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Lemma 1.2.6. Let \( \\left\\{ {{X}_{n}, n \\in \\mathbb{Z}}\\right\\} \) be an \( \\left( {\\alpha ,\\beta }\\right) \\) -mixing sequence, \( X \\in \) \( {L}_{p}\\left( {\\mathcal{F}}_{-\\infty }^{k}\\right) \) and \( Y \\in {L}_{q}\\left( {\\mathcal{F}}_{k + n}^{\\infty }\\right) \) with \( p, q \\geq 1 \) and \( 1/p... | Proof. Without loss of generality, assume that \( {\\alpha p} \\geq 1 \), which implies that \( {\\beta q} \\leq 1 \) . Put\n\n\[ \n{Y}_{1} = {YI}\\left( {\\left| Y\\right| \\leq C}\\right) ,\\;{Y}_{2} = Y - {Y}_{1}\n\]\n\nwhere \( C \) is a positive constant specified later on. Write\n\n\[ \n\\left| {{EXY} - {EXEY}}\\... | Yes |
Lemma 1.2.10. Let \( \\left\\{ {{X}_{n}, n \\in \\mathbb{Z}}\\right\\} \) be a \( \\varphi \) -mixing sequence, \( X \\in {\\mathcal{F}}_{-\\infty }^{k} \) and \( Y \\in {\\mathcal{F}}_{k + n}^{\\infty } \) with \( E\\left| X\\right| < \\infty \) and \( \\left| Y\\right| \\leq C \) . Then | \[ \\left| {{EXY} - {EXEY}}\\right| \\leq {2C\\varphi }\\left( n\\right) E\\left| X\\right| . \] | Yes |
Lemma 1.2.11. Let \( \\left\\{ {{X}_{n}, n \\in \\mathbb{Z}}\\right\\} \) be a \( \\psi \) -mixing sequence, \( X \\in {\\mathcal{F}}_{-\\infty }^{k} \) and \( Y \\in {\\mathcal{F}}_{k + n}^{\\infty } \) with \( E\\left| X\\right| < \\infty \) and \( E\\left| Y\\right| < \\infty \) . Then \( E\\left| {XY}\\right| < \\i... | Proof. At first, we assume that \( X \) and \( Y \) are non-negative simple functions. We have\n\n\[\\left| {{EXY} - {EXEY}}\\right| = \\left| {\\mathop{\\sum }\\limits_{{i, j}}{a}_{i}{b}_{j}\\left( {P\\left( {{A}_{i}{B}_{j}}\\right) - P\\left( {A}_{i}\\right) P\\left( {B}_{j}\\right) }\\right) }\\right|\]\n\n\[\\leq \... | Yes |
Theorem 2.1.1.\n\n\[ \operatorname{Var}{S}_{n} = \mathop{\sum }\limits_{{\left| j\right| < n}}\left( {n - \left| j\right| }\right) R\left( j\right) \] | The proof of the theorem can be found in the book of Ibragimov and Linnik (1971) and is not presented here. | No |
Theorem 2.1.3. Let \( \left\{ {X}_{n}\right\} \) be a strictly stationary \( \alpha \) -mixing sequence satisfying that \( E{X}_{1} = 0, E{X}_{1}^{2} < \infty ,{\sigma }_{n}^{2} = E{S}_{n}^{2} \rightarrow \infty \) and \( \left\{ {{S}_{n}^{2}/{\sigma }_{n}^{2}, n \geq 1}\right\} \) is integrable uniformly. Then the con... | Proof. By the proof of Theorem 2.1.2 and Remark 2.1.1, it suffices to show the following facts.\n\n1. \( {\sigma }_{n}^{2} \rightarrow \infty \) ;\n\n2. for any \( \varepsilon > 0 \), there exist \( p = p\left( \varepsilon \right), N = N\left( \varepsilon \right) \) such that\n\n\[ \left| {E{S}_{n}\mathop{\sum }\limits... | Yes |
Lemma 2.2.1. Let \( \\left\\{ {{X}_{n}, n \\geq 1}\\right\\} \) be an \( \\alpha \) -mixing sequence. For any given integers \( p, q \) and \( k \), let \( {\\xi }_{j} \) be \( {\\mathcal{F}}_{\\left( {j - 1}\\right) \\left( {p + q}\\right) + 1}^{{jp} + \\left( {j - 1}\\right) q} \) measurable, \( j = 1,2,\\cdots, k \)... | Proof. Let events\n\n\[ \nA = \\left\\{ {\\mathop{\\max }\\limits_{{1 \\leq l \\leq k}}\\left| {{\\xi }_{1} + \\cdots + {\\xi }_{l}}\\right| > {2C}}\\right\\} ,\\;B = \\left\\{ {\\left| {{\\xi }_{1} + \\cdots + {\\xi }_{k}}\\right| > C}\\right\\} , \n\]\n\n\[ \n{A}_{1} = \\left\\{ {\\left| {\\xi }_{1}\\right| > {2C}}\\... | Yes |
Lemma 2.2.7. Let \( \left\{ {{X}_{n}, n \geq 1}\right\} \) be a \( \varphi \) -mixing sequence, \( 0 < \eta < 1 \) . Suppose that there exists an integer \( p,1 \leq p \leq n \), a number \( A > 0 \) such that\n\n\[ \varphi \left( p\right) + \mathop{\max }\limits_{{p \leq i \leq n}}P\left\{ {\left| {{S}_{n} - {S}_{i}}\... | Proof. Put \( {E}_{i} = \left\{ {\mathop{\max }\limits_{{1 \leq j < i}}\left| {S}_{j}\right| < a + A + b \leq \left| {S}_{i}\right| }\right\} \) . Then\n\n\[ P\left\{ {\mathop{\max }\limits_{{1 \leq i \leq n}}\left| {S}_{i}\right| \geq a + A + b}\right\} \]\n\n\[ \leq P\left( {\left| {S}_{n}\right| \geq a}\right) + \ma... | Yes |
Lemma 2.2.8. Let \( \\left\\{ {{X}_{n}, n \\geq 1}\\right\\} \) be a \( \\varphi \) -mixing sequence with \( E{X}_{n} = 0 \) and \( \\mathop{\\sup }\\limits_{n}E{\\left| {X}_{n}\\right| }^{2 + \\delta } < \\infty \) for some \( \\delta > 0 \) . Suppose that | Proof. It is easy to see that for \( r \\geq 1 \) and \( x \\geq 0 \) | No |
Lemma 2.2.9. Let \( \\left\\{ {{X}_{n}, n \\geq 1}\\right\\} \) be a \( \\varphi \) -mixing sequence satisfying (2.2.17) and \( q > 0 \) satisfying \( \\eta {4}^{q} < 1 - \\eta \) . Then\n\n\[ E\\mathop{\\max }\\limits_{{1 \\leq i \\leq n}}{\\left| {S}_{i}\\right| }^{q} \\leq {\\left( 1 - \\eta - \\eta {4}^{q}\\right) ... | Proof. By Lemma 2.2.7, we have for \( x \\geq {8A} \)\n\n\[ P\\left\\{ {\\mathop{\\max }\\limits_{{1 \\leq i \\leq n}}\\left| {S}_{i}\\right| \\geq x}\\right\\} \]\n\n\[ \\leq \\frac{1}{1 - \\eta }\\left( {P\\left\\{ {\\left| {S}_{n}\\right| \\geq \\frac{5}{8}x}\\right\\} + P\\left\\{ {\\mathop{\\max }\\limits_{{1 \\le... | Yes |
Lemma 2.2.10. Let \( \\left\\{ {{X}_{n}, n \\geq 1}\\right\\} \) be a \( \\varphi \) -mixing sequence. Suppose that there exists an array \( \\left\\{ {c}_{kn}\\right\\} \) of positive numbers such that\n\n\[ \n\\mathop{\\max }\\limits_{{1 \\leq i \\leq n}}E{S}_{k}^{2}\\left( i\\right) \\leq {c}_{kn} \n\]\n\n\( \\left(... | Proof. Take \( \\eta = {4}^{-{2q}},{A}^{2} = 2{c}_{kn}/\\eta \) . There exists a \( {p}_{0} \) such that \( \\varphi \\left( {p}_{0}\\right) \\leq \\eta /2 \) since \( \\varphi \\left( p\\right) \\rightarrow 0 \) as \( p \\rightarrow \\infty \) . Using (2.2.23) we can verify that (2.2.17) is satisfied. Hence, we get (2... | Yes |
Theorem 3.1.2. Suppose that \( E{X}_{1} = 0, E{X}_{1}^{2} < \infty \) and \( {\sigma }_{n}^{2} \rightarrow \infty \) as \( n \rightarrow \infty \) . Then in order that \( \left\{ {X}_{n}\right\} \) obeys the CLT, it is necessary and sufficient that \( \left\{ {{S}_{n}^{2}/{\sigma }_{n}^{2}, n \geq 1}\right\} \) is inte... | Proof. Necessity. Suppose that\n\n\[ \n{S}_{n}/{\sigma }_{n}\overset{d}{ \rightarrow }N\left( {0,1}\right) \n\]\n\nwhere \( N\left( {0,1}\right) \) stands for a standard normal variable. Therefore, for any \( \varepsilon > 0 \) there exists a \( K > 0 \) such that\n\n\[ \n\mathop{\lim }\limits_{{n \rightarrow \infty }}... | Yes |
Theorem 3.1.3. Suppose that \( E{X}_{1} = 0, E{X}_{1}^{2} = 1,{\sigma }_{n}^{2} = {nh}\left( n\right) \), where \( h\left( n\right) \) is a slowly varying function. Then in order that the distributions of \( \left\{ {{S}_{n}/{\sigma }_{n}, n \geq 1}\right\} \) tend to the standard normal distribution \( \Phi \left( x\r... | In order to prove this theorem, we need the following lemma.\n\nAt first, we introduce some notations. Let integer \( p \) and real number \( g \) satisfy\n\n\[ 2 \leq g \leq {\alpha }^{-1/4}\left( {\sigma }_{p}^{1/4}\right) \land {\sigma }_{p}^{1/4} \]\n\nwhere \( \alpha \left( x\right) = \alpha \left( \left\lbrack x\... | Yes |
Lemma 3.2.1. Let \( {\xi }_{1},\cdots ,{\xi }_{n} \) be random variables. Put\n\n\[ \n\alpha = \mathop{\max }\limits_{{1 \leq k \leq n - 1}}\sup \left\{ {\left| {P\left( {AB}\right) - P\left( A\right) P\left( B\right) }\right| : A \in \sigma \left( {{\xi }_{1},\cdots ,{\xi }_{k}}\right) ,}\right.\n\]\n\n\[ \n\left. {B ... | Proof. Put\n\n\[ \n{A}_{1} = \left\{ {\left| {\xi }_{1}\right| > {2\varepsilon }}\right\} \]\n\n\[ \n{A}_{k} = \left\{ {\left| {\mathop{\sum }\limits_{{j = 1}}^{k}{\xi }_{j}}\right| > {2\varepsilon },\left| {\mathop{\sum }\limits_{{j = 1}}^{l}{\xi }_{j}}\right| \leq {2\varepsilon },1 \leq l \leq k - 1}\right\} ,\;1 < k... | Yes |
Lemma 3.2.2. Let \( \\left\\{ {{X}_{n}, n \\geq 1}\\right\\} \) be as in Theorem 3.2.3, and satisfy\n\n\[ \n\\mathop{\\sup }\\limits_{{n \\geq 1, m \\geq 0}}E{\\left( {S}_{m + n} - {S}_{m}\\right) }^{2}/n < \\infty .\n\]\n\n\\( \\left( {3.2.15}\\right) \\)\n\nAssume that there exist positive integers \\( p = p\\left( n... | Proof. Denote \\( k = \\left\\lbrack {n/\\left( {p + q}\\right) }\\right\\rbrack .k \\rightarrow \\infty \\) as \\( n \\rightarrow \\infty \\) by (3.2.16). Put\n\n\[ \n{\\xi }_{j} = \\mathop{\\sum }\\limits_{{i = j\\left( {p + q}\\right) + 1}}^{{j\\left( {p + q}\\right) + p}}{X}_{i}\n\]\n\n\[ \n{\\eta }_{j} = \\mathop{... | Yes |
Theorem 4.1.1. Let \( \left\{ {{X}_{n}, n \geq 1}\right\} \) be a \( \rho \) -mixing sequence of random variables with \( E{X}_{n} = 0, E{X}_{n}^{2} < \infty \) and\n\n(i) \( \mathop{\lim }\limits_{{n \rightarrow \infty }}E{S}_{n}^{2}/n = {\sigma }^{2} > 0 \) ,\n\n(ii) \( \left\{ {{X}_{n}^{2}, n \geq 1}\right\} \) is u... | Proof of Theorem 4.1.1. We need only to check the conditions \( {\left( ii\right) }^{\prime } \) and \( \left( {iv}\right) \) in Lemma 4.1.1.\n\n1) We prove that \( \left\{ {{S}_{k}^{2}\left( n\right) /n, n \geq 1, k \geq 0}\right\} \) is uniformly integrable. Let \( N > 0 \) be specified later on. Denote\n\n\[ \n{X}_{... | Yes |
Lemma 4.3.1. Let \( f\left( {k, m}\right) \) be a non-negative function satisfying (4.1.2). Suppose that there exist \( \alpha > 0, r \geq 1 \) such that\n\n\[ E{\left| {S}_{k}\left( l\right) \right| }^{r} \leq f\left( {k, l}\right) \left\{ {{f}^{\alpha }\left( {k, l}\right) {w}_{1}\left( {f\left( {k, l}\right) }\right... | Proof. From (4.1.2) and (4.3.2), we have for \( l \leq n \)\n\n\[ {f}^{\beta }\left( {k, l}\right) {w}_{1}\left( {f\left( {k, l}\right) }\right) \leq a{f}^{\beta }\left( {k, n}\right) {w}_{1}\left( {f\left( {k, n}\right) }\right) .\n\nTherefore, by (4.3.1), for every \( k \geq 0,1 \leq l \leq n \)\n\n\[ E{\left| {S}_{k... | Yes |
Lemma 4.3.2. Let \( \\left\\{ {{X}_{n}, n \geq 1}\\right\\} \) be a \( \\rho \) -mixing sequence with \( E{X}_{n} = 0 \) , and \( {q}_{1},{q}_{2} \geq 2 \) . Suppose that the non-negative function \( h\\left( n\\right) \) satisfies:\n\n\[ \n\\max \\left( {h\\left( \\left\\lbrack \\frac{n}{2}\\right\\rbrack \\right), h\... | Proof. For simplicity, we assume that \( {X}_{n}, n \geq 1 \) have a common\ndistribution. Denote\n\n\[ \n{X}_{i1} = {X}_{i}I\\left( {\\left| {X}_{i}\\right| \leq B}\\right) - E{X}_{i}I\\left( {\\left| {X}_{i}\\right| \leq B}\\right) , \n\]\n\n\[ \n{X}_{i2} = {X}_{i}I\\left( {B < \\left| {X}_{i}\\right| < A}\\right) - ... | Yes |
Lemma 4.3.3. Let \( 0 < \delta \leq 1 \) . Suppose that the non-negative function \( h\left( n\right) \) satisfies the following conditions: there exist integer \( {n}_{0} > 0,0 < \theta < \) \( {2}^{\delta /\left( {2 + \delta }\right) },0 < {\delta }^{\prime } < \delta \) and \( a > 0 \) such that for any \( n \geq {n... | Proof. Denote\n\n\[ {X}_{i1} = {X}_{i}I\left( {\left| {X}_{i}\right| < B}\right) - E{X}_{i}I\left( {\left| {X}_{i}\right| < B}\right) ,\;{S}_{l1}\left( i\right) = \mathop{\sum }\limits_{{j = l + 1}}^{{l + i}}{X}_{j1}, \]\n\n\[ {X}_{i2} = {X}_{i}I\left( {\left| {X}_{i}\right| \geq B}\right) - E{X}_{i}I\left( {\left| {X}... | Yes |
Lemma 4.4.1. If condition (iii) is satisfied, for any \( 0 < a < 1 \) ,\n\n\[ \n{t}_{\left\lbrack na\right\rbrack }^{2}/{t}_{n}^{2} \rightarrow a\;\text{ as }n \rightarrow \infty .\n\] | Proof. (4.4.2) implies\n\n\[ \n{t}_{n}^{2}H\left( {t}_{\left\lbrack na\right\rbrack }\right) /\left( {{t}_{\left\lbrack na\right\rbrack }^{2}H\left( {t}_{n}\right) }\right) \rightarrow 1/a\;\text{ as }n \rightarrow \infty .\n\]\n\nWe show that there is a \( M > 0 \) such that\n\n\[ \n\mathop{\limsup }\limits_{{n \right... | Yes |
Lemma 4.4.2.\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{nP}\left( {\left| {X}_{1}\right| > {t}_{n}}\right) = 0 \] \n\nand \n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{\left( n/H\left( {t}_{n}\right) \right) }^{1/2}E\left| {X}_{1}\right| I\left( {\left| {X}_{1}\right| > {t}_{n}}\right) = 0. \] | The proof of this lemma can be found in Bradley (1988) and will be not presented here. | No |
Theorem 5.1.1. Let \( \\left\\{ {{X}_{n}, n \\geq 1}\\right\\} \) be a \( \\varphi \) -mixing sequence with \( E{X}_{n} = \n\n\( 0, E{X}_{n}^{2} < \\infty \) satisfying\n\n(i) \( {\\sigma }_{n}^{2} = {nh}\\left( n\\right) \), where \( h\\left( n\\right) \) is slowly varying,\n\n(ii) \( \\mathop{\\lim }\\limits_{{n \\ri... | Proof. We are going to verify the conditions in Theorem 4.0.4. By the definition of \( \\varphi \) -mixing,\n\n\[ \n\\left| {P\\left\\{ {\\mathop{\\bigcap }\\limits_{{i = 1}}^{r}{E}_{i}}\\right\\} - \\mathop{\\prod }\\limits_{{i = 1}}^{r}P\\left\\{ {E}_{i}\\right\\} }\\right| \\leq {r\\varphi }\\left( \\left\\lbrack {n... | Yes |
Lemma 5.1.1. Let \( \\left\\{ {{X}_{n}, n \\geq 1}\\right\\} \) be a \( \\varphi \) -mixing sequence. For any given positive integer \( q \) and \( a > 0, m \\geq 0, r \\geq q + 1 \), we have\n\n\[ \n\\left( {1 - \\varphi \\left( q\\right) - \\mathop{\\max }\\limits_{{q \\leq j \\leq r}}P\\left\\{ {\\left| {{S}_{m + r}... | Proof. Denote \( {A}_{1} = \\left\\{ {\\left| {X}_{m + 1}\\right| > {3a}}\\right\\} \), \n\n\[ \n{A}_{j} = \\left\\{ {\\left| {{S}_{m + j} - {S}_{m}}\\right| > {3a},\\;\\left| {{S}_{m + i} - {S}_{m}}\\right| \\leq {3a},\\;1 \\leq i \\leq j - 1}\\right\\} ,\\;2 \\leq j \\leq r, \n\]\n\n\[ \n{B}_{j} = \\left\\{ {\\left| ... | Yes |
Corollary 5.1.2. Let \( \left\{ {{X}_{n}, n \geq 1}\right\} \) be a strictly stationary \( \varphi \) -mixing sequence with \( E{X}_{1} = 0, E{X}_{1}^{2} < \infty \) and \( {\sigma }_{n}^{2} \rightarrow \infty \) . Denote\n\n\[ \n{Y}_{n}\left( t\right) = \frac{1}{{\sigma }_{n}}\left( {{S}_{\left\lbrack nt\right\rbrack ... | Proof. Obviously, (a) and (b) are equivalent and (b) implies (c). From Theorem 5.4 of Billingsley (1968), it follows that (c) implies (d). Finally \ | No |
Lemma 5.1.3. Let \( \\left\\{ {{Y}_{n}, n \\geq 1}\\right\\} \) be a sequence of random variables satisfying (5.1.15). Then for every \( A \\geq {a}_{0}^{2} \) we have\n\n\[ \n{E}_{{\\left( 1 + 2b\\right) }^{2}A}{T}_{n}^{2} \\leq {\\left( 1 + 2b\\right) }^{2}\\frac{\\eta }{1 - \\eta }{E}_{A}{T}_{n}^{2} \n\]\n\n\[ \n+ {... | Proof. By (5.1.14) and a change of variables one obtains\n\n\[ \n{E}_{{\\left( 1 + 2b\\right) }^{2}A}{T}_{n}^{2} = {\\left( 1 + 2b\\right) }^{2}{AP}\\left\\{ {{T}_{n}^{2} > {\\left( 1 + 2b\\right) }^{2}A}\\right\\} \n\]\n\n\[ \n+ {\\left( 1 + 2b\\right) }^{2}{\\int }_{A}^{\\infty }P\\left\\{ {{T}_{n}^{2} > {\\left( 1 +... | Yes |
Lemma 5.1.4. Let \( \\left\\{ {{X}_{n}, n \\geq 1}\\right\\} \) be a centered sequence such that \( {\\varphi }^{ * } < \) \( 1/4 \) and \( \\left\\{ {\\mathop{\\max }\\limits_{{1 \\leq i \\leq n}}E{X}_{i}^{2}/{\\sigma }_{n}^{2}, n \\geq 1}\\right\\} \) is bounded. Then\n\n\[ \n\\left\\{ {\\mathop{\\max }\\limits_{{1 \... | Proof. Let \( p \) be an integer such that \( \\varphi \\left( p\\right) < 1/4 \) . We have\n\n\[ \n\\mathop{\\max }\\limits_{{1 \\leq i \\leq n}}E{\\left( {S}_{n} - {S}_{i}\\right) }^{2} \\leq \\mathop{\\max }\\limits_{{1 \\leq i < n - p}}E{\\left( {S}_{n} - {S}_{i}\\right) }^{2} + {p}^{2}\\mathop{\\max }\\limits_{{1 ... | Yes |
Lemma 5.1.5. Let \( \left\{ {{X}_{n}, n \geq 1}\right\} \) be a centered sequence with \( {\phi }^{ * } < 1/4 \) . Then \( \left\{ {\mathop{\max }\limits_{{1 \leq i \leq n}}{S}_{i}^{2}/{\sigma }_{n}^{2}, n \geq 1}\right\} \) is uniformly integrable if and only if \( \left\{ {\mathop{\max }\limits_{{1 \leq i \leq n}}{X}... | proof. First, because\n\n\[ P\left\{ {\mathop{\max }\limits_{{1 \leq i \leq n}}\left| {X}_{i}\right| > {2x}{\sigma }_{n}}\right\} \leq P\left\{ {\mathop{\max }\limits_{{1 \leq i \leq n}}\left| {S}_{i}\right| > x{\sigma }_{n}}\right\} \]\n\n(5.1.19)\n\nfor any \( x > 0 \), one of the implications follows by the relation... | No |
Assume that \( \left\{ {{\eta }_{n}, n \geq 1}\right\} \) be a strictly stationary \( \varphi \) -mixing sequence with \( E{\eta }_{n} = 0, E{\eta }_{n}^{2} < \infty ,{\tau }_{n}^{2} = E{\left( \mathop{\sum }\limits_{{i = 1}}^{n}{\eta }_{i}\right) }^{2} \rightarrow \infty \) , \( \liminf {\tau }_{n}^{2}/n = 0 \) . If \... | \[ P\left\{ {{\alpha }_{0} = {a}_{k}{\tau }_{{n}_{k}}}\right\} = {b}_{k}{n}_{k}^{-1}\;\text{ for }k \in \mathbb{N}, \] \[ P\left\{ {{\alpha }_{0} = 0}\right\} = 1 - \mathop{\sum }\limits_{k}{b}_{k}{n}_{k}^{-1} \] (5.2.1) where the positive integers \( {n}_{1} < {n}_{2} < {n}_{3} < \cdots \), and the sequences of real n... | Yes |
Theorem 6.1.1. Let \( \left\{ {{X}_{\mathbf{t}},\mathbf{t} \in {\mathbb{Z}}^{d}}\right\} \) be a strictly stationary \( {\alpha }_{ * } \) -mixing random field with \( E{X}_{\mathbf{t}} = 0, E{\left| {X}_{\mathbf{t}}\right| }^{2 + \delta } < \infty \) for some \( \delta > 0 \) . If for some \( \tau > 0 \)\n\n(i) \( {\a... | The proof of Theorem 6.1.1 will need some lemmas. It is clear from the proof of Lemma 1.2.3 that we have\n\nLemma 6.1.1. Let random | No |
Lemma 6.1.2. Let \( {\xi }_{1},\cdots ,{\xi }_{n} \) be a sequence of random vectors, \( \mid E\mathop{\prod }\limits_{{j = i}}^{n} \) \( \times {\xi }_{j}\left| { < \infty ,\;i = 1,\cdots, n - 1,\;}\right| E{\xi }_{i}| \leq 1,\;i = 1,\cdots, n.\; \) Then\n\n\[ \left| {E\mathop{\prod }\limits_{{s = 1}}^{n}{\xi }_{s} - ... | Proof. Obviously, we have\n\n\[ \left| {E\mathop{\prod }\limits_{{s = 1}}^{n}{\xi }_{s} - \mathop{\prod }\limits_{{s = 1}}^{n}E{\xi }_{s}}\right| \leq \mathop{\sum }\limits_{{i = 1}}^{{n - 1}}\left| {E{\xi }_{i}\mathop{\prod }\limits_{{j = i + 1}}^{n}{\xi }_{j} - E{\xi }_{i}E\mathop{\prod }\limits_{{j = i + 1}}^{n}{\xi... | Yes |
Theorem 6.1.2. Let \( \\left\\{ {{X}_{\\mathbf{t}},\\mathbf{t} \\in {\\mathbb{Z}}^{d}}\\right\\} \) be an \( \\alpha \) -mixing random field with \( E{X}_{\\mathbf{t}} = 0, E{X}_{\\mathbf{t}}^{2} < \\infty \) . If there exists a \( g \\in \\mathcal{G} \) such that\n\n\[ \n\\mathop{\\sup }\\limits_{\\mathbf{t}}{Eg}\\lef... | The proof of Theorem 6.1.2 is similar to that of Theorem 3.2.3. | No |
Theorem 6.1.3. Let \( \left\{ {{X}_{\mathbf{t}},\mathbf{t} \in {\mathbb{Z}}^{d}}\right\} \) be a centered strictly stationary random field with \( 0 < E{X}_{0}^{2} < \infty ,{\rho }^{ * }\left( r\right) \rightarrow 0 \) as \( r \rightarrow \infty \) and the continuous positive spectral density \( f\left( \cdot \right) ... | The proof of Theorem 6.1.3 will not be presented here. | No |
Theorem 6.2.2. Let \( \\left\\{ {{\\xi }_{\\mathbf{t}},\\mathbf{t} \\in {\\mathbb{Z}}^{d}}}\\right\\} \) be a strictly stationary nonuniform \( \\varphi \) -mixing random field and satisfy\n\n(i) there exists a non-negative function \( \\varphi \\left( \\cdot \\right) \) on \( {R}^{1} \), such that for any \( \\Lambda ... | By a direct calculation, Lu (1995) proved that the conclusion of Theorem 6.2.2 holds for a more general indexed set \( \\mathcal{A} \) (with the metric entropy exponent \( r,0 < r < 1 \) ) and the condition for the rate of nonuniform \( \\varphi \) - mixing was weakened. | Yes |
Lemma 6.2.1. There exist \( {C}_{0}, q \), depending only on \( d \), such that for all \( p \) satisfying \( 0 < p < 1/d \), and for every measurable \( A \subset {R}^{d} \) of finite measure, we can find a slice \( S \) that bisects \( A \), has thickness \( {\left( \left| A\right| /2\right) }^{p} \), and is such tha... | The proof of Lemma 6.2.1 was given in Goldie and Greenwood (1986b). | No |
There exist constants \( a, b \), depending only on \( d \), such that\n\n\[ \parallel Z\left( A\right) {\parallel }_{2} \leq a{e}^{b\rho }\sigma {\left| A\right| }^{1/2},\;A \in {\mathcal{B}}^{d}. \] | Proof. Without loss of generality we assume \( \sigma = 1 \) . Set\n\n\[ \sigma \left( h\right) = \mathop{\sup }\limits_{{A \in {\mathcal{B}}^{d},\left| A\right| = h}}\parallel Z\left( A\right) {\parallel }_{2},\;\bar{\sigma }\left( h\right) = \mathop{\sup }\limits_{{{h}^{\prime } \leq h}}\sigma \left( {h}^{\prime }\ri... | Yes |
Lemma 6.2.4. Let \( \\left\\{ {{X}_{\\mathbf{i}},\\mathbf{i} \in {\\mathbb{Z}}^{d}}\\right\\} \) be a \( \\rho \) -mixing random field. If \( \\left\\{ {{X}_{\\mathbf{i}}^{2},\\mathbf{i} \in {\\mathbb{Z}}^{d}}\\right\\} \) is uniformly integrable and \( {\\rho }^{\\prime } < \\infty \), then the set of random variables... | \[ E\\left( {\\frac{{Z}^{2}\\left( A\\right) }{\\left| A\\right| }I\\left( {\\frac{{Z}^{2}\\left( A\\right) }{\\left| A\\right| } > y}\\right) }\\right) < \\left\\{ {C\\left( {1 \\land {y}^{-1}}\\right) + {Cg}\\left( {y}^{1/4}\\right) }\\right\\} {e}^{C{\\rho }^{\\prime }}. \]\n\n\( \\left( {6.2.20}\\right) \) | Yes |
Lemma 6.2.7. Let \( \left\{ {Z}_{n}\right\} \) be a sequence of additive processes on \( {\mathcal{B}}^{d} \) such that the set \( \left\{ {{Z}_{n}^{2}\left( A\right) /\left| A\right|, A \in {\mathcal{B}}^{d}, n \geq 1}\right\} \) is uniformly integrable, and satisfying Lemma 6.2.6 (i), (ii), (iii). Then the finite dim... | The proofs of Lemmas 6.2.4-6.2.7 are given in Goldie and Greenwood \( \left( {{1986}\mathrm{a},\mathrm{b}}\right) \) . | No |
Theorem 6.2.4. Let \( \left\{ {{\xi }_{n,\mathbf{j}}\mathbf{j} \in {\mathcal{J}}_{n}}\right\} \), be a \( \rho \) -mixing triangular array. The smoothed partial-sum processes \( {Z}_{n} \) are defined by (6.2.1). Suppose that\n\n(i) \( E{\xi }_{n,\mathbf{j}} = 0 \) for any \( n \geq 1,\mathbf{j} \in {\mathcal{J}}_{n} \... | Proof. By Lemma 6.2.4 we have uniform integrability of the set \( \left\{ {{Z}_{n}^{2}\left( A\right) /\left| A\right|, n \geq 1, A \in {\mathcal{B}}^{d}}\right\} \) . Because \( {\rho }_{n}\left( x\right) \) is non-increasing in \( x \), condition (iii) implies \( {\rho }_{n}\left( x\right) \rightarrow 0\left( {n \rig... | Yes |
Lemma 6.3.1. Let \( \left\{ {{\xi }_{\mathbf{t}},\mathbf{t} \in {\mathbb{Z}}^{d}}\right\} ,\left\{ {{Z}_{n}\left( A\right), A \in {\mathcal{B}}^{d}}\right\} \) be as in Theorem 6.2.3 with \( \varphi \left( x\right) = O\left( {x}^{-\left( {d + 1 + \theta }\right) }\right) \), for some \( \theta > 0 \), instead of \( \va... | Proof. First, we prove that for \( \delta = 0 \)\n\n\[ \n\sigma \left( {2m}\right) \leq {2}^{1/2}{\left( 1 + 2{m}^{1/2}{\varphi }^{1/2}\left( {d}^{-1/2}{m}^{p} - 2\right) \right) }^{1/2}\sigma \left( m\right) + 3\overline{\sigma }\left( {c{m}^{r}}\right) , \n\]\n\nwhere \( \sigma \left( m\right) = \mathop{\sup }\limits... | Yes |
Lemma 7.1.1. We have\n\n\[ \ni\mathop{\sum }\limits_{{j = 1}}^{n}{a}_{j}^{\left( 1\right) } = - t + ⏲{\left( 1 + {12}\alpha \right) }^{1/2}{d}^{2/s}{\sigma }_{n}{\left( \alpha \left( h + 1\right) \right) }^{\left( {s - 2}\right) /{2s}}\left| t\right| /{c}_{0}^{3/2} \]\n\n\[ + \Theta \left( {{2}^{3 - s}/\left( {s - 1}\r... | Proof. Let \( \tau \) be a random variable uniformly distributed on the set \( \{ 1,2,\cdots, n\} \), and independent of \( \left\{ {{X}_{1},{X}_{2},\cdots ,{X}_{n}}\right\} \) . It is easy to see that for \( i = 0,1,\cdots, k \) and any \( b \geq 1 \)\n\n\[ E{\left| {Z}_{\tau }^{\left( l\right) }\right| }^{b} \leq {\l... | Yes |
If (7.1.23) is satisfied, then\n\n\[ \mathop{\sum }\limits_{{r = 2}}^{k}\mathop{\sum }\limits_{{j = 1}}^{n}\left| {E{Y}_{j}\mathop{\prod }\limits_{{l = 1}}^{{r - 1}}{\xi }_{j}^{\left( l\right) }{e}^{{it}{z}_{j}^{\left( r\right) }} - E\left( {{Y}_{j}\mathop{\prod }\limits_{{l = 1}}^{{r - 1}}{\xi }_{j}^{\left( l\right) }... | Proof. We only prove the second inequality since the first can be proved analogously. From the definition of \( {z}_{j}^{\left( r\right) } \), we have that for all \( r = 2,3,\cdots, k,\;{\widehat{{z}_{j}}}^{\left( r\right) } \) and \( {\widetilde{{z}_{j}}}^{\left( r\right) } \) cannot be equal to zero simultaneously. ... | Yes |
Theorem 7.2.1. Let \( \\left\\{ {{X}_{n}, n \\geq 1}\\right\\} \) be a strictly stationary \( \\varphi \) -mixing sequence of random variables with \( E{X}_{1} = 0, E{X}_{1}^{2} = 1 \) and \( E{\\left| {X}_{1}\\right| }^{2 + \\delta } < \\infty \) for some \( 0 < \\delta < 3 \) . Suppose\n\n\[ \n\\varphi \\left( n\\rig... | By using Lemma 2.2.10, Lu (1993) weakened the conditions and proved the following theorem. | No |
Lemma 8.1.1. Let \( \\left\\{ {{X}_{n}, n \geq 1}\\right\\} \) be a \( \\varphi \) -mixing sequence. Denote by\n\n\( m\\left( Y\\right) \) the median of a random variable \( Y \) . Then we have\n\n\[ \nP\\left\\{ \\right. \\mathop{\\max }\\limits_{{1 \\leq j \\leq n}}\\left| {{S}_{j} - m\\left( {{S}_{j} - {S}_{n + k - ... | Proof. Take \( k \) so large that \( C \\mathrel{\\text{:=}} \\varphi \\left( k\\right) < 1/2 \) . Let\n\n\[ T = \\left\\{ \\begin{array}{l} \\mathop{\\min }\\limits_{{1 \\leq j \\leq n}}\\left\\{ {j : {S}_{j} - m\\left( {{S}_{j} - {S}_{n + k - 1} + {S}_{j}\\left( {k - 1}\\right) }\\right) \\geq \\varepsilon }\\right\\... | Yes |
Lemma 8.1.3. Let \( \\left\\{ {{X}_{n}, n \geq 1}\\right\\} \) be as in Lemma 8.1.2. If \( {S}_{n} \) almost surely converge to a random variable as \( n \rightarrow \\infty \), then \( E\\mathop{\\max }\\limits_{{1 \\leq k \\leq n}}{S}_{k}^{2} \) are convergent. | Proof. Denote \( {D}_{n, n + m} = \\mathop{\\bigcup }\\limits_{{i = n}}^{{n + m}}\\left\\{ {\\left| {{S}_{i}\\left( {n + m - i}\\right) }\\right| > \\varepsilon }\\right\\} \). If \( \\varphi \\left( M\\right) < 1/6 \), by Lemma 8.1.2\n\n\[ P\\left\\{ {D}_{n, n + m}\\right\\} \]\n\n\[ \\geq \\frac{\\left( {1 - {6\\varp... | Yes |
Lemma 8.1.4. Let \( B = \\left\\{ {\\mathop{\\max }\\limits_{{1 \\leq j \\leq n}}\\left| {X}_{j}\\right| \\geq \\varepsilon }\\right\\} ,{T}_{n} = \\mathop{\\sum }\\limits_{{i = 1}}^{n}I\\left( {\\left| {X}_{i}\\right| \\geq \\varepsilon }\\right) \) . If \( \\varphi \\left( M\\right) < 1/2 \), we have\n\n\[ P\\left( B... | Proof. Denote\n\n\[ {T}_{0} = 0,{T}_{n}\\left( m\\right) = {T}_{n + m} - {T}_{n} \]\n\n\[ {B}_{n} = \\left\\{ {\\left| {X}_{n}\\right| \\geq \\varepsilon }\\right\\} \]\n\n\[ {B}_{j} = \\left\\{ {\\mathop{\\max }\\limits_{{j < i \\leq n}}\\left| {X}_{i}\\right| < \\varepsilon ,\\left| {X}_{j}\\right| \\geq \\varepsilon... | Yes |
Corollary 8.2.1. Let \( \\left\\{ {{X}_{n}, n \geq 1}\\right\\} \) be a \( \\varphi \) -mixing sequence of identically distributed random variables. Put \( {S}_{n} = \\mathop{\\sum }\\limits_{{k = 1}}^{n}{X}_{k} \) . If \( {S}_{n}/n \\rightarrow b \) a.s., where \( b \) is a finite constant, then \( E\\left| {X}_{1}\\r... | Proof. From \( {S}_{n}/n \\rightarrow b \) (a.s.) we have\n\n\[ \n\\frac{{X}_{n}}{n} = \\frac{{S}_{n}}{n} - \\frac{{S}_{n - 1}}{n - 1} \\cdot \\frac{n - 1}{n} \\rightarrow 0\\;\\text{ a.s. }\n\]\n\nTherefore \( P\\left\\{ {\\left| {{X}_{n}/n}\\right| \\geq \\varepsilon }\\right. \\), i.o. \( \\} = 0 \) for any \( \\var... | Yes |
Theorem 8.2.2. Let \( \\left\\{ {{X}_{n}, n \\geq 1}\\right\\} \) be a \( \\varphi \) -mixing ( \( \\rho \) -mixing) sequence of identically distributed random variables with\n\n\[ \n\\mathop{\\sum }\\limits_{{n = 1}}^{\\infty }{\\varphi }^{1/2}\\left( {2}^{n}\\right) < \\infty \\left( {\\mathop{\\sum }\\limits_{{n = 1... | Theorem 8.2.2 is an immediate consequence of Corollary 8.3.4 (Corollary 8.4.2), so we omit its proof here. | No |
Theorem 8.3.3. Suppose that \( l\left( x\right) \) is strictly monotone. Let \( \left\{ {{X}_{n}, n \geq }\right. \) \( 1\} \) be a \( \varphi \) -mixing sequence with a common distribution. If (8.3.8) is satisfied then (8.3.2) holds and | \[ \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{l\left( n\right) - l\left( \left\lbrack {n/2}\right\rbrack \right) }{n}P\left\{ {\mathop{\sup }\limits_{{i \geq n}}\frac{\beta \left( {S}_{i}\right) }{i} > \varepsilon }\right\} < \infty ,\;\text{ for any }\varepsilon > 0. \] | No |
Corollary 8.3.3. Let \( \\left\\{ {{X}_{n}, n \\geq 1}\\right\\} \) be a \( \\varphi \) -mixing sequence with a common distribution and \( E{X}_{1} = 0, E{\\left| {X}_{1}\\right| }^{t} < \\infty 1 \\leq t < 2 \) . If \( \\mathop{\\sum }\\limits_{{n = 1}}^{\\infty }{\\varphi }^{1/2}\\left( {2}^{n}\\right) < \\) \( \\inf... | \[ \\mathop{\\sum }\\limits_{{n = 1}}^{\\infty }\\frac{1}{n}P\\left\\{ {\\mathop{\\max }\\limits_{{1 \\leq i \\leq n}}\\left| {S}_{i}\\right| \\geq \\varepsilon {n}^{1/t}}\\right\\} < \\infty \n\]\n\nfor any \( \\varepsilon > 0 \) . | Yes |
Lemma 8.4.1. Let \( \\left\\{ {{\\xi }_{n}, n \\geq 1}\\right\\} \) be a \( \\rho \) -mixing sequence with \( E{\\xi }_{n} = 0 \) , \( {Eg}\\left( \\left| {\\xi }_{n}\\right| \\right) < \\infty \), where \( g\\left( x\\right) \) is a function for which there exists a constant \( 0 < C < \\infty \\;{such}\\{that}\\;\\ma... | Proof. For simplicity, we assume that \( \\left\\{ {\\xi ,{\\xi }_{i}, i \\geq 1}\\right\\} \) have a common distribution. Put \[ {\\xi }_{i1} = {\\xi }_{i}I\\left( {\\left| {\\xi }_{i}\\right| \\leq B}\\right) - E{\\xi }_{i}I\\left( {\\left| {\\xi }_{i}\\right| \\leq B}\\right) ; \] \[ {\\xi }_{i2} = {\\xi }_{i}I\\lef... | Yes |
Lemma 8.4.2. Let \( h\left( x\right) \) be a positive slowly varying function. Then \( {x}^{\varepsilon }h\left( x\right) \) is a quasi-monotone non-decreasing function for any \( \varepsilon > 0 \) . | Proof. By the property of a slowly varying function, we have\n\n\[ \mathop{\lim }\limits_{{N \rightarrow \infty }}\mathop{\sup }\limits_{{{2}^{N} \leq x \leq {2}^{N + 1}}}\frac{h\left( x\right) }{h\left( {2}^{N + 1}\right) } = \mathop{\lim }\limits_{{N \rightarrow \infty }}\mathop{\inf }\limits_{{{2}^{N} \leq x \leq {2... | Yes |
Lemma 8.6.1. If for any \( x > 0 \)\n\n\[ \mathop{\liminf }\limits_{{n \rightarrow \infty }}\mathop{\inf }\limits_{{k \leq n - 1}}P\left\{ {{S}_{n} - {S}_{k} \geq - x}\right\} > {\varphi }^{ * }\left( 1\right) \]\n\nthen there exists a constant \( C > 0 \) such that for any \( y \) and \( n \in \mathbb{N} \)\n\n\[ P\le... | Proof. By the definition of \( {\varphi }^{ * }\left( 1\right) \) and \( \left( {8.6.11}^{\prime }\right) \) we have\n\n\[ P\left\{ {{S}_{n} \geq y - x}\right\} \]\n\n\[ \geq P\left\{ {{S}_{n} \geq y - x,\mathop{\max }\limits_{{k \leq n}}{S}_{k} \geq y}\right\} \]\n\n\[ = \mathop{\sum }\limits_{{k = 1}}^{n}P\left\{ {{S... | Yes |
Lemma 8.6.3. Let \( \left\{ {X}_{n}\right\} \) be a sequence of random variables, \( H\left( x\right) \) satisfy (8.6.4) and \( {n\psi }\left( n\right) > \left\lbrack {n/2}\right\rbrack \psi \left( \left\lbrack {n/2}\right\rbrack \right), n = 1,2,\cdots \) . Then (8.6.7) \( \Rightarrow \) \( \left( {8.6.8}^{\prime }\ri... | Proof. Noting that \( {n\psi }\left( n\right) \uparrow \) and \( H\left( n\right) /H\left( {2n}\right) \geq \delta \) we have\n\n\[ \mathop{\sum }\limits_{{n = 2}}^{\infty }\frac{{n\psi }\left( n\right) - \left\lbrack {n/2}\right\rbrack \psi \left( \left\lbrack {n/2}\right\rbrack \right) }{n}P\left\{ {\mathop{\sup }\li... | Yes |
Proposition 9.1.1. Let \( \left\{ {{X}_{n}, n \geq 1}\right\} \) be a sequence of random variables with \( E{X}_{n} = 0,\mathop{\sup }\limits_{n}E{\left| {X}_{n}\right| }^{2 + \delta } < \infty \left( {0 < \delta \leq 2}\right) \) . Let \( {\mathcal{F}}_{n} = \) \( \sigma \left\{ {{X}_{k},1 \leq k \leq n}\right\} \) be... | Proof. 1) We first prove that for each \( \varepsilon > 0 \) ,\n\n\[ \nS\left( t\right) - \mathop{\sum }\limits_{{k \leq t}}{Y}_{k} = O\left( {{t}^{1/\left( {2 + \delta }\right) }{\left( \log t\right) }^{\left( {1 + \varepsilon }\right) /\left( {2 + \delta }\right) }}\right) \;\text{ a.s. }\n\]\n\n(9.1.9)\n\nIn fact, n... | Yes |
Lemma 9.2.1. Let \( \\left\\{ {X}_{n}\\right\\} \) be a sequence of strictly stationary random variables, \( E{X}_{1}^{2} < \\infty \) . Then\n\n\[ \n\\mathop{\\sum }\\limits_{{i = 1}}^{n}\\left\\{ {\\left| {X}_{i}\\right| I\\left( {\\left| {X}_{i}\\right| \\geq {i}^{1/2}}\\right) + E\\left| {X}_{i}\\right| I\\left( {\... | Proof. We have\n\n\[ \n\\mathop{\\sum }\\limits_{{i = 1}}^{\\infty }{i}^{-1/2}E\\left| {X}_{i}\\right| I\\left( {{X}_{i}^{2} \\geq i}\\right) \n\]\n\n\[ \n\\leq \\mathop{\\sum }\\limits_{{j = 1}}^{\\infty }\\mathop{\\sum }\\limits_{{i = 1}}^{j}{i}^{-1/2}E\\left| {X}_{1}\\right| I\\left( {j - 1 < {X}_{1}^{2} \\leq j}\\r... | Yes |
Lemma 9.2.2. If \( E\left| X\right| < \infty \), then for any \( 0 < b \leq 1,\varepsilon > 0 \)\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{\infty }{k}^{b - 1}{e}^{-\varepsilon {k}^{b}}E{\left| X\right| }^{1 + \varepsilon }I\left( {\left| X\right| < {e}^{{k}^{b}}}\right) < \infty . \]\n | Proof. It is easy to see that the left hand side of the above inequality equals to\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{\infty }{k}^{b - 1}{e}^{-\varepsilon {k}^{b}}E{\left| X\right| }^{1 + \varepsilon }I\left( {\left| X\right| < 1}\right) \]\n\n\[ + \mathop{\sum }\limits_{{k = 1}}^{\infty }{k}^{b - 1}{e}^{-\varepsil... | Yes |
Lemma 9.2.3. Let \( \\left\\{ {{c}_{n}, n \\geq 1}\\right\\} \) be a sequence of nonincreasing positive numbers. Then, for any real sequence \( \\left\\{ {{\\eta }_{n}, n \\geq 1}\\right\\} \), we have\n\n\[ \n\\mathop{\\max }\\limits_{{1 \\leq i \\leq n}}\\left| {\\mathop{\\sum }\\limits_{{j = 1}}^{i}{\\eta }_{j}}\\ri... | Proof. Denote \( {D}_{i} = \\mathop{\\sum }\\limits_{{j = 1}}^{i}{c}_{j}{\\eta }_{j} \) . We have\n\n\[ \n{\\eta }_{j} = \\left( {{D}_{j} - {D}_{j - 1}}\\right) /{c}_{j} = \\mathop{\\sum }\\limits_{{i = 1}}^{j}\\left( {\\frac{1}{{c}_{i}} - \\frac{1}{{c}_{i - 1}}}\\right) \\left( {{D}_{j} - {D}_{j - 1}}\\right) , \n\]\n... | Yes |
Lemma 9.2.4. Let \( \\left\\{ {{X}_{n}, n \\geq 1}\\right\\} \) be a \( \\rho \) -mixing sequence with \( E{X}_{n} = \) \( 0, E{X}_{n}^{2} < \\infty \) . Let \( {u}_{k},{v}_{k} \) be as above. Denote \( {u}_{k}\\left( n\\right) = \\mathop{\\sum }\\limits_{{i = k + 1}}^{{k + n}}{u}_{i},{v}_{k}\\left( n\\right) = \) \( \... | Proof. We only give the proof of (9.2.24). It is obvious for \( n = 1 \) . For \( n \\geq 2 \), denote \( {n}_{1} = n - \\left\\lbrack {n/2}\\right\\rbrack ,{n}_{2} = \\left\\lbrack {n/2}\\right\\rbrack \) . Then by the definition of \( \\rho \\left( \\cdot \\right) \) we have\n\n\[ \nE{u}_{k}^{2}\\left( n\\right) = E{... | Yes |
Lemma 9.2.5. Let \( \left\{ {{X}_{n}, n \geq 1}\right\} \) be as in Theorem 9.2.1, we have\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{n}{v}_{j} = o\left( {\exp \left( {{n}^{a}/3}\right) }\right) \;\text{ a.s. } \] | Proof. Using Lemma 2.2.5 with \( q = 2 \) and condition (ii), we have\n\n\[ E{\left( \mathop{\sum }\limits_{{j = 1}}^{n}{v}_{j}\right) }^{2} \leq {cn}\mathop{\max }\limits_{{1 \leq j \leq n}}E{v}_{j}^{2} \]\n\n\[ \leq c{n}^{a}\exp \left( {{n}^{a}/2}\right) . \]\n\nIt follows from the Borel-Cantelli lemma that (9.2.27) ... | Yes |
Lemma 9.2.6. Assume that the conditions of Theorem 9.2.1 are satisfied and\n\n\[ 0 < a < {\varepsilon }^{\prime }/\left( {8 + {\varepsilon }^{\prime }}\right) \]\n\n(9.2.28)\n\nThen\n\n\[ \mathop{\max }\limits_{{{N}_{k} < j \leq {N}_{k + 1}}}\left| {\mathop{\sum }\limits_{{i = {N}_{k}}}^{j}{\widehat{X}}_{i}}\right| = o... | Proof. By the Borel-Cantelli lemma, we need only to prove that for any \( \varepsilon > 0 \)\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{\infty }P\left\{ {\mathop{\max }\limits_{{{N}_{k} < j \leq {N}_{k + 1}}}\left| {\mathop{\sum }\limits_{{i = {N}_{k}}}^{j}{\widehat{X}}_{i}}\right| \geq \varepsilon \exp \left( {{k}^{a}/2}\... | Yes |
Lemma 9.2.8. Suppose that a satisfies (9.2.28), and condition (ii) of Theorem 9.2.1 is satisfied. Then for \( {\delta }_{1} = {\varepsilon }^{\prime }/4 \), we have\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{\infty }{e}^{-\left( {1 + {\delta }_{1}}\right) {k}^{a}}E{\left| {\xi }_{k}\right| }^{2 + 2{\delta }_{1}} < \infty .... | Proof. From Lemma 2.2.5, Lemma 9.2.2 and condition (ii), we have\n\n\[ E{\left| {\xi }_{k}\right| }^{2 + 2{\delta }_{1}} \leq {cE}{\left| {u}_{k}\right| }^{2 + 2{\delta }_{1}} \]\n\n\[ \leq c\left\{ {\left( {k}^{a - 1}\exp \left( {k}^{a}\right) \right) }^{1 + {\delta }_{1}}\right. \]\n\n\[ + \left. {{k}^{a - 1}\exp \le... | Yes |
Lemma 10.1.1. Let \( \left\{ {{Z}_{n},{\mathcal{F}}_{n}, n \geq 1}\right\} \) be a supermartingale with \( E{Z}_{n} = \) 0. Let \( {Z}_{0} = 0 \) and \( {U}_{i} = {Z}_{i} - {Z}_{i - 1} \) for \( i \geq 1 \) . Suppose \( {U}_{i} \leq C \) a.s. for some \( 0 \leq C < \infty \) and all \( i \geq 1 \) . Fix \( \lambda > 0 ... | \[ {M}_{n} = \exp \left( {\lambda {Z}_{n}}\right) \exp \left\{ {-\left( {{\lambda }^{2}/2}\right) \left( {1 + {\lambda C}/2}\right) \mathop{\sum }\limits_{{i = 1}}^{n}E\left( {{U}_{i}^{2} \mid {\mathcal{F}}_{i - 1}}\right) }\right\} \] for \( n \geq 1 \) and \( {M}_{0} = 1 \) a.s. Then \( \left\{ {{M}_{n},{\mathcal{F}}... | Yes |
Theorem 10.2.1. (Lin 1991) Suppose that \( \left\{ {X}_{n}\right\} \) defined above satisfies the following conditions:\n\n(i) \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\inf }\limits_{{m \geq 0}}E{\left( {X}_{m + 1} + \cdots + {X}_{m + n}\right) }^{2}/n > 0 \) ;\n\n(ii) there exist \( {t}_{0}, M > 0 \),... | Proof. From conditions (i)-(iii), there exist constants \( {w}_{1} \) and \( {w}_{2} \) , \( 0 < {w}_{1} \leq {w}_{2} < \infty \) such that\n\n\[ \n{w}_{1}n \leq E{\left( {X}_{m + 1} + \cdots + {X}_{m + n}\right) }^{2} \leq {w}_{2}n \n\]\n\n(10.2.7)\n\nfor every integer \( m \geq 0 \) and \( n \) large enough. The righ... | Yes |
Lemma 11.1.1. Let \( \left\{ {{X}_{\mathbf{j}},\mathbf{j} \in {\mathbb{N}}^{d}}\right\} \) be a \( \varphi \) -mixing random field with \( E{X}_{\mathbf{j}} = 0,\left| {X}_{\mathbf{j}}\right| \leq {\Delta }_{\mathbf{n}} \) a.s. \( \mathbf{1} \leq \mathbf{j} \leq \mathbf{n} \) . Denote \( {\sigma }_{\mathbf{n}} = \matho... | Proof. Put \( \mathbf{N} = \left( {{N}_{1},\cdots ,{N}_{d}}\right) \) such that \( {2m}\left( {{N}_{i} - 1}\right) \leq {n}_{i} \leq {2m}{N}_{i} \) , \( i = 1,\cdots, d \) . Denote \( V = \left\{ {\left( {{v}_{1},\cdots ,{v}_{d}}\right) : {v}_{i} = 1\text{or}2, i = 1,\cdots, d}\right\} \) . Define \[ {I}_{\mathbf{v},\m... | Yes |
Lemma 11.1.2. We have\n\n\[ \mathop{\sup }\limits_{{A \in \mathcal{A}}}\left| {{S}_{\mathbf{n}}\left( A\right) - {S}_{\mathbf{n}}^{\prime }\left( A\right) }\right| = O\left( 1\right) \;\text{ a.s. } \]\n\n(11.1.9)\n\n\[ \mathop{\sum }\limits_{{\mathbf{j} \leq n\mathbf{1}}}E\left| {{X}_{\mathbf{j}} - {X}_{\mathbf{j}}^{\... | Proof. Since\n\n\[ P\left\{ {{X}_{\mathbf{j}} \neq {X}_{\mathbf{j}}^{\prime }}\right\} = P\left\{ {\left| {X}_{\mathbf{j}}\right| \geq {\left| \mathbf{j}\right| }^{\left( {1 + \tau }\right) /\left( {2 + \delta }\right) }}\right\} \leq c{\left| \mathbf{j}\right| }^{-\left( {1 + \tau }\right) }, \]\n\nwe have \( P\left\{... | Yes |
Lemma 11.1.3. We have\n\n\[ \mathop{\sup }\limits_{A}\left| {{S}_{\mathbf{n}}^{\prime }\left( A\right) - {S}_{\mathbf{n}}^{\prime }\left( {A\left( {2}^{-b}\right) }\right) }\right| = O\left( {\left| n\right| }^{\frac{1}{2}\left( {1 - \frac{\tau \delta }{2 + \delta }}\right) }\right) \;\text{ a.s. } \] | Proof. For any \( A \in \mathcal{A} \), by conditions (11.0.6),(11.1.1) and (11.1.11), we have\n\n\[ \left| {{S}_{\mathbf{n}}^{\prime }\left( A\right) - {S}_{\mathbf{n}}^{\prime }\left( {A\left( {2}^{-b}\right) }\right) }\right| \]\n\n\[ \leq \mathop{\sum }\limits_{\mathbf{j}}\left| {\left| {\mathbf{n}A \cap {C}_{\math... | Yes |
Lemma 11.1.7. For \( {t}_{k} \in {G}_{\beta } \), we have\n\n\[ \mathop{\sup }\limits_{{A \in \mathcal{A}}}\mathop{\max }\limits_{{\mathbf{{t}_{k}} < \mathbf{n} \leq \mathbf{{t}_{k + 1}}}}\left| {\mathop{\sum }\limits_{\mathbf{j}}\left| {{\left( \mathbf{{t}_{k}}A\left( {2}^{-a}\right) \right) }_{ * } \cap {I}_{1} \cap ... | Proof. It is easy to see that\n\n\[ \mathop{\sup }\limits_{{A \in \mathcal{A}}}\mathop{\max }\limits_{{\mathbf{{t}_{k}} < \mathbf{n} \leq \mathbf{{t}_{k + 1}}}}\left| {\mathop{\sum }\limits_{\mathbf{j}}\left| {{\left( \mathbf{{t}_{k}}A\left( {2}^{-a}\right) \right) }_{ * } \cap {I}_{1} \cap {C}_{\mathbf{j}}}\right| {X}... | Yes |
Lemma 11.2.1. Let \( \\left\\{ {{X}_{\\mathbf{j}},\\mathbf{j} \\in {\\mathbb{N}}^{d}}\\right\\} \) be an \( \\alpha \) -mixing random field with \( E{X}_{\\mathbf{j}} = 0, E{\\begin{Vmatrix}{X}_{\\mathbf{j}}\\end{Vmatrix}}^{2 + \\delta } \\leq {C}_{1} \) and\n\n\[ \n{C}_{0} = \\mathop{\\sum }\\limits_{{r = 1}}^{\\infty... | Proof. By Lemma 1.2.4 we have\n\n\[ \n\\parallel E{X}_{\\mathbf{j}}{X}_{\\mathbf{k}}\\parallel \\leq {10\\alpha }{\\left( d\\left( \\mathbf{j},\\mathbf{k}\\right) \\right) }^{\\delta /\\left( {2 + \\delta }\\right) }\\parallel {X}_{\\mathbf{j}}{\\parallel }_{2 + \\delta }\\parallel {X}_{\\mathbf{k}}{\\parallel }_{2 + \... | Yes |
Lemma 11.2.4. If in (11.2.8) \( \beta \geq {6d} \) then\n\n\[ P\left\{ {\mathop{\sum }\limits_{{\mathbf{r} \notin L,\psi \left( {r}_{i}\right) \leq n}}\begin{Vmatrix}{\mathop{\sum }\limits_{{\mathbf{j} \in R\mathbf{r}}}{X}_{\mathbf{j}}}\end{Vmatrix} > {n}^{d/2 - 1/{16}}}\right\} \leq c{m}^{-2}, \]\n\n(11.2.18)\n\nwhere... | Proof. For \( r \notin L \) with \( \psi \left( {r}_{i}\right) \leq n, i = 1,\cdots, d \) we have by definition of \( L \) for some \( i \leq d,\psi {\left( {r}_{i}\right) }^{8d} < \mathop{\prod }\limits_{{i = 1}}^{d}\psi \left( {r}_{i}\right) \leq {n}^{d} \), hence \( \psi \left( {r}_{i}\right) \leq {n}^{1/8} \) . And... | Yes |
Theorem 12.1.2. Let \( \\left\\{ {{U}_{n}, n \\geq 1}\\right\\} \) be a sequence of strictly stationary \( \\alpha \) -mixing random variables uniformly distributed over \( \\left\\lbrack {0,1}\\right\\rbrack \) and\n\n\[ \n\\alpha \\left( n\\right) = O\\left( {n}^{-r}\\right) \\;r > 2.\n\]\n\nThen \( {\\alpha }_{n} \\... | By the following lemmas, from the proof of Theoem 12.1.1 it follows that Theorem 12.1.2 holds true. | No |
Lemma 12.1.2. If the conditions of Lemma 12.1.1 are satisfied with \( 2 < r < 3 \), then\n\n\[ E{S}_{n}^{4} \leq c\left( {{n}^{4 - r} + {n}^{2}{\tau }^{2{\theta }_{1}}}\right) \] | Proof. From Lemma 1.2.5 and (12.1.14) we have\n\n\[ \mathop{\sum }\limits_{{n\left( 3\right) }}^{\left( 1\right) } \leq 6\mathop{\sum }\limits_{{n\left( 3\right) }}^{\left( 1\right) }\alpha \left( {i}_{1}\right) \leq 6\mathop{\sum }\limits_{{{i}_{1} = 0}}^{n}{\left( {i}_{1} + 1\right) }^{2}\alpha \left( {i}_{1}\right) ... | Yes |
Lemma 12.2.1. Let \( \\left\\{ {{\\xi }_{i}, i \\geq 1}\\right\\} \) be a sequence of random variables and let \( {\\mathcal{F}}_{i} = \\sigma \\left( {{\\xi }_{j}, j \\leq i}\\right) \) . Then, for any \( p \\geq 2 \), there exists a constant \( D = D\\left( p\\right) \) such that\n\n\[ E{\\left| \\mathop{\\sum }\\lim... | Proof. Let \( {\\eta }_{i} = {\\xi }_{i} - E\\left( {{\\xi }_{i} \\mid {\\mathcal{F}}_{i - 1}}\\right) \) for \( 1 \\leq i \\leq n \) . Then, \( \\left\\{ {{\\eta }_{i},{\\mathcal{F}}_{i - 1},1 \\leq }\\right. \) \( i \\leq n\\} \) is a martingale difference sequence. By the well-known Burkholder\n\n(1973) inequality, ... | Yes |
Lemma 12.2.2. Let \( 2 < p < r \leq \infty ,2 < v \leq r \) and \( \left\{ {{X}_{n}, n \geq 1}\right\} \) be an \( \alpha \) -mixing sequence of random variables with \( E{X}_{n} = 0 \) and \( {\begin{Vmatrix}{X}_{n}\end{Vmatrix}}_{r} < \infty \) . Assume that\n\n\[ \n\alpha \left( n\right) \leq C{n}^{-\theta }\n\]\n\n... | Proof. For the sake of convenience of statement, we assume that \( \left\{ {X,{X}_{n}, n \geq 1}\right\} \) is a strictly stationary \( \alpha \) -mixing sequence. By a result of Rio (1993), there is \( {D}_{1} = {D}_{1}\left( v\right) \) such that\n\n\[ \nE{S}_{n}^{2} \leq {D}_{1}n{C}_{n}\parallel X{\parallel }_{v}^{2... | Yes |
Lemma 12.3.1. Let \( A \geq {\sigma }^{2} \) . We have\n\n\[ P\left\{ {\mathop{\sum }\limits_{{j \leq q}}E\left( {{Y}_{j}^{2} \mid {\mathcal{L}}_{j - 1}}\right) \geq {4An}}\right\} \leq c{A}^{-2}{p}^{2 - \rho }.\] | Proof. By the Hölder inequality we have\n\n\[ E\left( {{Y}_{j}^{2} \mid {\mathcal{L}}_{j - 1}}\right) \leq E\left( {{y}_{j}^{2} \mid {\mathcal{L}}_{j - 1}}\right) \]\n\n(12.3.14)\n\nBy Lemma 1.2.1, we have\n\n\[ {\begin{Vmatrix}E\left( {y}_{j}^{2} \mid {\mathcal{L}}_{j - 1}\right) - E{y}_{j}^{2}\end{Vmatrix}}_{2} \leq ... | Yes |
Lemma 12.3.2. For all \( R > 0 \) we have\n\n\[ P\left\{ {\left| {\mathop{\sum }\limits_{{j \leq q}}{Y}_{j}}\right| > {5R}{n}^{1/2}}\right\} \leq c\left\{ {\exp \left( {-{R}^{2}/A}\right) + {A}^{-2}{p}^{2 - \rho }}\right\} . | Proof. Let \( M \) be the index \( j \) of \( {H}_{j} \) or \( {I}_{j} \) containing \( n \) . Define\n\n\[ {U}_{k} = \left\{ \begin{array}{ll} \mathop{\sum }\limits_{{j \leq k}}{Y}_{j}, & \text{ if }k \leq M \\ {U}_{M}, & \text{ if }k > M \end{array}\right. \]\n\n\[ {s}_{k}^{2} = \left\{ \begin{array}{ll} \mathop{\sum... | Yes |
Lemma 12.3.3. We have\n\n\[ \nP\left\{ {\left| {\mathop{\sum }\limits_{{j \leq q}}{y}_{j}}\right| \geq {7R}{n}^{1/2}}\right\} \leq c\left( {\exp \left( {-{R}^{2}/A}\right) + {n}^{1 + {\rho \kappa } - \rho /2}\left( {{A}^{-2} + {R}^{-2}}\right) }\right) .\n\] | Proof. We have\n\n\[ \n\left| {\mathop{\sum }\limits_{{j \leq q}}{y}_{j}}\right| \leq \left| {\mathop{\sum }\limits_{{j \leq q}}{Y}_{j}}\right| + \left| {\mathop{\sum }\limits_{{j \leq q}}{v}_{j}}\right| \n\]\n\nBy the Chebyshev and Minkowski inequalities we obtain from (12.3.13) and expressing \( p \) and \( q \) in t... | Yes |
Theorem 12.3.3. Let \( \left\{ {{X}_{j}, j \geq 1}\right\} \) be a sequence of strictly stationary \( \alpha \) -mixing d-dimensional random vectors with a distribution \( F\left( x\right) \) and\n\n\[ \alpha \left( n\right) = O\left( {n}^{-4 - {2d}}\right) \]\n\nDenote \( {g}_{n}\left( s\right) = I\left( {{X}_{n} \leq... | Proof. Let \( P \) be the probability measure induced by \( F \) . Let \( {F}_{i}\left( {s}_{i}\right) \) , \( 1 \leq i \leq d \) be the \( i \) -th marginal of \( F\left( s\right), s = \left( {{s}_{1},\cdots ,{s}_{d}}\right) \) . Let \( r \geq 1 \) be given. We define\n\n\[ {s}_{ij} = \operatorname{inv}{F}_{i}\left( {... | Yes |
Theorem 12.4.2. Let \( \\left\\{ {{X}_{n}, n \\geq 1}\\right\\} \) be a sequence of strictly stationary \( \\varphi \) -mixing random variables with a common distribution \( F\\left( x\\right) \). Suppose that \( F\\left( x\\right) \) satisfies the 1-ulL condition and \( \\mathop{\\sum }\\limits_{{i = 1}}^{\\infty }{\\... | (12.4.3) | No |
Lemma 12.4.1. Let \( \left\{ {{X}_{n}, n \geq 1}\right\} \) be a \( \varphi \) -mixing sequence with \( E{X}_{n} = \) \( 0,\left| {X}_{n}\right| \leq d, E{X}_{n}^{2} \leq D \) and \( \mathop{\sum }\limits_{{i = 1}}^{\infty }{\varphi }^{1/2}\left( {2}^{i}\right) < \infty \) . Then there is \( {C}_{1} = \) \( {C}_{1}\lef... | \[ P\left\{ {\left| {\mathop{\sum }\limits_{{i = 1}}^{n}{X}_{i}}\right| > \varepsilon }\right\} \]\n\[ \leq \exp \left\{ {3\sqrt{e}n\frac{\varphi \left( m\right) }{m} - {\alpha \varepsilon } + {C}_{1}{\alpha }^{2}{Dn}}\right\} \] | Yes |
Theorem 13.1.1. Let \( h \) be a non-degenerate kernel. Assume that condition (13.1.3) is satisfied and \( {\sigma }_{n}^{2} \rightarrow \infty \) . Moreover, assume that for any \( \varepsilon > 0 \)\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{n}{{\sigma }_{n}^{2}}E{\widetilde{h}}_{1}{\left( {X}_{1}\rig... | Proof. Put\n\n\[ {\widehat{W}}_{n}\left( t\right) = \frac{nt}{m{\sigma }_{n}}\left( {{\widehat{U}}_{\left\lbrack nt\right\rbrack } - \theta }\right) ,\;0 \leq t \leq 1. \]\n\nBy Corollary 5.1.4, \( {\widehat{W}}_{n} \Rightarrow W \) as \( n \rightarrow \infty \) . Theorem 2.1.2 implies that \( {\sigma }_{n}^{2}/{n}^{1 ... | Yes |
Theorem 13.1.2. Let \( h \) be a non-degenerate kernel. Assume that\n\n\[ \mathop{\sup }\limits_{{1 \leq {t}_{1} < \cdots < {t}_{m}}}E{\left| h\left( {X}_{{t}_{1}},\cdots ,{X}_{{t}_{m}}\right) \right| }^{2 + \delta } < \infty \;\text{ for some }\;\delta > 0. \]\n\n(13.1.14)\n\nMoreover, assume that\n\n(i) \( {\sigma }_... | Proof. By Lemma 13.1.2, condition (13.1.14) implies that \( E{\left| {\widetilde{h}}_{1}\left( {X}_{1}\right) \right| }^{2 + \delta } < \infty \). Therefore from Remark 9.1.1, we conclude that we can redefine the sequence \( \left\{ {{\widetilde{h}}_{1}\left( {X}_{n}\right), n \geq 1}\right\} \) on a new probability sp... | Yes |
Theorem 13.2.2. Let the random error sequence \( \left\{ {e}_{i}\right\} \) with (13.2.1) be strictly stationary and \( \varphi \) -mixing. Suppose that \( E{\left| {e}_{1}\right| }^{8 + \delta } < \infty \) for some \( 0 < \delta \leq 1 \) and\n\n\[ \varphi \left( n\right) = o\left( {n}^{1 - 4\left( {2 + \varepsilon }... | In order to prove the theorem, we need a lemma: | No |
Theorem 13.3.1. Suppose that \( D \) is a compact subset of \( {R}^{d} \) and \( f \) is continuous on an \( \varepsilon \) -neighborhood of \( D \) . Suppose that \( K \) satisfies the following conditions:\n\n(1) \( K\left( \cdot \right) \) is a density on \( {R}^{d} \) ,\n\n(2) \( K\left( x\right) \leq {K}_{1} < \in... | Proof of Theorem 13.3.1.\n\nObviously by Lemma 2.2.2, under condition (13.3.4), (13.3.3) with \( {h}_{n} = O\left( {\left( {\log }^{2}n/n\right) }^{1/\left( {d + 2}\right) }\right) \) implies (13.3.5). We prove (13.3.3). Write\n\n\[ \mathop{\sup }\limits_{{x \in D}}\left| {{f}_{n}\left( x\right) - f\left( x\right) }\ri... | Yes |
Corollary 14.1.2. Suppose that the condition of Theorem 14.1.2 and condition (14.1.5) are satisfied (for some \( \beta > 0 \) ), and \( {B}_{N} = O\left( 1\right) ,\left| {a}_{k}\right| \downarrow 0 \) . Then when \( r = 0 \) we have | \[ S\left( t\right) - W\left( t\right) = O\left( {{\log }^{2}t}\right) \;\text{ a.s. } \] | Yes |
Lemma 14.1.1. Let \( {\xi }_{1},\cdots ,{\xi }_{n} \) be a sequence of random variables. Put\n\n\[ \n{S}_{k} = \mathop{\sum }\limits_{{j = 1}}^{k}{\xi }_{j},\;{M}_{k} = \mathop{\max }\limits_{{1 \leq j \leq k}}\left| {S}_{j}\right| .\n\]\n\nSuppose that there exists a sequence \( \left\{ {c}_{k}\right\} \) such that fo... | The proof refers to Theorem 2.4.1 in Stout (1974). | No |
Lemma 14.1.2. For any \( v \geq 0 \) ,\n\n\[ \mathop{\sum }\limits_{{j \geq k}}\frac{{A}_{j}^{v}}{{n}_{j}} = O\left( {\frac{{A}_{k}^{v}}{{n}_{k}}{k}^{r}}\right) \]\n\n(14.1.10)\n\n\[ \mathop{\sum }\limits_{{j > k}}\frac{{A}_{j}^{v}}{{n}_{j} - {n}_{k}} = O\left( {\frac{{A}_{k}^{v}}{{n}_{k}}{k}^{2r}}\right) \]\n\n(14.1.1... | Proof. (14.1.4) implies that there exists a constant \( C > 0 \) such that\n\n\[ {A}_{k + j} = O\left( {{\left( \frac{k + j}{j}\right) }^{C}{A}_{j}}\right) \]\n\nfor all \( k, j \geq 1 \) . And by \( {n}_{k + 1}/{n}_{k} \geq 1 + q/{k}^{r} \), we have\n\n\[ \frac{{n}_{k + j}}{{n}_{k}} \geq \mathop{\prod }\limits_{{i = 1... | Yes |
Lemma 14.1.3. Let \( W\left( t\right) \) be a Wiener process and \( \left\{ {t}_{n}\right\} \) be a sequence of random variables. Suppose that there exists a sequence \( \left\{ {b}_{n}\right\} \) of real numbers with \( {b}_{n} = o\left( n\right) \), such that\n\n\[ \n{t}_{n} - n = O\left( {b}_{n}\right) \;\text{ a.s.... | Proof. (14.1.13) implies that there exists a constant \( C \) such that\n\n\[ \n\left| {{t}_{n} - n}\right| \leq C{b}_{n}\;\text{ a.s.,} \n\]\n\nhence\n\n\[ \n\left| {W\left( {t}_{n}\right) - W\left( n\right) }\right| \leq \mathop{\sup }\limits_{{0 \leq t \leq n - C{b}_{n}}}\mathop{\sup }\limits_{{0 \leq s \leq {2C}{b}... | Yes |
Lemma 14.1.4. We have\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{\infty }\left| {{\xi }_{k} - {X}_{k}}\right| = O\left( 1\right) \] | Proof. We have from (14.1.15)\n\n\[ \left| {{\xi }_{k} - {X}_{k}}\right| = O\left( {{A}_{k}^{-p}{a}_{k}}\right) = O\left( {A}_{k}^{-p + 1}\right) . \]\n\nHence (14.1.17) follows from (14.1.5). | No |
Lemma 14.1.5. We have\n\n\[ \mathop{\sum }\limits_{{1 \leq j \leq N}}E{X}_{j}^{2} - {A}_{N}^{2} = O\left( 1\right) \] | Proof. Noting that\n\n\[ E{\xi }_{j}^{2} - {a}_{j}^{2}/2 = {a}_{j}^{2}\sin {2\pi }{n}_{j}/{2\pi }{n}_{j} \]\n\n\[ \mathop{\sum }\limits_{{j \leq N}}E{\xi }_{j}^{2} - {A}_{N}^{2} = O\left( {\mathop{\sum }\limits_{{j \leq N}}{a}_{j}^{2}/{n}_{j}}\right) = O\left( 1\right) \]\n\nand\n\n\[ \mathop{\sum }\limits_{{j \leq N}}... | Yes |
Lemma 14.1.7. We have\n\n\[ \n{u}_{j} = O\left( {{j}^{r}{A}_{j}^{1 - \delta }\log {A}_{j}}\right) \n\] | Proof. Using (14.1.16), (14.1.20) and Lemma 14.1.2, we find\n\n\[ \n{u}_{j} = \mathop{\sum }\limits_{{k = 0}}^{\infty }E\left( {{\xi }_{k + j} \mid {\mathcal{F}}_{j - 1}}\right) \n\]\n\n\[ \n= O\left( {\mathop{\sum }\limits_{{k = 0}}^{\infty }{A}_{k + j}^{1 - \delta }\left( {1 \land {A}_{j}^{p}{n}_{j}/{n}_{k + j}}\righ... | Yes |
Lemma 14.1.8. For each \( k, n\left( {k < n}\right) \) , \n\n\[ \mathop{\sum }\limits_{{k < i \leq n}}E\left( {{X}_{i}{u}_{n + 1} \mid {\mathcal{F}}_{i}}\right) = O\left( {{A}_{n}^{2 - {2\delta }}{n}^{2r}{\log }^{2}{A}_{n}}\right) \] \n\n\[ \mathop{\sum }\limits_{{k < i \leq n}}E\left( {{X}_{i}{u}_{k} \mid {\mathcal{F}... | Proof. By (14.1.20) we have \n\n\[ E\left( {{u}_{n + 1} \mid {\mathcal{F}}_{i}}\right) = \mathop{\sum }\limits_{{j = 0}}^{\infty }E\left( {{\xi }_{j + n + 1} \mid {\mathcal{F}}_{i}}\right) \] \n\n\[ = O\left( {\mathop{\sum }\limits_{{j = 0}}^{\infty }{A}_{j + n + 1}^{1 - \delta }\left( {1 \land {A}_{i}^{p}{n}_{i}/{n}_{... | Yes |
Lemma 14.1.9. We have\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{n}E{Y}_{j}^{2} - {A}_{n}^{2} = O\left( {{n}^{2r}{A}_{n}^{2 - {2\delta }}{\log }^{2}{A}_{n}}\right) . \] | Proof. Since \( \left\{ {Y}_{j}\right\} \) is a martingale difference sequence, we have\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{n}E{Y}_{j}^{2} = E{\left( \mathop{\sum }\limits_{{j = 1}}^{n}{Y}_{j}\right) }^{2} \]\n\n\[ = E{\left( \mathop{\sum }\limits_{{j = 1}}^{n}{X}_{j}\right) }^{2} + {2E}{u}_{n + 1}\mathop{\sum }\lim... | No |
Lemma 14.1.11. We have\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{n}\left( {E\left( {{Y}_{j}^{2} \mid {\mathcal{F}}_{j - 1}}\right) - {Y}_{j}^{2}}\right) = O\left( {{n}^{r}{A}_{n}^{2 - \delta }{\log }^{3}{A}_{n}}\right) \;\text{ a.s. } \] | Proof. Put \( {R}_{j} = {Y}_{j}^{2} - E\left( {{Y}_{j}^{2} \mid {\mathcal{F}}_{j - 1}}\right) \) . Then \( \left\{ {{R}_{j},{\mathcal{F}}_{j}}\right\} \) is a martingale defference sequence, and we have\n\n\[ E{R}_{k}^{2} = O\left( {E{Y}_{k}^{4}}\right) = O\left( {E{Y}_{k}^{2}{A}_{k}^{2 - {2\delta }}{k}^{2r}{\log }^{2}... | No |
Lemma 14.1.12. We have under the conditions of Theorem 14.1.1\n\n\\[ \mathop{\sum }\limits_{{j \leq n}}{T}_{j} - {A}_{n}^{2} = O\left( {{A}_{n}^{2 - \delta + r/\beta }{\log }^{3}{A}_{n}}\right) \\]\n\n(14.1.33) | Proof. Write\n\n\\[ \mathop{\sum }\limits_{{j \leq n}}{T}_{j} - {A}_{n}^{2} \\]\n\n\\[\n= \mathop{\sum }\limits_{{j \leq n}}\left( {{T}_{j} - E\left( {{T}_{j} \mid {\mathcal{F}}_{j - 1}}\right) }\right) + \mathop{\sum }\limits_{{j \leq n}}\left( {E\left( {{Y}_{j}^{2} \mid {\mathcal{F}}_{j - 1}}\right) - {Y}_{j}^{2}}\ri... | Yes |
Lemma 14.2.1. Let \( A \) be a real symmetric matrix of order \( n \) with eigenvalues \( {\lambda }_{1},\cdots ,{\lambda }_{n} \) . Denote \( \lambda = \mathop{\max }\limits_{{1 \leq i \leq n}}\left| {\lambda }_{i}\right| \) . Then for any row vector \( \mathbf{C} \) we have\n\n\[ \left| {\mathbf{{CAC}}}^{\prime }\rig... | Proof. We need only to prove that matrices \( {\lambda I} - A \) and \( {\lambda I} + A \) are non-negative definite. By the well-known property of a matrix, there exists a real orthogonal matrix \( U \) such that \( {U}^{\prime }{AU} \) is a diagonal matrix \( \Lambda \) with the diagonal elements which is just equal ... | Yes |
Lemma 14.2.2. At least one of the following inequalities is satisfied for eigenvalues of any matrix \( A = {\left( {a}_{ij}\right) }_{n \times n} \) :\n\n\[ \left| {\lambda - {a}_{ii}}\right| \leq \mathop{\sum }\limits_{{j = 1, j \neq i}}^{n}\left| {a}_{ij}\right| \;i = 1,\cdots, n. \] | This result, the so-called circle-plate theorem, is due to Gerschgorin (cf. Franklin 1968, p.161 Theorem 1). | No |
Lemma 14.2.3. If (14.2.1), (14.2.2) and (14.2.3) are satisfied, and for any \( k \geq 1 \)\n\n\[ E{X}_{k}^{2} \geq \mathop{\sum }\limits_{{j = 1, j \neq k}}^{\infty }\left| {E{X}_{k}{X}_{j}}\right| + 1 \]\n\nthen\n\n\[ {\begin{Vmatrix}{u}_{n}\end{Vmatrix}}_{2} = O\left( 1\right) \] | Proof. Let \( A \) be the covariance matrix of \( \left( {{X}_{1},\cdots ,{X}_{j}}\right) \) and \( \mathbf{C} = \) \( \left( {E{X}_{1}{X}_{j + k},\cdots, E{X}_{j}{X}_{j + k}}\right) \) . Then, by (5.22) in Philipp and Stout (1975), we have\n\n\[ E\left( {{E}^{2}\left( {{X}_{j + k} \mid {\mathcal{F}}_{j}}\right) }\righ... | Yes |
Lemma 14.3.2. Let \( {\phi }_{t}^{s} \) and \( {\psi }_{t}^{s} \) be the non-negative, strongly measurable, homogeneous additive functionals of \( \mathbf{X} \) with the finite \( {\alpha }_{\phi } = {E}_{\mu }{y}_{1},{\alpha }_{\psi } \) \( = {E}_{\mu }{\psi }_{{\tau }_{2}}^{{\tau }_{1}} \neq 0 \) . Then\n\n(a)\n\n\[ ... | Proof. By the strict stationarity and \( \varphi \) -mixing property of \( \left\{ {y}_{n}\right\} \) , \( \left\{ {y}_{n}\right\} \) is an ergodic sequence and its invariant \( \sigma \) -field \( \mathcal{U} \) is trivial. By (14.3.8) for any \( A \in \mathcal{U} \) \n\n\[ \n{P}_{a}\left( A\right) = {P}_{\mu }\left( ... | Yes |
Proposition 1. Let \( f : A \rightarrow B \) .\n\n(1) The map \( f \) is injective if and only if \( f \) has a left inverse.\n\n(2) The map \( f \) is surjective if and only if \( f \) has a right inverse.\n\n(3) The map \( f \) is a bijection if and only if there exists \( g : B \rightarrow A \) such that \( f \circ ... | Proof: Exercise. | No |
(1) If \( \sim \) defines an equivalence relation on \( A \) then the set of equivalence classes of \( \sim \) form a partition of \( A \) . | Proof: Omitted. | No |
Proposition 4. \( {\left( \mathbb{Z}/n\mathbb{Z}\right) }^{ \times } = \{ \bar{a} \in \mathbb{Z}/n\mathbb{Z} \mid \left( {a, n}\right) = 1\} \) . | It is easy to see that if any representative of \( \bar{a} \) is relatively prime to \( n \) then all representatives are relatively prime to \( n \) so that the set on the right in the proposition is well defined. | No |
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