq 5 \). Consider\n\n\[ \n{\int }_{{B}_{1}\left( 0\right) }{A}_{ij}^{\alpha \beta }\left( {x, u}\right) {D}_{\beta }{u}^{j}{D}_{\alpha }{\varphi }^{i}{dx} = 0,\;\forall \varphi \in {H}_{0}^{1}\left( {{B}_{1}\left( 0\right) ,{\mathbb{R}}^{... | Obviously, \( {A}_{ij}^{\alpha \beta }\left( {x, u}\right) {\xi }_{\alpha }^{i}{\xi }_{\beta }^{j} \geq {\left| \xi \right| }^{2} \), the \( {A}_{ij}^{\alpha \beta }\left( {x, u}\right) \) are sufficiently smooth, and \( {u}_{\gamma } = \frac{x}{{\left| x\right| }^{\gamma }} \in {H}^{1}\left( {{B}_{1}\left( 0\right) ,{... | Yes |
Lemma 3.2 (Caccioppoli’s inequality). If \( u \in {H}_{loc}^{1}\left( {\Omega ,{\mathbb{R}}^{N}}\right) \) is a weak solution of (3.1), where the \( {A}_{ij}^{\alpha \beta }\left( {x, u}\right) \) satisfy (3.2) and (3.3), then for all \( {x}^{0} \in \Omega \) and all \( \rho \) and \( R \) with \( 0 < \rho < R < \opera... | This lemma can be proved by following the same procedure as in the proof in Theorem 2.1 in Chapter 8. | No |
Lemma 3.3. Let \( {b}_{ij}^{\alpha \beta } \) be constants satisfying \( {b}_{ij}^{\alpha \beta }{\xi }_{\alpha }^{i}{\xi }_{\beta }^{j} \geq \lambda {\left| \xi \right| }^{2},\lambda > 0,\left| {b}_{ij}^{\alpha \beta }\right| \leq \Lambda \) \( \left( {i, j = 1,\cdots, N;\alpha ,\beta = 1,\cdots, n}\right) \) . If \( ... | The proof is similar to that of (2.5) in Lemma 2.2 in Chapter 9. | No |
Theorem 4.1 (Reverse Hölder inequality). Let \( B \) be a ball in \( {\mathbb{R}}^{n} \). Suppose that\n\n\( {1}^{ \circ }\;g \geq 0, g \in {L}^{q}\left( B\right), q > 1;f \geq 0, f \in {L}^{r}\left( B\right), r > q;\)\n\n\( {2}^{ \circ }\; \) for all \( {x}^{0} \in B \) and all \( R,0 < R < \operatorname{dist}\left( {... | For a proof, see Appendix 5 or [GQ1]. | No |
Theorem 4.3. Suppose that \( {A}_{i}^{\alpha },{B}_{i} \) satisfy the controllable structure conditions:\n\n(4.7)\n\n\[ \n{A}_{i}^{\alpha }\left( {x, u, p}\right) {p}_{\alpha }^{i} \geq \lambda {\left| p\right| }^{2} - \Lambda {\left| u\right| }^{r} - {f}^{2}\left( x\right) \n\]\n\n(4.8)\n\n\[ \n\left\{ \begin{array}{l... | Proof. For all \( B \subset \Omega \), all \( {x}^{0} \in B \), and all \( R,0 < R < \operatorname{dist}\left( {{x}^{0},\partial B}\right) \), we choose in the integral identity\n\n\[ \n{\int }_{\Omega }\left\lbrack {{A}_{i}^{\alpha }\left( {x, u,{Du}}\right) {D}_{\alpha }{\varphi }^{i} + {B}_{i}\left( {x, u,{Du}}\righ... | Yes |
Theorem 6.4. Suppose that \( \Omega \) is a bounded domain in \( {\mathbb{R}}^{n} \), that \( u \) is a weak solution of the elliptic system (3.1) (or (5.1), respectively), and that the assumptions of Theorem 3.1 (or Theorem 5.1, respectively) hold. Then there exists an open set \( {\Omega }_{0} \subset \Omega \) such ... | Proof. By Theorem \( {4.3}^{\prime } \), we deduce that \( \left| {Du}\right| \in {L}_{loc}^{p} \) for some \( p > 2 \) . If \( p > n \), then the embedding theorem implies that \( u \in {C}_{loc}^{0,\delta }\left( {\Omega ,{\mathbb{R}}^{N}}\right) ,\delta > 0 \), and hence \( {\Omega }_{0} = \Omega \) . If \( 2 < p \l... | Yes |
Proposition 1.4. Let \( u \in {W}^{k, p}\left( \Omega \right) ,\varphi \in {C}_{0}^{k}\left( \Omega \right) \) . Then \( {\varphi u} \in {W}^{k, p}\left( \Omega \right) \), and | \[ {D}^{\alpha }\left( {\varphi u}\right) = \mathop{\sum }\limits_{{\beta \leq \alpha }}\left( \begin{array}{l} \alpha \\ \beta \end{array}\right) {D}^{\beta }u{D}^{\alpha - \beta }\varphi ,\;\forall \left| \alpha \right| \leq k. \] | Yes |
Proposition 1.5. Suppose that \( f \) is continuous on \( \mathbb{R} \) and piecewise continuously differentiable, \( {f}^{\prime } \in {L}^{\infty }\left( \mathbb{R}\right) \) . If \( u \in {W}^{1}\left( \Omega \right) \), then \( f \circ u \in {W}^{1}\left( \Omega \right) \), and | \[ D\left( {f \circ u}\right) = \left\{ \begin{array}{l} {f}^{\prime }\left( u\right) {Du}\;\text{ if }u \notin L, \\ 0\;\text{ if }u \in L, \end{array}\right. \] where \( L \) is the set of all corner points of \( f \) (i.e., those points where \( f \) is not differentiable). | Yes |
Corollary 3.1. Let \( {B}_{R} \) be a ball in \( {\mathbb{R}}^{n} \) with radius \( R \). \( {1}^{ \circ }\; \) If \( u \in {W}_{0}^{1, p}\left( {B}_{R}\right) ,1 \leq p < + \infty \), then \[ {\int }_{{B}_{R}}{\left| u\right| }^{p}{dx} \leq C\left( {n, p}\right) {R}^{p}{\int }_{{B}_{R}}{\left| Du\right| }^{p}{dx}. \] | Proof. If \( R = 1 \), Theorem 3.1 implies the conclusion. If \( R \neq 1 \), the conclusion can be established through a rescaling. | No |
Lemma 5.1 (Covering lemma). Suppose that \( E \) is a measurable set covered by a family of balls \( \left\{ {B}_{j}\right\} \) with their radii bounded above. Then there exists a subsequence of disjoint balls \( {B}_{{j}_{1}},{B}_{{j}_{2}},\cdots \) (finitely many or infinitely many) such that\n\n\[ \left| E\right| \l... | Proof. Choose \( {B}_{{j}_{1}} \in \left\{ {B}_{j}\right\} \) such that\n\n\[ \operatorname{diam}{B}_{{j}_{1}} \geq \frac{1}{2}\mathop{\sup }\limits_{j}\operatorname{diam}{B}_{j} \]\n\nWe use induction. Suppose that \( {B}_{{j}_{1}},\cdots ,{B}_{{j}_{k}} \) have been chosen. We want to choose \( {B}_{{j}_{k + 1}} \) su... | Yes |
Theorem 1.1.1 (see Serfling (1988, p18)). A sequence of random vectors \( {\mathbf{X}}_{n} \) in \( {\mathbb{R}}^{k} \) converges in distribution to the random vector \( \mathbf{X} \) if and only if each linear combination of the component of \( {\mathbf{X}}_{n} \) converges in distribution to the same linear combinati... | Proof: \ | No |
Theorem 1.2.1. If \( \mathbf{Y} \sim {N}_{m}\left( {\mu ,\mathbf{\sum }}\right) \) and \( \left| \mathbf{\sum }\right| \neq 0 \), then PDF of \( \mathbf{Y} \) is\n\n\[ f\left( \mathbf{y}\right) = \frac{1}{{\sqrt{2\pi }}^{m}{\left| \mathbf{\sum }\right| }^{1/2}}\exp \left\{ {-\frac{1}{2}{\left( \mathbf{y} - \mu \right) ... | Proof: As \( \mathbf{\sum } > 0,\left( {\text{we can define}{\mathbf{\sum }}^{1/2} > 0\text{ such that }{\mathbf{\sum }}^{1/2}{\mathbf{\sum }}^{1/2} = \mathbf{\sum }}\right) \mathbf{\sum } = {\mathbf{U}}^{T}\operatorname{diag}\left( {{\lambda }_{1},\ldots ,{\lambda }_{m}}\right) \mathbf{U} \) where \( \mathbf{U} \) is ... | Yes |
Theorem 1.2.2. If \( \mathbf{Y} \sim {N}_{m}\left( {\mu ,\mathbf{\sum }}\right) \), then the characteristic function of \( \mathbf{Y}{\phi }_{Y}\left( \mathbf{t}\right) = \) \( \exp \left( {i{\mathbf{t}}^{T}\mu - \frac{1}{2}{\mathbf{t}}^{T}\mathbf{\sum }\mathbf{t}}\right) \) when \( t \equiv {\left( {t}_{1},\ldots ,{t}... | Proof: Let \( \mathbf{X}\overset{d}{ \sim }{N}_{m}\left( {0,{\mathbf{I}}_{m}}\right) \) . Then,\n\n\[{\phi }_{X}\left( \mathbf{t}\right) = E\left( {e}^{i{\mathbf{t}}^{T}\mathbf{X}}\right) = E\left( {e}^{i\mathop{\sum }\limits_{i}{t}_{i}{X}_{i}}\right) = E\left( {e}^{i{t}_{1}{X}_{1}}\right) \cdots E\left( {e}^{i{t}_{m}{... | Yes |
Theorem 1.2.3. A multivariate random vector \( \mathbf{Y} \) is normally distributed if and only if \( {\mathbf{a}}^{T}\mathbf{Y} \) is univariately normally distributed for any \( \mathbf{a} \in {\mathbb{R}}^{m} \) . | Proof: \ | No |
Theorem 1.2.4. If \( \mathbf{Y} \sim {N}_{m}\left( {\mu ,\mathbf{\sum }}\right) \), then (i) \( {\mathbf{Y}}^{\left( \mathbf{1}\right) } \sim {N}_{q}\left( {{\mu }^{\left( \mathbf{1}\right) },{\mathbf{\sum }}_{\mathbf{{11}}}}\right) \) ,(ii) \( {\mathbf{Y}}^{\left( \mathbf{2}\right) } \sim {N}_{m - q}\left( {{\mu }^{\l... | Proof : Let\n\n\[ \mathbf{B} = \left( {{\mathbf{I}}_{q},{\mathbf{0}}_{q \times \left( {m - q}\right) }}\right) \]\n\nbe a partition of \( \mathbf{B} \) into two blocks of matrices. Clearly \( {\mathbf{Y}}^{\left( 1\right) } = \mathbf{{BY}} \) . According to Lemma 1.2.1, \( {\mathbf{Y}}^{\left( \mathbf{1}\right) } \sim ... | Yes |
Theorem 1.2.6. Assume \( \mathbf{Y} \sim {N}_{m}\left( {\mathbf{\mu },\mathbf{\sum }}\right) \) with \( \left| \mathbf{\sum }\right| \neq 0 \), then \( {\mathbf{Y}}^{\left( 1\right) } \) and \( {\mathbf{Y}}^{\left( 2\right) } \) are independent if and only if \( {\mathbf{\sum }}_{\mathbf{{12}}} = \mathbf{0} \) . | Proof : On the one hand, if \( {\mathbf{Y}}^{\left( 1\right) } \) and \( {\mathbf{Y}}^{\left( 2\right) } \) are independent, then \( \operatorname{Cov}\left( {{\mathbf{Y}}^{\left( 1\right) },{\mathbf{Y}}^{\left( 2\right) }}\right) = 0 \) . From Theorem 2.4.1 (iii), \( {\mathbf{\sum }}_{\mathbf{{12}}} = \mathbf{0} \) . ... | Yes |
Corollary 1.2.1 If \( \mathbf{X} \sim {N}_{n}\left( {\mathbf{0},\mathbf{I}}\right) ,\mathbf{Y} = {\mathbf{A}}_{p \times n}\mathbf{X} + {\mu }_{p \times 1},\mathbf{Z} = {\mathbf{B}}_{q \times n}\mathbf{X} + {\nu }_{q \times 1} \), then \( \mathbf{Y} \) and \( \mathbf{Z} \) are independent if and only if \( {\mathbf{{AB}... | Proof : Let\n\n\[ \mathbf{W} = \left( \begin{array}{l} \mathbf{Y} \\ \mathbf{Z} \end{array}\right) = \left( \begin{array}{l} \mathbf{A} \\ \mathbf{B} \end{array}\right) \mathbf{X} + \left( \begin{array}{l} \mu \\ \nu \end{array}\right) \]\n\nBy the definition of \( \mathbf{Y} \) and \( \mathbf{Z} \) ,\n\n\[ \mathbf{W} ... | Yes |
Theorem 1.2.7. If \( \mathbf{X} \sim N\left( {\mu ,\mathbf{\sum }}\right) ,\mathbf{\sum } > 0 \), for the partition of \( \mathbf{X} \) given in (1.2.6). The conditional distribution of \( {\mathbf{X}}^{\left( 1\right) } \) given \( {\mathbf{X}}^{\left( 2\right) } \) is q-variate normal with the conditional mean and va... | \[ E\left( {{\mathbf{X}}^{\left( \mathbf{1}\right) } \mid {\mathbf{X}}^{\left( \mathbf{2}\right) } = {\mathbf{x}}^{\left( \mathbf{2}\right) }}\right) = {\mu }^{\left( \mathbf{1}\right) } + {\mathbf{\sum }}_{12}{\mathbf{\sum }}_{22}^{-1}\left( {{\mathbf{x}}^{\left( \mathbf{2}\right) } - {\mu }^{\left( \mathbf{2}\right) ... | Yes |
Theorem 1.2.8. \( {\mathbf{X}}^{\left( \mathbf{1} \cdot \mathbf{2}\right) } \) is uncorrelated with \( {\mathbf{X}}^{\left( \mathbf{2}\right) } \) . | Proof: \( \operatorname{Cov}\left( {{\mathbf{X}}^{\left( \mathbf{1} \cdot \mathbf{2}\right) },{\mathbf{X}}^{\left( \mathbf{2}\right) }}\right) = \operatorname{Cov}\left( {{\mathbf{X}}^{\left( \mathbf{1}\right) } - {\mu }^{\left( \mathbf{1}\right) } - \beta \left( {{\mathbf{X}}^{\left( \mathbf{2}\right) } - {\mu }^{\lef... | Yes |
For any \( \alpha \in {\mathbb{R}}^{p - q} \), Var \( \left( {X}_{i}^{\left( 1 : 2\right) }\right) \leq \operatorname{Var}\left( {{X}_{i} - {\alpha }^{T}{\mathbf{X}}^{\left( \mathbf{2}\right) }}\right) \) | Proof: Clearly \( E\left( {X}_{i}^{\left( 1 \cdot 2\right) }\right) = 0, E\left( {{X}_{i} - {\alpha }^{T}{\mathbf{X}}^{\left( \mathbf{2}\right) }}\right) = {\mu }_{i} - {\alpha }^{T}{\mu }^{\left( \mathbf{2}\right) } \) . Hence,\n\n\[ \operatorname{Var}\left( {{X}_{i} - {\alpha }^{T}{\mathbf{X}}^{\left( \mathbf{2}\righ... | Yes |
Theorem 1.2.10. For any \( \alpha \in {\mathbb{R}}^{p - q} \) , \( \operatorname{Corr}\left( {{X}_{i},{\beta }_{i}^{T}{\mathbf{X}}^{\left( \mathbf{2}\right) }}\right) \geq \operatorname{Corr}\left( {{X}_{i},{\alpha }^{T}{\mathbf{X}}^{\left( \mathbf{2}\right) }}\right) \) | Proof: For any constant \( c \in \mathbb{R},\operatorname{Corr}\left( {{X}_{i},{\alpha }^{T}{\mathbf{X}}^{\left( \mathbf{2}\right) }}\right) = \operatorname{Corr}\left( {{X}_{i}, c{\alpha }^{T}{\mathbf{X}}^{\left( \mathbf{2}\right) }}\right) \) . i.e. the correlation coefficient is unaffected by a change of unit. Also,... | Yes |
Lemma 1.2.2 Let \( f\left( {\cdot , \cdot }\right) \) be a bivariate function, \( \mathbf{a} = {\left( {a}_{1},\ldots ,{a}_{p}\right) }^{T} \neq \mathbf{0} \) . Then,\n\n\[ \n{\int }_{{\mathbb{R}}^{p}}f\left( {\mathop{\sum }\limits_{{i = 1}}^{p}{a}_{i}{x}_{i},\mathop{\sum }\limits_{{i = 1}}^{p}{x}_{i}^{2}}\right) d{x}_... | Proof :\n\nLet \( {\gamma }_{1} = {\left( \frac{{a}_{1}}{\parallel a\parallel },\ldots ,\frac{{a}_{p}}{\parallel a\parallel }\right) }^{T} \neq \mathbf{0} \), so that \( {\gamma }_{1}^{T}{\gamma }_{1} = 1 \) . By Gram-Schmidt orthogonalization procedure, we can construct \( {\gamma }_{2},\ldots ,{\gamma }_{p} \) such t... | Yes |
Theorem 1.2.12. If \( \phi \in {\Phi }_{p} \) iff \( \phi \left( t\right) = {\int }_{0}^{\infty }{\Omega }_{p}\left( {t{r}^{2}}\right) {dF}\left( r\right) \) where \( F\left( \cdot \right) \) is a cumulative distribution function (cdf) over \( \lbrack 0,\infty ) \) . | Proof : Let us first establish \ | No |
Corollary 1.2.5 \( \mathbf{X} \sim {\mathbf{S}}_{p}\left( \phi \right) \) if and only if \( \mathbf{X}\overset{d}{ = }\mathbf{\Gamma }\mathbf{X} \) for any orthogonal matrix \( \mathbf{\Gamma } \) . | Proof: \ | No |
Theorem 1.2.13. If \( \mathbf{X} \triangleq R{\mathbf{U}}^{\left( p\right) } \sim {\mathbf{S}}_{p}\left( \phi \right) \) for a \( \phi \in {\Phi }_{p} \) where \( R \) and \( {\mathbf{U}}^{\left( p\right) } \) are independent, and \( P\left( {\mathbf{X} = \mathbf{0}}\right) = 0 \), then \( \parallel \mathbf{X}\parallel... | Proof : As \( P\left( {\mathbf{X} = \mathbf{0}}\right) = 0, P\left( {\parallel \mathbf{X}\parallel = 0}\right) = 0 \) . Choose \( {f}_{1}\left( \mathbf{X}\right) = \sqrt{{\mathbf{X}}^{\mathbf{T}}\mathbf{X}} \) and \( {f}_{2}\left( \mathbf{X}\right) = \frac{X}{\parallel \mathbf{X}\parallel } \) . As \( \mathbf{X}\overse... | Yes |
Corollary 1.2.6 \( E{\mathbf{U}}^{\left( p\right) } = \mathbf{0} \) and \( \operatorname{Var}\left( {\mathbf{U}}^{\left( p\right) }\right) = \frac{1}{p}{\mathbf{I}}_{p} \) . | Proof: Let \( \mathbf{X}\overset{d}{ \sim }N\left( {\mathbf{0},{\mathbf{I}}_{p}}\right) \) . Corollary 1.2.4 says that \( \mathbf{X}\overset{d}{ = }R{\mathbf{U}}^{\left( p\right) } \), and \( R \) and \( {\mathbf{U}}^{\left( p\right) } \) are independent. Hence\n\n\[ \mathbf{0} = E\mathbf{X} = {ER} \cdot E{\mathbf{U}}^... | Yes |
Theorem 1.2.14. Suppose \( \mathbf{X}\overset{d}{ = }R{\mathbf{U}}^{\left( p\right) }\overset{d}{ = }{\mathbf{S}}_{p}\left( \phi \right) \) . Then, \( \mathbf{X} \) has a pdff \( \left( \cdot \right) \) if and only if \( R \) has a density \( g\left( \cdot \right) \) and\n\n\[ g\left( r\right) = \frac{2{\pi }^{\frac{p}... | Proof: \ | No |
Theorem 1.2.15. A univariate function \( \phi \left( \cdot \right) \) can be used to define an elliptically contoured (EC) distribution, \( E{C}_{p}\left( {\mu ,\mathbf{\Lambda },\phi }\right) \) for every \( \mu \in {\mathbb{R}}^{p} \) and \( \mathbf{\Lambda } \geq 0 \) with \( \operatorname{rank}\left( \mathbf{\Lambd... | Proof : To see the \ | No |
Corollary 1.2.7 \( \mathbf{Y}\overset{d}{ \sim }E{C}_{p}\left( {\mu ,\mathbf{\Lambda },\phi }\right) \), with \( \operatorname{rank}\left( \mathbf{\Lambda }\right) = k \) if and only if \( \mathbf{Y}\overset{d}{ = }\mu + \) \( R{\mathbf{A}}^{T}{U}^{\left( k\right) } \) where \( R \) and \( {U}^{\left( k\right) } \) are... | Proof: For \ | No |
Corollary 1.2.16. If \( \mathbf{Y}\overset{d}{ \sim }E{C}_{p}\left( {\mu ,\mathbf{\Lambda },\phi }\right) \), then \( {\mathbf{r}}_{m \times 1} + {\mathbf{B}}_{m \times n}\mathbf{Y}\overset{d}{ \sim }E{C}_{m}\left( {r + \mathbf{B}\mu ,\mathbf{B}\mathbf{\Lambda }{\mathbf{B}}^{\mathbf{T}},\phi }\right) \) . | Hence the EC family is closed under the linear transformation, which is like the Gaussian family shown in Lemma 1.2.1. | No |
Theorem 1.2.17. Suppose \( \mathbf{Y}\overset{d}{ \sim }E{C}_{p}\left( {\mu ,\Lambda ,\phi }\right) \), rank \( \left( \Lambda \right) = k > 0 \) . Then, any marginal of \( \mathbf{Y} \) is normally distributed if and only if \( \mathbf{Y}\overset{d}{ \sim }{N}_{p}\left( {\mu ,\Lambda }\right) \) . | Proof : Only prove the necessary part as the sufficient part is obvious. Suppose \( \mathbf{Y} = {\left( {\mathbf{Y}}_{p}^{\left( 1\right) },{\mathbf{Y}}_{n - p}^{\left( 2\right) }\right) }^{T},{\mathbf{Y}}^{\left( 1\right) } \sim {N}_{p}\left( {{\mu }^{\left( 1\right) },{\Lambda }_{11}}\right) \) and \( \operatorname{... | Yes |
Theorem 1.2.18. Let \( \mathbf{Y} = {\left( {Y}_{1},\ldots ,{Y}_{p}\right) }^{T}\overset{d}{ \sim }E{C}_{p}\left( {\mu ,\Lambda ,\phi }\right) \) and \( \Lambda = \operatorname{diag}\left( {{\gamma }_{11},\ldots ,{\gamma }_{nn}}\right) \) with \( {\gamma }_{ii} > 0 \) . Then the followings are equivalent:\n\n(a) \( \ma... | Proof: Clearly (a) implies (b), and (b) implies (c) under the assumed form of \( \Lambda = \) \( \operatorname{diag}\left( {{\gamma }_{11},\ldots ,{\gamma }_{nn}}\right) \) . Let us prove that (c) implies (a).\n\nSuppose \( {Y}_{i} \) and \( {Y}_{j} \) are independent for \( 1 \leq i < j \leq n \) . From Corollary 1.2.... | Yes |
Theorem 2.1.1. Let \( \left( {{X}_{1},{Y}_{1}}\right) ,\cdots ,\left( {{X}_{n},{Y}_{n}}\right) \) be i.i.d. samples from a bivariate normal distribution \( \left( {X, Y}\right) \sim N\left( {{\mu }_{1},{\mu }_{2},{\sigma }_{1}^{2},{\sigma }_{2}^{2},\rho }\right) \) where \( \rho \) is correlation coefficient. The sampl... | Proof: It is derived by the delta method under the normality assumption, this is a homework to readers. | No |
Corollary 2.1.1 For large \( n \), the asymptotical behaviour of \( Z \) statistic sequence \( \left\{ {Z}_{n}\right\} \) is\n\n\[ \sqrt{n}\left( {\operatorname{arctanh}{r}_{n} - \operatorname{arctanh}\rho }\right) \overset{d}{ \rightarrow }N\left( {0,1}\right) \] | Moreover, the standard error of \( {r}_{n} \) is approximately \( 1/\sqrt{n - 3} \), see ?,? for detailed proof. | No |
The height and the weight of a randomly chosen Amsterdam woman is denoted by \( X \) and \( Y \) respectively. We make a presumption that \( \left( {X, Y}\right) \) has (approximately) a bivariate normal distribution with correlation coefficient \( \rho \) . At a level of significance of \( 5\% \) one wants to test\n\n... | To this end a sample of size 40 is drawn from this population. After the experimen- \( \mathrm{t} \) is completed the statistic \( {r}_{n} = 0 : {690} \) which give the value of Fisher’s \( \mathrm{Z} \) statistic: \( Z = \frac{1}{2}\log \left( \frac{1 + {0.69}}{1 - {0.69}}\right) = {0.848} \) . By Corollary, the \( \m... | Yes |
Theorem 2.2.1. Let \( C \) be a p-dimensional copula. Then, for all \( u, v \in {\left\lbrack 0,1\right\rbrack }^{p} \) , \[ \left| {C\left( u\right) - C\left( v\right) }\right| \leq \mathop{\sum }\limits_{{k = 1}}^{p}\left| {{u}_{k} - {v}_{k}}\right| \] and hence, \( C \) is uniformly continuous on \( {\left\lbrack 0,... | Proof: Apply Lemma 2.2.1. | No |
Corollary 2.2.1 All the \( {k}^{\text{th }} \) and diagonal sections of \( C \) are non-decreasing and uniformly continuous on \( I \) . | Proof : Readily from Lemma ?? and Theorem 2.2.1. | No |
Lemma 2.2.2 Let \( C \) be a copula and \( u = \left( {{u}_{1},\ldots ,{u}_{p}}\right) \in {I}^{p} \) . Then,\ni) \( \frac{\partial C\left( u\right) }{\partial {u}_{k}} \) exists for almost all \( {u}_{k} \in I \) and \( 0 \leq \frac{\partial C\left( u\right) }{\partial {u}_{k}} \leq 1 \) .\nii) The function \( {u}_{\l... | Proof : i) As all the \( k \) -sections of \( C \) is non-decreasing and continuous, it is differential almost everywhere (a.e.) with respect to the Lebesgue measure, that is, \( \frac{\partial C\left( u\right) }{\partial {u}_{k}} \) exists a.e.. For any \( 0 \leq {t}_{1} \leq {t}_{2} \leq 1 \), as \( C \) is \( k \) -... | No |
Theorem 2.3.1. (Sklar 1959, Schweiser and Sklar 1974)\n\nLet \( F \) be a p-dimensional distribution function with margins \( {F}_{1},\ldots ,{F}_{p} \) . Then, there exists a p-dimensional copula \( C \) such that for all \( x \in {\overline{\mathbb{R}}}^{p} \) ,\n\n\[ F\left( {{x}_{1},\ldots ,{x}_{p}}\right) = C\left... | Proof : As \( F \) is a distribution function, it satisfies the conditions of Lemma 2.2.1. So, for all \( x = \left( {{x}_{1},\ldots ,{x}_{p}}\right) \) and \( y = \left( {{y}_{1},\ldots ,{y}_{p}}\right) \in {\mathbb{R}}^{p} \) ,\n\n\[ \left| {F\left( x\right) - F\left( y\right) }\right| \leq \mathop{\sum }\limits_{{k ... | No |
Corollary 2.3.1 Let \( F \) be a p-dimensional distribution with continuous margins \( {F}_{i} \) and copula \( C \) . Then, for all \( u = \left( {{u}_{1},\ldots ,{u}_{p}}\right) \in {\left\lbrack 0,1\right\rbrack }^{p} \) , \[ C\left( u\right) = F\left( {{F}_{1}^{-1}\left( {u}_{1}\right) ,\ldots ,{F}_{p}^{-1}\left( {... | Proof : As \( {F}_{i} \) is continuous, \( C \) is unique and \( \operatorname{Ran}{F}_{i} = I \) . Hence, for all \( {u}_{i} \in I \), let \( {x}_{i} = {F}_{i}^{-1}\left( {u}_{i}\right) \) . Then, Sklar Theorem implies \[ F\left( {{x}_{1},\ldots ,{x}_{p}}\right) = F\left( {{F}_{1}^{-1}\left( {u}_{1}\right) ,\ldots ,{F... | Yes |
Lemma 2.3.1 For any p-dimensional sub-copula \( {C}^{\prime } \), any \( u \in \operatorname{Dom}{C}^{\prime } \) , \[ {W}_{p}\left( u\right) \leq {C}^{\prime }\left( u\right) \leq {M}_{p}\left( u\right) \] | Proof : For any \( u = \left( {{u}_{1},\ldots ,{u}_{p}}\right) \in \operatorname{Dom}{C}^{\prime } \), and \( k = 1,\ldots, p \) , \[ {C}^{\prime }\left( {{u}_{1},\ldots ,{u}_{p}}\right) \leq {C}^{\prime }\left( {1,\ldots ,1,{u}_{k},1,\ldots ,1}\right) = {u}_{k}. \] Thus, \( {C}^{\prime }\left( {{u}_{1},\ldots ,{u}_{p}... | Yes |
Theorem 2.3.2. For any \( p \geq 3 \) and fixed \( u \in {I}^{p} \), there is a copula \( {C}_{u} \) which depends on \( u \) such that \( {C}_{u}\left( u\right) = {W}_{p}\left( u\right) \) . | Proof : See Nelson (1998). | No |
Theorem 2.4.4. Let \( \left( {{X}_{1},\ldots ,{X}_{p}}\right) \) be a continuous random vector with copula \( {C}_{{X}_{1},\ldots ,{X}_{p}} \) and \( {\alpha }_{1},\ldots ,{\alpha }_{p} \) be strictly monotone on \( \operatorname{Ran}\left( {X}_{1}\right) ,\ldots ,\operatorname{Ran}\left( {X}_{p}\right) \), respectivel... | When at least one of random variables \( {\alpha }_{1}\left( {X}_{1}\right) ,{\alpha }_{2}\left( {X}_{2}\right) \) is strictly decreasing, we enjoy the fact that the \( {C}_{{\alpha }_{1}\left( {X}_{1}\right) ,{\alpha }_{2}\left( {X}_{2}\right) }\left( {{u}_{1},{u}_{2}}\right) \) is a simple transformation of \( {C}_{{... | Yes |
Theorem 2.5.1. Let \( \\left( {X, Y}\\right) \) and \( \\left( {\\widetilde{X},\\widetilde{Y}}\\right) \) be independent continuous random vectors with joint distributions \( H \) and \( \\widetilde{H} \), and copulae \( C \) and \( \\widetilde{C} \), respectively, and \( X \) and \( \\widetilde{X}, Y \) and \( \\widet... | Proof : Note that\n\n\[ \nQ = {2P}\\{\\left( {X - \\widetilde{X}}\\right) \\left( {Y - \\widetilde{Y}}\\right) > 0\\} - 1 \n\] \n\n\[ \n= 2\\left\\{ {P\\left( {X > \\widetilde{X}, Y > \\widetilde{Y}}\\right) + P\\left( {X < \\widetilde{X}, Y < \\widetilde{Y}}\\right) }\\right\\} - 1. \n\] \n\n1510 By conditioning and t... | Yes |
Corollary 2.5.1 If \( F \) has copula \( C \), then\n\n\[ \n{\tau }_{F} = Q\left( {C, C}\right) = 4\iint C\left( {u, v}\right) {dC}\left( {u, v}\right) - 1 = {4E}\left\lbrack {C\left( {U, V}\right) }\right\rbrack - 1 \]\n\nwhere \( \left( {U, V}\right) \sim C \) . | Example: Consider a copula \( {C}_{\theta }\left( {u, v}\right) = {uv} + {\theta uv}\left( {1 - u}\right) \left( {1 - v}\right) \) where \( \theta \in ( - 1,1\rbrack \) . Then,\n\n\[ \nd{C}_{\theta }\left( {u, v}\right) = \frac{{\partial }^{2}{C}_{\theta }\left( {u, v}\right) }{\partial u\partial v}{dudv} = \left( {1 +... | Yes |
Lemma 3.1.1 Suppose the \( p \times p \) symmetric matrix \( \mathbf{C} = {\mathbf{C}}^{T} \) has a spectral decomposition:\n\n\[ \mathbf{C} = \mathop{\sum }\limits_{{j = 1}}^{p}{\alpha }_{j}{\gamma }_{j}{\gamma }_{j}^{T} = \mathbf{\Gamma }\left( \begin{array}{lll} {\alpha }_{1} & & \\ & \ddots & \\ & & {\alpha }_{p} \... | Proof: Let \( \mathbf{Y} = {\mathbf{\Gamma }}^{T}\mathbf{X} \) . Then, \( \mathbf{Y}\overset{d}{ \sim }N\left( {{\mathbf{\Gamma }}^{T}\mathbf{\mu },{\mathbf{I}}_{p}}\right) = N\left( {\mathbf{\lambda },{\mathbf{I}}_{p}}\right) \) where \( \mathbf{\lambda } = {\left( {\lambda }_{1},\ldots ,{\lambda }_{p}\right) }^{T} = ... | Yes |
Theorem 3.1.1. Suppose \( {\mathbf{C}}^{T} = \mathbf{C} \) and \( \mathbf{X} \sim N\left( {\mathbf{\mu },{\mathbf{I}}_{p}}\right) \) . Then, \( {\mathbf{X}}^{T}\mathbf{C}\mathbf{X} \sim {\chi }_{r}^{2}\left( {{\mathbf{\mu }}^{T}\mathbf{C}\mathbf{\mu }}\right) \) iff \( {\mathbf{C}}^{2} = \mathbf{C} \) and \( \operatorn... | Proof: \ | No |
Corollary 3.1.2. Suppose that \( {\mathbf{C}}^{T} = \mathbf{C} \) and \( \mathbf{X} \sim {N}_{p}\left( {\mathbf{\mu },{\mathbf{I}}_{p}}\right) \) . Then, \( {\mathbf{X}}^{T}\mathbf{C}\mathbf{X} \sim {\chi }_{p}^{2} \) iff \( {\mathbf{C}}^{2} = \mathbf{C},\operatorname{rank}\left( \mathbf{C}\right) = p \) and \( \mathbf... | Proof: \ | No |
Theorem 3.1.3. Let \( {\mathbf{C}}_{i}^{T} = {\mathbf{C}}_{i}, i = 1,2,\cdots, k \) and \( \mathbf{X} \sim {N}_{p}\left( {\mathbf{\mu },{\mathbf{I}}_{p}}\right) \) . Then, \( {\left\{ {\mathbf{X}}^{T}{\mathbf{C}}_{j}\mathbf{X}\right\} }_{j = 1}^{k} \) are independent if and only if \( {\mathbf{C}}_{i}{\mathbf{C}}_{j} =... | Proof: \ | No |
Theorem 3.1.5. Let \( {\mathbf{C}}^{T} = \mathbf{C} \) and \( \mathbf{X} \sim {N}_{p}\left( {\mathbf{\mu },\mathbf{\sum }}\right) \) with \( > 0 \) . Then, \( {\mathbf{X}}^{T}\mathbf{C}\mathbf{X} \sim {\chi }_{r}^{2}\left( {{\mu }^{T}\mathbf{C}\mathbf{\mu }}\right) \) if and only if \( {\left( \mathbf{C}\mathbf{\sum }\... | Proof: Let \( \mathbf{Y} \sim {N}_{p}\left( {{\mathbf{\sum }}^{-1/2}\mathbf{\mu },{\mathbf{I}}_{p}}\right) \) . Then by Theorem 3.1.1, \( {\mathbf{X}}^{T}\mathbf{C}\mathbf{X} = {\mathbf{Y}}^{T}{\mathbf{\sum }}^{1/2}\mathbf{C}{\mathbf{\sum }}^{1/2}\mathbf{Y} \) . | No |
Theorem 3.2.1. Under the above assumptions on the linear model, let the rank of \( \\mathbf{X} \) be \( p \), then\n\n(i) Let \( \\mathbf{A} \) be a \( p \\times n \) matrix, then \( \\mathrm{E}\\parallel \\mathbf{A}\\mathbf{\\varepsilon }{\\parallel }^{2} = \\mathrm{E}\\left( {{\\mathbf{\\varepsilon }}^{T}{\\mathbf{A}... | Proof: (i) As \( \\parallel {A\\varepsilon }\\parallel = {\\varepsilon }^{T}{\\mathbf{A}}^{T}\\mathbf{A}\\varepsilon \) is scalar, we have\n\n\[ {\\varepsilon }^{T}{\\mathbf{A}}^{T}\\mathbf{A}\\varepsilon = \\operatorname{tr}\\left( {{\\varepsilon }^{T}{\\mathbf{A}}^{T}\\mathbf{A}\\varepsilon }\\right) = \\operatorname... | Yes |
Theorem 3.2.2. Let \( \mathbf{Y} \sim {N}_{n}\left( {\mathbf{X}\beta ,{\sigma }^{2}{\mathbf{I}}_{n}}\right) \) with a \( n \times m \) design matrix \( \mathbf{X},\operatorname{rank}\left( \mathbf{X}\right) = p \) and \( {\beta }_{m \times 1} \) is unknown. Then,\n\n\[ \n{R}_{0}^{2} = \mathop{\min }\limits_{\beta }{\le... | Proof: Let \( {Y}_{i} = {\mathbf{X}}_{i}^{\mathbf{T}}\beta + {\epsilon }_{i} \) for \( i = 1,\ldots, n \) with \( {\epsilon }_{i}\overset{i.i.d.}{ \sim }N\left( {0,{\sigma }^{2}}\right) \) . Then, \( {R}_{0}^{2} \) is the least square in the residuals and \( {R}_{0}^{2} = \sum {\widehat{\epsilon }}_{i}^{2} \) where \( ... | No |
Theorem 3.2.3. Under the same assumptions as Theorem 3.2.2, let \( {\mathbf{H}}_{p \times k} \) have a rank \( k, L\left( \mathbf{H}\right) \subset L\left( {\mathbf{X}}^{\mathbf{T}}\right) \), and \( {R}_{1}^{2} = \mathop{\min }\limits_{{{\mathbf{H}}^{\mathbf{T}}\beta = \xi }}{\left( \mathbf{Y} - \mathbf{X}\beta \right... | Proof: The solution of \( {\mathbf{H}}^{\mathbf{T}}\beta = \xi \) has a general form of \( \beta = {\beta }_{0} + \gamma \) where \( {\beta }_{0} \) is a specific solution of \( {\mathbf{H}}^{\mathbf{T}}\beta = \xi \) and \( \gamma \) is a solution of \( {\mathbf{H}}^{\mathbf{T}}\gamma = \mathbf{0} \) . A general form ... | Yes |
Theorem 3.3.2. If \( \mathbf{A} \sim {\mathbf{W}}_{p}\left( {n,\mathbf{\sum },\tau }\right) \), then \( {\ell }^{T}\mathbf{A}\ell \sim {\sigma }_{\ell }^{2}{\chi }_{n}^{2}\left( \lambda \right) \) for a constant vector \( \ell \in {\mathbb{R}}^{p} \) where \( {\sigma }_{\ell }^{2} = {\ell }^{\mathbf{T}}\sum \ell \) and... | Proof :\n\n\[ \mathbf{A} = \mathop{\sum }\limits_{{i = 1}}^{n}{\mathbf{X}}_{i}{\mathbf{X}}_{i}^{T} = {\mathbf{X}}^{\mathbf{T}}\mathbf{X} \]\n\n\[ {\ell }^{\mathbf{T}}\mathbf{A}\ell = {\ell }^{\mathbf{T}}{\mathbf{X}}^{\mathbf{T}}\mathbf{X}\ell = {\left( \mathbf{X}\ell \right) }^{\mathbf{T}}\left( {\mathbf{X}\ell }\right... | No |
Theorem 3.3.3. Let \( {\mathbf{C}}^{\mathbf{T}} = \mathbf{C} \) . Then, \( {\mathbf{X}}^{T}\mathbf{C}\mathbf{X} \sim {\mathbf{W}}_{p}\left( {r,\mathbf{\sum },\delta }\right) \) if and only if for all \( \ell \in {\mathbb{R}}^{p} \) such that \( {\mathbf{\ell }}^{T}{\mathbf{X}}^{T}\mathbf{{CX}}\mathbf{\ell } \sim {\sigm... | Proof: \ | No |
Theorem 3.3.4. If \( {\mathbf{A}}_{1} \sim {\mathbf{W}}_{p}\left( {{n}_{1},\mathbf{\sum },{\tau }_{1}}\right) \) and \( {\mathbf{A}}_{2} \sim {\mathbf{W}}_{p}\left( {{n}_{2},\mathbf{\sum },{\tau }_{2}}\right) \), and \( {\mathbf{A}}_{1} \) and \( {\mathbf{A}}_{2} \) are independent, then \( {\mathbf{A}}_{1} + {\mathbf{... | Proof: HW | No |
Theorem 3.3.5. Suppose that \( \mathbf{A} = \left( \begin{array}{ll} {\mathbf{A}}_{11} & {\mathbf{A}}_{12} \\ {\mathbf{A}}_{21} & {\mathbf{A}}_{22} \end{array}\right) \sim {\mathbf{W}}_{p}\left( {n,\mathbf{\sum },\tau }\right) \) where \( {\mathbf{A}}_{11} \) is a \( q \times q \) matrix and \( {\mathbf{A}}_{22} \) is ... | Proof: HW | No |
Corollary 3.3.8. Suppose that \( {\mathbf{C}}_{1} \) and \( {\mathbf{C}}_{2} \) are both projection matrix (i.e. \( {\mathbf{C}}_{i}^{2} = {\mathbf{C}}_{i} \) ). Then, \( {\mathbf{X}}^{\mathbf{T}}{\mathbf{C}}_{1}\mathbf{X} \) and \( {\mathbf{X}}^{\mathbf{T}}{\mathbf{C}}_{2}\mathbf{X} \) are independent if and only if \... | Proof: HW | No |
Theorem 3.3.10. (Basu’s Theorem, see Proposition 2.1 of Shao (2003)). If \( \mathbf{T} \) is boundedly complete sufficient for \( \mathcal{P} \), and \( \mathbf{U} \) is ancillary, then \( \mathbf{T} \) and \( \mathbf{U} \) are independently distributed on every \( \mathbf{\theta } \in \Theta \) . | By Basu’s Theorem, the proof of Corollary 3.3.9 is obvious, since \( \overline{\mathbf{X}} \) is complete and \( {\mathbf{S}}_{n} \) is ancillary for every fixed \( \mathbf{\sum } \) . Equivalently, \( \mathbf{S} = \overline{\mathbf{X}} \) and \( \mathbf{U} = {\mathbf{S}}_{n} \) are independently distributed for all \(... | No |
\[ \frac{1}{n}\mathrm{E}{\begin{Vmatrix}\widetilde{\mathbf{X}}\left( \widehat{\mathbf{\beta }} - {\mathbf{\beta }}^{ * }\right) \end{Vmatrix}}^{2} \geq \frac{p{\sigma }^{2}}{n} = \frac{1}{n}\mathrm{E}{\begin{Vmatrix}\mathbf{X}\left( \widehat{\mathbf{\beta }} - {\mathbf{\beta }}^{ * }\right) \end{Vmatrix}}^{2} = \text{ ... | Proof: Using the Matrix Jensen’s inequality that \( E{\left( {\mathbf{X}}^{T}\mathbf{X}\right) }^{-1} - {\left( E{\mathbf{X}}^{T}\mathbf{X}\right) }^{-1} \) is positive semi-definite in ?. From the property of trace operator, we have \( \widehat{\beta } - {\beta }^{ * } = {\left( {\mathbf{X}}^{T}\mathbf{X}\right) }^{-1... | Yes |
Theorem 3.3.12.\n\n\[ \frac{1}{n}\mathrm{E}{\begin{Vmatrix}\widetilde{\mathbf{X}}\left( \widehat{\mathbf{\beta }} - {\mathbf{\beta }}^{ * }\right) \end{Vmatrix}}^{2} = \frac{p{\sigma }^{2}}{n - p - 1}. \] | Proof: Follow the Lemma 3.3.11, we have\n\n\[ \frac{1}{n}\mathrm{E}\left\lbrack {\widetilde{\mathbf{X}}{\left( \widehat{\mathbf{\beta }} - {\mathbf{\beta }}^{ * }\right) }^{2}}\right\rbrack = \frac{{\sigma }^{2}}{n}\operatorname{tr}\left( {\mathrm{E}{\left( {\mathbf{X}}^{T}\mathbf{X}\right) }^{-1}\mathrm{E}{\widetilde{... | Yes |
Lemma 4.1.1. (Chernoff bound): \( P\left( {X \geq a}\right) \leq \mathop{\min }\limits_{{t > 0}}\left\{ {{e}^{-{ta}} \cdot E{e}^{tX}}\right\} \) . | Proof: By Markov inequality, we have\n\n\[ P\left( {X \geq a}\right) \leq \frac{E{e}^{tx}}{{e}^{ta}}\text{ for }t > 0, \]\n\nand then optimize \( t \) over \( t > 0 \) . | Yes |
Lemma 4.1.2. (Hoeffding lemma) Let \( {X}_{1},\cdots ,{X}_{n} \) be independent centralized random variables on \( R \) satisfying bound condition\n\n\[ \mathrm{E}{X}_{i} = 0,\left| {X}_{i}\right| \leq {c}_{i}\text{for}i = 1,2,\cdots, n\text{.} \]\n\n(4.1.1)\n\nThen\n\n\[ \operatorname{E}\exp \left\{ {\lambda \mathop{\... | Proof: By Jensen’s inequality for \( f\left( x\right) = {e}^{x} \), due to the convex combination of two point for any \( 0 \leq \alpha \leq 1 \), we have\n\n\[ {e}^{{\alpha \lambda z} + \left( {1 - \alpha }\right) {\lambda y}} \leq \alpha {e}^{\lambda z} + \left( {1 - \alpha }\right) {e}^{\lambda y} \]\n\nNext, we wan... | Yes |
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