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Proposition 10.28.\n\n(a) For lazy random walk on a simple graph with \( m \) edges and \( n \) vertices,\n\n\[ \n{t}_{\text{hit }} \leq {4nm} \leq 2{n}^{3}, \]\n\nand\n\n\[ \n{t}_{\operatorname{mix}}^{\left( \infty \right) } \leq {16nm} + 1 \leq 8{n}^{3}\text{, so}{t}_{\operatorname{mix}} \leq {8nm} + 1 \leq 4{n}^{3}\... | Proof. Since \( {t}_{\text{hit }} \leq \mathop{\max }\limits_{{a, b}}{t}_{a \leftrightarrow b} \), this result follows from Proposition 10.16 together with Theorem 10.22. (The extra factor of 2 comes from the laziness of the walk.) | No |
Proposition 10.29. Let \( H \) be the graph obtained by gluing together two copies of \( G \) at the vertex \( {v}_{ \star } \) as defined above. Let \( {\tau }_{\text{couple }}^{G} \) be the time for a coupling of two random walks on \( G \) to meet. Then there is a coupling of two random walks on \( H \) which has a ... | Outline of PROOF. Given a starting point in \( H \), a random walk in \( G \) can be lifted to a random walk in \( H \) in a unique way. (At \( {v}_{ \star } \), the particle moves to each copy with equal probability.) Applying this lifting to the given coupling in \( G \) yields a coupling in \( H \) where at time \( ... | Yes |
Proposition 11.4. Let \( A \subset \mathcal{X} \) . Set \( {t}_{\min }^{A} = \mathop{\min }\limits_{{a, b \in A, a \neq b}}{\mathbf{E}}_{a}\left( {\tau }_{b}\right) \) . Then\n\n\[ \n{t}_{\text{cov }} \geq \mathop{\max }\limits_{{A \subseteq \mathcal{X}}}{t}_{\min }^{A}\left( {1 + \frac{1}{2} + \cdots + \frac{1}{\left|... | Proof. Fix an initial state \( x \in A \) and let \( \sigma \) be a uniform random permutation of the elements of \( A \), chosen independently of the chain trajectory. Let \( {T}_{k} \) be the first time at which all of \( \sigma \left( 1\right) ,\sigma \left( 2\right) ,\ldots ,\sigma \left( k\right) \) have been visi... | Yes |
Proposition 11.9. The cover time satisfies\n\n\[ \n{t}_{\text{cov }} \geq \left( {\log 2}\right) k{2}^{k}\left( {1 + o\left( 1\right) }\right) .\n\] | Proof. Fix \( j = \left\lceil {{\log }_{2}k}\right\rceil \) and let \( A \subseteq \{ 0,1{\} }^{k} \) consist of those bitstrings that end with \( j \) zeroes followed by a 1 . Fix \( a, b \in A \), where \( a \neq b \) . By Lemma 11.8, we have\n\n\[ \n{\mathbf{E}}_{a}{\tau }_{b}^{ + } \geq \left( {1 - \theta }\right) ... | Yes |
Proposition 11.10. The cover time satisfies\n\n\[ \n{t}_{\text{cov }} \leq \left( {\log 2}\right) k{2}^{k}\left( {1 + o\left( 1\right) }\right) .\n\] | Proof. We partition the state space \( \{ 0,1{\} }^{k} \) into two sets. Fix \( j = \left\lceil {{\log }_{2}k}\right\rceil \) and let \( B \) be the set of all strings \( b \in \{ 0,1{\} }^{k} \) with the following property: any bitstring that is both a suffix and a prefix of \( b \) must have length less than \( k - j... | Yes |
Theorem 13.10 (Sinclair and Jerrum (1989), Lawler and Sokal (1988)). Let \( {\lambda }_{2} \) be the second largest eigenvalue of a reversible transition matrix \( P \), and let \( \gamma = 1 - {\lambda }_{2} \) . Then\n\n\[\n\frac{{\Phi }_{ \star }^{2}}{2} \leq \gamma \leq 2{\Phi }_{ \star }\n\]\n\n(13.6) | Proof of the upper bound in Theorem 13.10. By Lemmas 13.7 and 13.6,\n\n\[ \n\gamma = \mathop{\min }\limits_{\substack{{f \neq 0} \\ {{E}_{\pi }\left( f\right) = 0} }}\frac{\mathop{\sum }\limits_{{x, y \in \mathcal{X}}}\pi \left( x\right) P\left( {x, y}\right) {\left\lbrack f\left( x\right) - f\left( y\right) \right\rbr... | Yes |
Corollary 13.21 (Method of Canonical Paths). Let \( P \) be a reversible and irreducible transition matrix with stationary distribution \( \pi \) . Suppose \( {\Gamma }_{xy} \) is a choice of \( E \) -path for each \( x \) and \( y \), and let\n\n\[ \nB = \mathop{\max }\limits_{{e \in E}}\frac{1}{Q\left( e\right) }\mat... | Proof. Let \( \widetilde{P}\left( {x, y}\right) = \pi \left( y\right) \), and observe that the stationary measure for \( \widetilde{P} \) is clearly \( \widetilde{\pi } = \pi \) . For \( f \in {\mathbb{R}}^{\mathcal{X}} \) such that \( 0 = {E}_{\pi }\left( f\right) = \langle f,\mathbf{1}{\rangle }_{\pi } \), \n\n\[ \n\... | Yes |
Proposition 13.35. When \( \left\{ {G}_{n}\right\} \) is a \( \left( {d,\alpha }\right) \) -expander family, the lazy random walks on \( \left\{ {G}_{n}\right\} \) satisfy \( {t}_{\operatorname{mix}}\left( {G}_{n}\right) = O\left( {\log \left| {V\left( {G}_{n}\right) }\right| }\right) \) . | Proof. Theorem 13.10 implies that for all \( {G}_{n} \) the spectral gap for the simple random walk satisfies \( \gamma \geq {\alpha }^{2}/2 \) . Since each \( {G}_{n} \) is regular, the stationary distribution of the lazy random walk is uniform, and Theorem 12.4 tells us that for the lazy walk \( {t}_{\operatorname{mi... | Yes |
Proposition 13.37. For the family \( \left\{ {G}_{n}\right\} \) of random multigraphs described in (13.38), \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\mathbf{P}\left\{ {{\Phi }_{ \star }\left( {G}_{n}\right) > {0.01}}\right\} = 1. \] | Proof. Assume that \( \delta < {0.03} \) . We first show that with probability tending to 1 as \( n \rightarrow \infty \), every subset of \( A \) of size \( k \leq n/2 \) has more than \( \left( {1 + \delta }\right) k \) neighbors. Note that every edge in \( {G}_{n} \) connects a vertex in \( A = \left\{ {{a}_{1},\ldo... | No |
Proposition 15.8. Let \( G = \left( {V, E}\right) \) have maximum degree \( \Delta \), where \( \left| V\right| = n \) , and let \( \widetilde{G} = \left( {V,\widetilde{E}}\right) \), where \( \widetilde{E} \subset E \) . Let \( r = \left| {E \smallsetminus \widetilde{E}}\right| \) . If \( \gamma \) is the spectral gap... | Proof. We have for any \( \sigma \in \{ - 1,1{\} }^{V} \), \n\n\[ \pi \left( \sigma \right) = \frac{{e}^{\beta \mathop{\sum }\limits_{{\{ v, w\} \in \widetilde{E}}}\sigma \left( v\right) \sigma \left( w\right) + \beta \mathop{\sum }\limits_{{\{ v, w\} \in E \smallsetminus \widetilde{E}}}\sigma \left( v\right) \sigma \l... | Yes |
Theorem 17.6 (Optional Stopping Theorem, Version 1). If \( \left( {M}_{t}\right) \) is a martingale with respect to the filtration \( \left\{ {\mathcal{F}}_{t}\right\} \) and \( \tau \) is a stopping time for \( \left\{ {\mathcal{F}}_{t}\right\} \), then \( \left( {M}_{t \land \tau }\right) \) is a martingale with resp... | Proof of Theorem 17.6. If \( {A}_{t} \mathrel{\text{:=}} {\mathbf{1}}_{\{ \tau \geq t\} } \), then\n\n\[ \n{A}_{t} = 1 - {\mathbf{1}}_{\{ \tau \leq t - 1\} } \in {\mathcal{F}}_{t - 1} \n\] \n\nwhence \( \left( {A}_{t}\right) \) is previsible. By Theorem 17.4, \n\n\[ \n\mathop{\sum }\limits_{{s = 1}}^{t}{A}_{s}\left( {{... | Yes |
Lemma 17.15. For \( \alpha \in \left\lbrack {0,1/2}\right\rbrack \) , \n\n\[ \n\frac{\sqrt{1 + {2\alpha }} + \sqrt{1 - {2\alpha }}}{2} \leq \sqrt{1 - {\alpha }^{2}} \leq 1 - \frac{{\alpha }^{2}}{2}. \n\] | Squaring proves the right-hand inequality, and reduces the left-hand inequality to \n\n\[ \n\sqrt{1 - 4{\alpha }^{2}} \leq 1 - 2{\alpha }^{2} \n\] \n\nwhich is the right-hand inequality with \( {2\alpha } \) replacing \( \alpha \) . | No |
Let \( \left( {M}_{t}\right) \) be a non-negative martingale with respect to \( \left\{ {\mathcal{F}}_{t}\right\} \), and define\n\n\[ \n{T}_{h} \mathrel{\text{:=}} \min \left\{ {t \geq 0 : {M}_{t} = 0\text{ or }{M}_{t} \geq h}\right\} .\n\]\n\nAssume that\n\n(i) \( \operatorname{Var}\left( {{M}_{t + 1} \mid {\mathcal{... | Proof. For \( h \geq {M}_{0} \), we have that \( \{ \tau \geq t\} \subseteq \left\{ {{T}_{h} \geq t}\right\} \cup \left\{ {{M}_{{T}_{h}} \geq h}\right\} \), whence\n\n\[ \n\mathbf{P}\{ \tau \geq t\} \leq \mathbf{P}\left\{ {{T}_{h} \geq t}\right\} + \mathbf{P}\left\{ {{M}_{{T}_{h}} \geq h}\right\}\n\]\n\n(17.34)\n\nWe f... | Yes |
Proposition 17.20. Let \( {\left( {Z}_{t}\right) }_{t \geq 0} \) be a non-negative supermartingale with respect to \( \left\{ {\mathcal{F}}_{t}\right\} \), and let \( \tau \) be a stopping time for \( \left\{ {\mathcal{F}}_{t}\right\} \) . Suppose that\n\n(i) \( {Z}_{0} = k \) ,\n\n(ii) there exists \( B \) such that \... | The proof follows the same outline as the proof of Proposition 17.19 and is left to the reader in Exercise 17.4. | No |
Proposition 18.4. For a sequence of irreducible aperiodic reversible Markov chains with relaxation times \( \left\{ {t}_{\text{rel }}^{\left( n\right) }\right\} \) and mixing times \( \left\{ {t}_{\text{mix }}^{\left( n\right) }\right\} \), if there is a pre-cutoff, then \( {t}_{\text{mix }}^{\left( n\right) }/\left( {... | Proof. If \( {t}_{\text{mix }}^{\left( n\right) }/\left( {{t}_{\text{rel }}^{\left( n\right) } - 1}\right) \) does not tend to infinity, then there is an infinite set of integers \( J \) and a constant \( {c}_{1} > 0 \) such that \( \left( {{t}_{\text{rel }}^{\left( n\right) } - 1}\right) /{t}_{\text{mix }}^{\left( n\r... | Yes |
Proposition 21.3. Suppose that \( P \) is the transition matrix of an irreducible Markov chain \( \left( {X}_{t}\right) \) . Define \( G\left( {x, y}\right) \mathrel{\text{:=}} {\mathbf{E}}_{x}\left( {\mathop{\sum }\limits_{{t = 0}}^{\infty }{\mathbf{1}}_{\left\{ {X}_{t} = y\right\} }}\right) = \mathop{\sum }\limits_{{... | Proof. (i) \( \Leftrightarrow \) (iii). Every time the chain visits \( x \), it has the same probability of eventually returning to \( x \), independent of the past. Thus the number of visits to \( x \) is a geometric random variable with success probability \( 1 - {\mathbf{P}}_{x}\left\{ {{\tau }_{x}^{ + } < \infty }\... | Yes |
Proposition 21.6. Let \( \langle G,\{ c\left( e\right) \} \rangle \) be a network. The following are equivalent:\n\n(i) The weighted random walk on the network is transient.\n\n(ii) There is some node a with \( \mathcal{C}\left( {a \leftrightarrow \infty }\right) > 0 \) . (Equivalently, \( \mathcal{R}\left( {a \leftrig... | Proof. That (i) and (ii) are equivalent follows from (21.2), and (21.3) implies the equivalence of (ii) and (iii). | No |
Proposition 21.7 (Nash-Williams). If there exist disjoint edge-cutsets \( \left\{ {\Pi }_{n}\right\} \) that separate a from \( \infty \) and satisfy\n\n\[ \mathop{\sum }\limits_{n}{\left( \mathop{\sum }\limits_{{e \in {\Pi }_{n}}}c\left( e\right) \right) }^{-1} = \infty \]\n\n(21.4)\n\nthen the weighted random walk on... | Proof. Recall the definition of \( {z}_{n} \) given in the beginning of this section. The assumption (21.4) implies that \( \mathcal{R}\left( {a \leftrightarrow {z}_{n}}\right) \rightarrow \infty \) . Consequently, by Proposition 9.5, \( {\mathbf{P}}_{a}\left\{ {{\tau }_{{z}_{n}} < {\tau }_{a}^{ + }}\right\} \rightarro... | Yes |
Proposition 21.15. Let \( P \) be irreducible and suppose the Markov chain with transition matrix \( P \) is recurrent. Let \( \pi \) and \( \mu \) be two measures satisfying \( \pi = {\pi P} \) and \( \mu = {\mu P} \) . Then \( \mu = {c\pi } \) for some constant \( c \) . | Proof. Let \( h = \mu /\pi \) . Then \( h \) is harmonic for \( \widehat{P} \), the time-reversal of \( P \) . Since \( {\widehat{P}}^{t}\left( {x, x}\right) = {P}^{t}\left( {x, x}\right) \) for all \( t \geq 1 \), the \( \widehat{P} \) -chain is also recurrent. The conclusion follows from the fact that all such functi... | Yes |
Proposition 22.7. The following conditions are equivalent:\n\n(i) \( P \) is a monotone chain.\n\n(ii) If \( \mu \preccurlyeq \nu \), then \( {\mu P} \preccurlyeq {\nu P} \) .\n\n(iii) For every pair of comparable states \( x, y \in \mathcal{X} \) with \( x \preccurlyeq y \), there exists a coupling \( \left( {X, Y}\ri... | Proof. (i) \( \Rightarrow \) (ii). Let \( f \) be an increasing function. Then \( {Pf} \) is increasing, so\n\n\[ \left( {\mu P}\right) f = \mu \left( {Pf}\right) \leq \nu \left( {Pf}\right) = \left( {\nu P}\right) f. \]\n\n(ii) \( \Rightarrow \) (iii). If \( x \preccurlyeq y \), then \( {\delta }_{x}P \preccurlyeq {\d... | Yes |
Proposition 23.1. Let \( {G}_{n} \) denote the \( n \) -path with loops at 1 and \( n \) . Let \( {t}_{\text{rel }} \) be the relaxation time for the interchange process on the \( n \) -path, and let \( {t}_{\text{rel }} \) (single) be the relaxation time for the random walk on \( {G}_{n} \) with delay probability \( 1... | Proof. Let \( \varphi \left( j\right) = \cos \left( {\pi \left( {{2j} - 1}\right) /{2n}}\right) \) for \( j = 1,2,\ldots, n \) be the second eigenfunction for the simple random walk on \( {G}_{n} \), with eigenvalue \( {\lambda }_{2} = \cos \left( {\pi /n}\right) \) . (See (12.21).) If \( {\sigma }_{1} \) is the permut... | No |
Proposition 24.4. For all chains we have that\n\n\[ \n{t}_{\mathrm{G}} \leq 4{t}_{\text{stop }} + 1 \n\] | Proof of Proposition 24.4. We fix \( x \) . Let \( \tau \) be a stationary time, so that the distribution of \( {X}_{\tau } \) when started from \( x \) is \( \pi \) . Then \( \tau + s \) is also a stationary time for all \( s \geq 1 \) . Hence, if \( {Z}_{t} \) is a geometric random variable independent of \( \tau \) ... | Yes |
Proposition 24.8. For reversible chains,\n\n(i)\n\n\[ \n{t}_{\text{stop }} \leq 8{t}_{\mathrm{G}} \n\]\n\n(ii)\n\n\[ \n{t}_{\text{stop }} \leq 4\left( {{t}_{\text{Ces }} + 1}\right) \n\] | Proof. Consider the chain with transition matrix \( R\left( {x, y}\right) = {\mathbf{P}}_{x}\left\{ {{X}_{G} = y}\right\} \) , where \( G \) is geometric with mean \( t \) . Set \( t = {t}_{\mathrm{G}} \) so that\n\n\[ \n{\begin{Vmatrix}{\mathbf{P}}_{x}\left\{ {X}_{G} = \cdot \right\} - \pi \end{Vmatrix}}_{\mathrm{{TV}... | Yes |
Proposition 24.22. Given an irreducible Markov chain \( \left( {X}_{t}\right) \) with finite state space \( \mathcal{X} \) and stationary distribution \( \pi \), let \( A, C \subseteq \mathcal{X} \) with \( A \cap C = \varnothing \) . Then\n\n\[ \pi \left( A\right) \leq \frac{{d}^{ + }\left( {A, C}\right) }{{d}^{ + }\l... | Proof. Define\n\n\[ \tau = \min \left\{ {t > {\tau }_{C} : {X}_{t} \in A}\right\} .\n\]\n\nConsider a Markov chain on \( A \) defined as follows: for each \( x, y \in A \), let \( Q\left( {x, y}\right) = \) \( {\mathbf{P}}_{x}\left\{ {{X}_{\tau } = y}\right\} \) . Let \( \mu \) denote a stationary distribution of this ... | Yes |
Proposition 24.23. If \( P \) is an irreducible transition matrix, then for any positive eigenvalue \( \lambda > 0 \) , \[ 1 - \lambda \geq \frac{1}{{t}_{\mathrm{G}} + 1} \] In particular, for reversible lazy chains, \[ {t}_{\text{rel }} \leq {t}_{\mathrm{G}} + 1 \] | Proof. Let \( K\left( {x, y}\right) = {\mathbf{P}}_{x}\left\{ {{X}_{Z} = y}\right\} \), where \( Z \) is geometric with mean \( t = {t}_{\mathrm{G}} \) . Any eigenvalue \( \lambda \) of \( P \) gives the eigenvalue for the \( K \) -chain \[ \widetilde{\lambda } = \mathop{\sum }\limits_{{k = 1}}^{\infty }{\lambda }^{k}{... | Yes |
Proposition 24.24. Let \( P \) be an irreducible transition matrix on the state space \( \mathcal{X} \) and let \( \widetilde{P}\left( {x, y}\right) = \theta \left( x\right) P\left( {x, y}\right) + \left( {1 - \theta \left( x\right) }\right) {\delta }_{x}\left( y\right) \) . Assume that \( \theta \left( x\right) \geq {... | Proof. Note that one can construct the \( \widetilde{P} \) -chain from the \( P \) -chain \( \left( {X}_{t}\right) \) by repeating the state \( {X}_{t} \) for \( {D}_{t} \) steps, where the conditional distribution of \( {D}_{t} \), given \( {X}_{t} = x \), is geometric \( \left( {\theta \left( x\right) }\right) \) .\n... | Yes |
Proposition 2.2 (Characterization of Granger-Noncausality)\n\nLet \( {y}_{t} \) be a VAR process as in (2.3.4)/(2.3.5) with canonical MA operator \( \Phi \left( z\right) \) . Then\n\n\[ \n{z}_{t}\left( {1 \mid \left\{ {{y}_{s} \mid s \leq t}\right\} }\right) = {z}_{t}\left( {1 \mid \left\{ {{z}_{s} \mid s \leq t}\right... | Because we have just used the MA representation (2.3.4) and not its finite order VAR form, the proposition is not only valid for VAR processes but more generally for processes having a canonical MA representation such as (2.3.4). From (2.2.10) it is obvious that equality of the 1-step predictors implies equality of the... | Yes |
Proposition 2.3 (Characterization of Instantaneous Causality)\n\nLet \( {y}_{t} \) be as in (2.3.5)/(2.3.15) with nonsingular innovation covariance matrix \( {\sum }_{u} \) . Then there is no instantaneous causality between \( {z}_{t} \) and \( {x}_{t} \) if and only if\n\n\[ E\left( {{u}_{1t}{u}_{2t}^{\prime }}\right)... | This proposition provides a condition for instantaneous causality which is easy to check if the process is given in MA or VAR form. For instance, for the investment/income/consumption system with white noise covariance matrix (2.1.33),\n\n\[ {\sum }_{u} = \left\lbrack \begin{matrix} {2.25} & 0 & 0 \\ 0 & {1.0} & {.5} \... | Yes |
Proposition 2.4 (Zero Impulse Responses)\n\nIf \( {y}_{t} \) is a \( K \) -dimensional stable \( \operatorname{VAR}\left( p\right) \) process, then, for \( j \neq k \) ,\n\n\[ \n{\phi }_{{jk}, i} = 0\;\text{ for }i = 1,2,\ldots \n\]\n\nis equivalent to\n\n\[ \n{\phi }_{{jk}, i} = 0\;\text{ for }i = 1,\ldots, p\left( {K... | Proof of Proposition 2.4:\n\nReturning to the lag operator notation of Section 2.1.2, we have\n\n\[ \n\Phi \left( L\right) = {\left( {\phi }_{jk}\left( L\right) \right) }_{j, k} = A{\left( L\right) }^{-1} = A{\left( L\right) }^{adj}/\det \left( {A\left( L\right) }\right) ,\n\]\n\nwhere \( A{\left( L\right) }^{adj} = {\... | Yes |
Proposition 2.5 (Zero Orthogonalized Impulse Responses)\n\nIf \( {y}_{t} \) is a \( K \) -dimensional stable \( \operatorname{VAR}\left( p\right) \) process, then, for \( j \neq k \) ,\n\n\[ \n{\theta }_{{jk}, i} = 0\;\text{ for }i = 0,1,2,\ldots \n\]\n\nis equivalent to\n\n\[ \n{\theta }_{{jk}, i} = 0\;\text{ for }i =... | The proof of this result is analogous to that of Proposition 2.4 and is left as an exercise (see Problem 2.2). | No |
In the United States of Wonderland the growth rates of income (GNP) and the money stock (M2) as well as an interest rate (IR) are related as in the following \( \operatorname{VAR}\left( 2\right) \) model:\n\n\[ \left\lbrack \begin{matrix} {\mathrm{{GNP}}}_{t} \\ {\mathrm{{M2}}}_{t} \\ {\mathrm{{IR}}}_{t} \end{matrix}\r... | (a) Show that the process \( {y}_{t} = {\left( {\mathrm{{GNP}}}_{t},{\mathrm{{M2}}}_{t},{\mathrm{{IR}}}_{t}\right) }^{\prime } \) is stable. | Yes |
Proposition 3.1 (Asymptotic Properties of the LS Estimator)\n\nLet \( {y}_{t} \) be a stable, \( K \) -dimensional \( \operatorname{VAR}\left( p\right) \) process as in (3.1.1) with standard white noise residuals, \( \widehat{B} = Y{Z}^{\prime }{\left( Z{Z}^{\prime }\right) }^{-1} \) is the LS estimator of the VAR coef... | Proof: Using (3.2.10),\n\n\[ \operatorname{plim}\left( {\widehat{B} - B}\right) = \operatorname{plim}\left( \frac{U{Z}^{\prime }}{T}\right) \operatorname{plim}{\left( \frac{Z{Z}^{\prime }}{T}\right) }^{-1} = 0 \]\n\nby Lemma 3.1, because (3.2.14) implies plim \( U{Z}^{\prime }/T = 0 \) . Thus, the consistency of \( \wi... | No |
Proposition 3.2 (Asymptotic Properties of the White Noise Covariance Matrix Estimators)\n\nLet \( {y}_{t} \) be a stable, \( K \) -dimensional VAR \( \\left( p\\right) \) process as in (3.1.1) with standard white noise innovations and let \( \\bar{B} \) be an estimator of the VAR coefficients \( B \) so that \( \\sqrt{... | Proof:\n\n\[ \n\\frac{1}{T}\\left( {Y - \\bar{B}Z}\\right) {\\left( Y - \\bar{B}Z\\right) }^{\\prime } = \\left( {B - \\bar{B}}\\right) \\left( \\frac{Z{Z}^{\\prime }}{T}\\right) {\\left( B - \\bar{B}\\right) }^{\\prime } + \\left( {B - \\bar{B}}\\right) \\frac{Z{U}^{\\prime }}{T}\n\]\n\n\[ \n+ \\frac{U{Z}^{\\prime }}{... | Yes |
Proposition 3.3 (Asymptotic Properties of the Sample Mean) If the \( \operatorname{VAR}\left( p\right) \) process \( {y}_{t} \) given in (3.3.1) is stable and \( {u}_{t} \) is standard white noise, then\n\n\[ \sqrt{T}\left( {\bar{y} - \mu }\right) \overset{d}{ \rightarrow }\mathcal{N}\left( {0,{\sum }_{\bar{y}}}\right)... | The proposition follows from (3.3.10), (3.3.11), and Proposition C. 15 of Appendix C. The limiting distribution in (3.3.11) holds even in small samples for Gaussian white noise \( {u}_{t} \) . | Yes |
Proposition 3.4 (Asymptotic Properties of ML Estimators)\n\nLet \( {y}_{t} \) be a stationary, stable Gaussian \( \operatorname{VAR}\left( p\right) \) process as in (3.3.1). Then the ML estimators \( \widetilde{\mu },\widetilde{\mathbf{\alpha }} \), and \( \widetilde{\mathbf{\sigma }} = \operatorname{vech}\left( {\wide... | The covariance matrices are\n\n\[ \n{\sum }_{\widetilde{\mu }} = {\left( {I}_{K} - \mathop{\sum }\limits_{i}{A}_{i}\right) }^{-1}{\sum }_{u}{\left( {I}_{K} - \mathop{\sum }\limits_{i}{A}_{i}^{\prime }\right) }^{-1}, \n\]\n\n\[ \n{\sum }_{\widetilde{\mathbf{\alpha }}} = {\Gamma }_{Y}{\left( 0\right) }^{-1} \otimes {\sum... | Yes |
Proposition 3.5 (Asymptotic Distribution of the Wald Statistic)\n\nSuppose (3.6.2) holds. Furthermore, \( \operatorname{plim}\left( {Z{Z}^{\prime }/T}\right) = \Gamma \), plim \( {\widehat{\sum }}_{u} = {\sum }_{u} \) are both nonsingular and \( {H}_{0} : C\mathbf{\beta } = c \) is true, with \( C \) being an \( \left(... | In practice, it may be useful to make adjustments to the statistic or the critical values of the test to compensate for the fact that the matrix \( {\Gamma }^{-1} \otimes {\sum }_{u} \) is unknown and has been replaced by an estimator. Working in that direction, we note that\n\n\[ \n{NF}\left( {N, T}\right) \underset{T... | Yes |
Proposition 3.6 (Asymptotic Distributions of Impulse Responses)\n\nSuppose\n\n\[ \sqrt{T}\left\lbrack \begin{matrix} \widehat{\mathbf{\alpha }} - \mathbf{\alpha } \\ \widehat{\mathbf{\sigma }} - \mathbf{\sigma } \end{matrix}\right\rbrack \overset{d}{ \rightarrow }\mathcal{N}\left( {0,\left\lbrack \begin{matrix} {\sum }... | \[ \sqrt{T}\operatorname{vec}\left( {{\widehat{\Psi }}_{n} - {\Psi }_{n}}\right) \overset{d}{ \rightarrow }\mathcal{N}\left( {0,{F}_{n}{\sum }_{\widehat{\mathbf{\alpha }}}{F}_{n}^{\prime }}\right) ,\;n = 1,2,\ldots ,\n\]\n\n(3.7.6)\n\nwhere \( {F}_{n} \mathrel{\text{:=}} {G}_{1} + \cdots + {G}_{n} \) . | Yes |
Show that Equation (3.5.10) holds. | Hint: Define\n\n\[ \n{Z}_{t}\left( h\right) \mathrel{\text{:=}} \left\lbrack \begin{matrix} 1 \\ {y}_{t}\left( h\right) \\ \vdots \\ {y}_{t}\left( {h - p + 1}\right) \end{matrix}\right\rbrack \n\]\n\nand show \( {Z}_{t}\left( h\right) = \mathbf{B}{Z}_{t}\left( {h - 1}\right) \) by induction. | No |
In the context of Section 3.5, suppose that \( {y}_{t} \) is a stable Gaussian \( \operatorname{VAR}\left( p\right) \) process which is estimated by ML in mean-adjusted form. Show that the forecast MSE correction term has the form\n\n\[ \Omega \left( h\right) = E\left( {\frac{\partial {y}_{t}\left( h\right) }{\partial ... | with\n\n\[ \frac{\partial {y}_{t}\left( h\right) }{\partial {\mu }^{\prime }} = {I}_{K} - J{\mathbf{A}}^{h}\left\lbrack \begin{matrix} {I}_{K} \\ \vdots \\ {I}_{K} \end{matrix}\right\rbrack \]\nand\n\n\[ \frac{\partial {y}_{t}\left( h\right) }{\partial {\mathbf{\alpha }}^{\prime }} = \mathop{\sum }\limits_{{i = 0}}^{{h... | Yes |
Proposition 4.1 (Asymptotic Distribution of the LR Statistic)\n\nLet \( {y}_{t} \) be a stationary, stable \( \operatorname{VAR}\left( p\right) \) process as in (4.1.1) with standard white noise \( {u}_{t} \) (see Definition 3.1). Suppose the true parameter vector \( \mathbf{\beta } \) satisfies linear constraints \( C... | \[ {\lambda }_{LR} = T\left( {\ln \left| {\widetilde{\sum }}_{u}^{r}\right| - \ln \left| {\widetilde{\sum }}_{u}\right| }\right) \] \n\n\( \left( {{4.2.13}\mathrm{a}}\right) \)\n\n\[ = {\left( {\widetilde{\mathbf{\beta }}}_{r} - \widetilde{\mathbf{\beta }}\right) }^{\prime }\left( {Z{Z}^{\prime } \otimes {\widetilde{\m... | Yes |
Under the conditions of Proposition 4.2, if \( M > p \) , \[ \mathop{\lim }\limits_{{T \rightarrow \infty }}\Pr \{ \widehat{p}\left( \mathrm{{AIC}}\right) < p\} = 0\;\text{ and }\;\lim \Pr \{ \widehat{p}\left( \mathrm{{AIC}}\right) > p\} > 0 \] and the same holds for \( \widehat{p}\left( \mathrm{{FPE}}\right) \) . | Proof: By (4.3.10) and Corollary 4.2.2, \[ \Pr \{ \widehat{p}\left( \mathrm{{AIC}}\right) < p\} \leq \Pr \{ \widehat{p}\left( \mathrm{{SC}}\right) < p\} \rightarrow 0. \] Because AIC is not consistent by Corollary 4.2.1, \( \lim \Pr \{ \widehat{p}\left( \mathrm{{AIC}}\right) = p\} \) 1. Hence (4.3.14) follows. The same... | No |
Proposition 4.4 (Asymptotic Distributions of White Noise Autocovariances and Autocorrelations)\n\nLet \( {u}_{t} \) be a \( K \) -dimensional identically distributed standard white noise process, that is, \( {u}_{t} \) and \( {u}_{s} \) have the same multivariate distribution with nonsingular covariance matrix \( {\sum... | Proof: The result (4.4.6) follows from an appropriate central limit theorem. The i.i.d. assumption for the \( {u}_{t} \) implies that\n\n\[ {w}_{t} = \operatorname{vec}\left( {{u}_{t}{u}_{t - 1}^{\prime },\ldots ,{u}_{t}{u}_{t - h}^{\prime }}\right) \]\n\nis a stationary white noise process with covariance matrix \( E\... | Yes |
Proposition 4.5 (Asymptotic Distributions of Residual Autocovariances)\n\nLet \( {y}_{t} \) be a stationary, stable, \( K \) -dimensional \( \operatorname{VAR}\left( p\right) \) process as in (4.1.1) with identically distributed standard white noise process \( {u}_{t} \) and let the coefficients be estimated by multiva... | Proof: Using Lemma 4.2, \( \sqrt{T}{\widehat{\mathbf{c}}}_{h} \) is known to have the same asymptotic distribution as\n\n\[ \sqrt{T}{\mathbf{c}}_{h} - \sqrt{T}G\operatorname{vec}\left( {\widehat{B} - B}\right) \]\n\n\[ = \left\lbrack {-{\widetilde{G}}^{\prime } \otimes {I}_{K} : I}\right\rbrack \left\lbrack \begin{matr... | Yes |
Proposition 4.6 (Asymptotic Distributions of Residual Autocorrelations)\n\nLet \( D \) be the \( \\left( {K \\times K}\\right) \) diagonal matrix with the square roots of \( {\\sum }_{u} \) on the diagonal and define \( {G}_{0} \\mathrel{\\text{:=}} \\widetilde{G}\\left( {{I}_{h} \\otimes {D}^{-1}}\\right) \) . Then, u... | Proof: Noting that \n\n\[ \n{\\widehat{\\mathbf{r}}}_{h} = \\operatorname{vec}\\left( {\\widehat{\\mathbf{R}}}_{h}\\right) = \\operatorname{vec}\\left\\lbrack {{\\widehat{D}}^{-1}{\\widehat{\\mathbf{C}}}_{h}\\left( {{I}_{h} \\otimes {\\widehat{D}}^{-1}}\\right) }\\right\\rbrack \n\] \n\n\[ \n= \\left( {{I}_{h} \\otimes... | Yes |
Lemma 4.4\n\nLet \( {z}_{t} = {\left( {z}_{1t},\ldots ,{z}_{Kt}\right) }^{\prime } \) be a Gaussian white noise process with mean \( {\mu }_{z} \) and covariance matrix \( {I}_{K} \), i.e., \( {z}_{t} \sim \mathcal{N}\left( {{\mu }_{z},{I}_{K}}\right) \). Furthermore, let\n\n\[ \bar{z} = {\left( {\bar{z}}_{1},\ldots ,{... | The proof of this lemma is easily obtained, for instance, from results of Gasser (1975). | No |
Under the conditions of Proposition 4.10, \[ \operatorname{plim}\left\lbrack {\frac{1}{\sqrt{T}}\mathop{\sum }\limits_{{t = 1}}^{T}{\widehat{u}}_{t} \otimes {\widehat{u}}_{t} \otimes {\widehat{u}}_{t} - \frac{1}{\sqrt{T}}\mathop{\sum }\limits_{{t = 1}}^{T}\left( {{u}_{t} - \bar{u}}\right) \otimes \left( {{u}_{t} - \bar... | Proof: A proof for the special case of a \( \operatorname{VAR}\left( 1\right) \) process \( {y}_{t} \) is given and the generalization is left to the reader. Also, we just show the first result. The second one follows with analogous arguments. For the special VAR(1) case, \[ {\widehat{u}}_{t} = \left( {{y}_{t} - \bar{y... | No |
Prove Lemma 4.1. | [Hint: Suppose \( k < n \) . Then\n\n\[ \n{c}_{n} + {a}_{n} = {c}_{n} + \left( {{a}_{n} - {a}_{n - 1}}\right) + \cdots + \left( {{a}_{k + 1} - {a}_{k}}\right) + {a}_{k} \]\n\n\[ \n> {c}_{n} + \left( {{b}_{n} - {b}_{n - 1}}\right) + \cdots + \left( {{b}_{k + 1} - {b}_{k}}\right) + {a}_{k} \]\n\n\[ \n\geq {c}_{k} + {b}_{... | No |
Show (4.4.14) and (4.4.15). | \[ \operatorname{vec}\left( {\sqrt{T}{UF}\left\lbrack {{I}_{h} \otimes {Z}^{\prime }{\left( \widehat{B} - B\right) }^{\prime }}\right\rbrack /T}\right) \]\n\[ = \sqrt{T}\operatorname{vec}\left\lbrack {\frac{1}{T}{UF}\left( {{I}_{h} \otimes {Z}^{\prime }}\right) \left( {{I}_{h} \otimes {\left( \widehat{B} - B\right) }^{... | Yes |
Proposition 5.1 (Asymptotic Properties of the GLS Estimator)\n\nSuppose the conditions of Proposition 3.1 are satisfied, that is, \( {y}_{t} \) is a \( K \) - dimensional stable, stationary \( \operatorname{VAR}\left( p\right) \) process and \( {u}_{t} \) is independent white noise with bounded fourth moments. If \( \m... | Proof: Under the conditions of the proposition, \( \operatorname{plim}\left( {Z{Z}^{\prime }/T}\right) = \Gamma \) and\n\n\[ \frac{1}{T}\operatorname{vec}\left( {U{Z}^{\prime }}\right) \overset{d}{ \rightarrow }\mathcal{N}\left( {0,\Gamma \otimes {\sum }_{u}}\right) \]\n\n(see Lemma 3.1). Hence, by results stated in Ap... | Yes |
Under the conditions of Proposition 5.1, if plim \( {\bar{\sum }}_{u} = {\sum }_{u} \), the EGLS estimator \( \widehat{\widehat{\gamma }} \) in (5.2.9) is asymptotically equivalent to the GLS estimator \( \widehat{\gamma } \) in (5.2.6), that is, plim \( \widehat{\widehat{\gamma }} = \gamma \) and | \[ \sqrt{T}\left( {\widehat{\widehat{\gamma }} - \mathbf{\gamma }}\right) \overset{d}{ \rightarrow }\mathcal{N}\left( {0,{\left\lbrack {R}^{\prime }\left( \Gamma \otimes {\sum }_{u}^{-1}\right) R\right\rbrack }^{-1}}\right) . \] \( \left( {5.2.10}\right) \) | Yes |
Proposition 5.4 (Asymptotic Properties of the White Noise Covariance Estimator)\n\nUnder the conditions of Proposition 5.1, \( {\sum }_{u} \) is consistent and\n\n\[ \operatorname{plim}\sqrt{T}\left( {{\breve{\sum }}_{u} - U{U}^{\prime }/T}\right) = 0. \] | In (5.2.15), \( T \) may be replaced by \( T - {Kp} - 1 \) without affecting the consistency of the covariance matrix estimator. However, there is little justification for subtracting \( {Kp} + 1 \) from \( T \) in the present situation because, due to zero restrictions, some or all of the \( K \) equations of the syst... | No |
Proposition 5.5 (Asymptotic Properties of the Restricted ML Estimators) Let \( {y}_{t} \) be a Gaussian stable \( K \) -dimensional \( \operatorname{VAR}\left( p\right) \) process as in (5.1.1) and \( \mathbf{\beta } = \operatorname{vec}\left( B\right) = R\mathbf{\gamma } + r \) as in (5.2.2). Then the ML estimators \(... | \[ \sqrt{T}\left\lbrack \begin{matrix} \widetilde{\mathbf{\beta }} - \mathbf{\beta } \\ \widetilde{\mathbf{\sigma }} - \mathbf{\sigma } \end{matrix}\right\rbrack \overset{d}{ \rightarrow }\mathcal{N}\left( {0,\left\lbrack \begin{matrix} R{\left\lbrack {R}^{\prime }\left( \Gamma \otimes {\sum }_{u}^{-1}\right) R\right\r... | Yes |
Proposition 5.7 (Asymptotic Distributions of Residual Autocovariances and Autocorrelations)\n\nSuppose \( {y}_{t} \) is a stable, stationary, \( K \) -dimensional \( \operatorname{VAR}\left( p\right) \) process with identically distributed standard white noise \( {u}_{t} \) and the parameter vector \( \mathbf{\beta } \... | Proof: The proof is similar to that of Propositions 4.5 and 4.6. Defining \( \widetilde{G} \) as in Lemma 4.2, the lemma implies that \( \sqrt{T}{\widehat{\mathbf{c}}}_{h} \) is known to have the same asymptotic distribution as\n\n\[ \sqrt{T}{\mathbf{c}}_{h} - \sqrt{T}G\operatorname{vec}\left( {\widehat{\widehat{B}} - ... | Yes |
Proposition 5.8 (Approximate Distribution of the Portmanteau Statistic) Suppose the conditions of Proposition 5.7 are satisfied and there are no restrictions linking the intercept terms to the \( {A}_{1},\ldots ,{A}_{p} \) coefficients, that is,\n\n\[ R = \left\lbrack \begin{matrix} {R}_{\left( 1\right) } & 0 \\ 0 & {R... | Proof: Under the conditions of the proposition, the covariance matrix of the asymptotic distribution in (5.2.32) is\n\n\[ {\sum }_{\mathbf{c}}^{r}\left( h\right) = {I}_{h} \otimes {\sum }_{u} \otimes {\sum }_{u} - G{R}_{\left( 2\right) }{\left\{ {R}_{\left( 2\right) }^{\prime }\left\lbrack {\Gamma }_{Y}\left( 0\right) ... | Yes |
Proposition 5.9 (Asymptotic Distribution of LM Statistic for Residual Autocorrelation of Restricted VAR)\n\nUnder the conditions of Proposition 5.7,\n\n\[ {\lambda }_{LM}\left( h\right) \overset{d}{ \rightarrow }{\chi }^{2}\left( {h{K}^{2}}\right) \] | Notice that unlike for the portmanteau test, the asymptotic distribution of the LM statistic is identical to that obtained for unrestricted VARs in Proposition 4.8. However, \( {\lambda }_{LM}\left( h\right) \) is in general not exactly an LM statistic because the restricted estimator \( \widehat{\gamma } \) is not ide... | No |
Consider the recursive system of Section 5.2.5,\n\n\[ \n{y}_{t} = \eta + {A}_{0}^{ * }{y}_{t} + \cdots + {A}_{p}^{ * }{y}_{t - p} + {\varepsilon }_{t}\n\]\n\nwhere \( {\varepsilon }_{t} \) has a diagonal covariance matrix \( {\sum }_{\varepsilon } \) . Show that \( \mathop{\sum }\limits_{t}{\varepsilon }_{t}^{\prime }{... | Hint: Note that\n\n\[ \n\mathop{\sum }\limits_{{t = 1}}^{T}{\varepsilon }_{t}^{\prime }{\sum }_{\varepsilon }^{-1}{\varepsilon }_{t} = \mathop{\sum }\limits_{{k = 1}}^{K}\mathop{\sum }\limits_{{t = 1}}^{T}{\varepsilon }_{kt}^{2}/{\sigma }_{{\varepsilon }_{k}}^{2}\n\]\n\nand consider the partial derivatives with respect... | No |
(1) \( {T}^{-1}\mathop{\sum }\limits_{{t = 1}}^{T}{z}_{t - 1}^{\left( 1\right) }{z}_{t - 1}^{\left( 1\right) \prime } = {T}^{-1}\mathop{\sum }\limits_{{t = 1}}^{T}{\beta }^{\prime }{y}_{t - 1}{y}_{t - 1}^{\prime }\beta \overset{p}{ \rightarrow }{\Gamma }_{z}^{\left( 1\right) } \) . | The proof follows Ahn & Reinsel (1990). Lemma 7.1(1) is implied by a standard weak law of large numbers (see, e.g., Proposition C.12(7)) because \( {z}_{t - 1}^{\left( 1\right) } \) contains stationary components only. | Yes |
Result 1\n\nLet\n\n\[ D = \left\lbrack \begin{matrix} {T}^{1/2} & 0 \\ 0 & T \end{matrix}\right\rbrack \]\n\nThen\n\n\[ \operatorname{vec}\left\lbrack {Q\left( {\widehat{\mathbf{\Pi }} - \mathbf{\Pi }}\right) {Q}^{-1}D}\right\rbrack \]\n\n\[ \overset{d}{ \rightarrow }\left\lbrack \begin{array}{l} \mathcal{N}\left( {0,{... | Proof:\n\n\[ Q\left( {\widehat{\mathbf{\Pi }} - \mathbf{\Pi }}\right) {Q}^{-1}D \]\n\n\[ = \left\lbrack {{T}^{-1/2}\mathop{\sum }\limits_{{t = 1}}^{T}{v}_{t}{z}_{t - 1}^{\left( 1\right) \prime } : {T}^{-1}\mathop{\sum }\limits_{{t = 1}}^{T}{v}_{t}{z}_{t - 1}^{\left( 2\right) \prime }}\right\rbrack \]\n\n\[ \times D{\le... | Yes |
Proposition 7.1 (Asymptotic Properties of the LS Estimator for a VECM) Consider the VECM (7.2.1). The LS estimator given in (7.2.4) is consistent and | \[ \sqrt{T}\operatorname{vec}\left( {\left\lbrack {\widehat{\mathbf{\Pi }} : \widehat{\mathbf{\Gamma }}}\right\rbrack - \left\lbrack {\mathbf{\Pi } : \mathbf{\Gamma }}\right\rbrack }\right) \overset{d}{ \rightarrow }\mathcal{N}\left( {0,{\sum }_{\mathrm{{co}}}}\right) ,\] (7.2.6) where \[ {\sum }_{\mathrm{{co}}} = \lef... | Yes |
Consider the VECM (7.2.1) with cointegration matrix \( \beta \) normalized as in (7.1.10). Suppose \( \widehat{\alpha } \) and \( {\widehat{\sum }}_{u} \) are consistent estimators of \( \alpha \) and \( {\sum }_{u} \), respectively. Then the EGLS estimator of \( {\beta }_{\left( K - r\right) }^{\prime } \) given in (7... | \[ \left. {T\left( {{\widehat{\widehat{\mathbf{\beta }}}}_{\left( K - r\right) }^{\prime } - {\mathbf{\beta }}_{\left( K - r\right) }^{\prime }}\right) \overset{d}{ \rightarrow }{\left( {\int }_{0}^{1}{\mathbf{W}}_{K - r}^{\# }d{\mathbf{W}}_{r}^{\# \prime }\right) }^{\prime }\left( {{\int }_{0}^{1}{\mathbf{W}}_{K - r}^... | Yes |
Proposition 7.3 (ML Estimators of a VECM)\n\nLet \( M \mathrel{\text{:=}} {I}_{T} - \Delta {X}^{\prime }{\left( \Delta X\Delta {X}^{\prime }\right) }^{-1}{\Delta X},{R}_{0} \mathrel{\text{:=}} {\Delta YM} \) and \( {R}_{1} \mathrel{\text{:=}} {Y}_{-1}M \), as before, and define\n\n\[ \n{S}_{ij} \mathrel{\text{:=}} {R}_... | Proof: From Chapter 3, Section 3.4, it is known that for any fixed \( \alpha \) and \( \beta \) the maximum of \( \ln l \) is attained for\n\n\[ \n\widetilde{\mathbf{\Gamma }}\left( {\alpha {\beta }^{\prime }}\right) = \left( {{\Delta Y} - \alpha {\beta }^{\prime }{Y}_{-1}}\right) \Delta {X}^{\prime }{\left( \Delta X\D... | Yes |
Proposition 7.4 (Asymptotic Properties of the ML Estimators of a VECM) The ML estimators for the VECM (7.2.1) given in Proposition 7.3 have the following asymptotic properties: | \[ \sqrt{T}\operatorname{vec}\left( {\left\lbrack {\widetilde{\alpha }{\widetilde{\beta }}^{\prime } : \widetilde{\mathbf{\Gamma }}}\right\rbrack - \left\lbrack {\mathbf{\Pi } : \mathbf{\Gamma }}\right\rbrack }\right) \overset{d}{ \rightarrow }\mathcal{N}\left( {0,{\sum }_{\mathrm{{co}}}}\right) ,\] (7.2.21) where \( {... | Yes |
Proposition 7.6 (Asymptotic Properties of the Restricted EGLS Estimator) Suppose \( {y}_{t} \) is generated by the VECM (7.2.1) and \( \beta \) satisfies the restrictions in (7.3.1). Then \[ {\left\lbrack {\mathbf{R}}^{\prime }\left( {R}_{1}^{\left( 2\right) }{R}_{1}^{\left( 2\right) \prime } \otimes {\widehat{\mathbf{... | Thus, standard inference procedures can be based on the transformed estimator. It can also be shown that \( \widehat{\widehat{\gamma }} - \gamma = {O}_{p}\left( {T}^{-1}\right) \) . In other words, the estimator is superconsistent. Clearly, consistent estimators of \( \mathbf{\alpha } \) and \( {\sum }_{u} \) are readi... | No |
Show that, in the proof of Result 6 of Section 7.1,\n\n\[ \n{T}^{-1}\mathop{\sum }\limits_{{t = 1}}^{T}\left( {{u}_{t}^{ * } - {u}_{t}}\right) {y}_{t - 1}^{\left( 2\right) \prime } = {o}_{p}\left( 1\right) \n\] | (Hint: Use\n\n\[ \n\left. {{T}^{-1}\mathop{\sum }\limits_{{t = 1}}^{T}\left( {{u}_{t}^{ * } - {u}_{t}}\right) {y}_{t - 1}^{\left( 2\right) \prime } = \left( {\widehat{\alpha } - \alpha }\right) {T}^{-1}\mathop{\sum }\limits_{{t = 1}}^{T}{\beta }^{\prime }{y}_{t - 1}{y}_{t - 1}^{\left( 2\right) \prime }}\right) \n\] | No |
Prove that \( \sqrt{T}\left\lbrack {\widetilde{\mathbf{\alpha }}{\widetilde{\mathbf{\beta }}}^{\prime } - \widetilde{\mathbf{\alpha }}\left( \mathbf{\beta }\right) {\mathbf{\beta }}^{\prime }}\right\rbrack = {o}_{p}\left( 1\right) \) holds in the proof of Lemma 7.3. | \[ \left. {\widetilde{\alpha }{\widetilde{\beta }}^{\prime } - \widetilde{\alpha }\left( \beta \right) {\beta }^{\prime } = \widetilde{\alpha }\left\lbrack {{\widetilde{\beta }}^{\prime } - {\beta }^{\prime }}\right\rbrack + \left\lbrack {\widetilde{\alpha } - \widetilde{\alpha }\left( \beta \right) }\right\rbrack {\be... | No |
Proposition 8.1 (Consistent VAR Order Estimation)\n\nLet\n\n\\[ \n{y}_{t} = \\nu + {A}_{1}{y}_{t - 1} + \\cdots + {A}_{p}{y}_{t - p} + {u}_{t} \n\\]\n\nbe a \\( K \\) -dimensional \\( \\operatorname{VAR}\\left( p\\right) \\) process with \\( {A}_{p} \\neq 0 \\) and standard white noise \\( {u}_{t} \\) and suppose that ... | This proposition extends Proposition 4.2 to processes with integrated variables. It implies that AIC is not a consistent criterion while HQ and SC are both consistent. Thus, if consistent estimation is the objective, we may apply \\( \\mathrm{{HQ}} \\) and \\( \\mathrm{{SC}} \\) for stationary and integrated processes. | No |
Proposition 8.3 (Limiting Distributions of GLS-LR Tests for the Cointe-grating Rank)\n\nUnder the conditions of Proposition 8.2, the GLS-LR test statistics have the following limiting null distributions:\n\n\[{\lambda }_{LR}^{GLS}\left( {{r}_{0}, K}\right) \overset{d}{ \rightarrow }\operatorname{tr}\left( \mathcal{D}\r... | Proofs of these results can be found in Saikkonen & Lütkepohl (2000b, d) and Lütkepohl et al. (2001). | No |
Lemma 8.1\n\n\[ \n{\widetilde{C}}_{i} - {\widehat{C}}_{i} = {O}_{p}\left( {T}^{-1}\right) \;\text{ for }\;i = 1,2,\ldots \n\] | Although Brüggemann et al. (2004) showed this result for full VECMs estimated by reduced rank ML or unrestricted LS, it is clear from their proof that it also applies for other asymptotically equivalent estimation methods. The lemma enables us to get the asymptotic distributions of residual autoco-variances, for exampl... | Yes |
Consider the model\n\n\[ \left\lbrack {\Delta {Y}_{\left( 1\right) } : \Delta {Y}_{\left( 2\right) }}\right\rbrack = \alpha {\beta }^{\prime }{Y}_{-1} + \left\lbrack {{\mathbf{\Gamma }}_{\left( 1\right) } : {\mathbf{\Gamma }}_{\left( 2\right) }}\right\rbrack \left\lbrack \begin{matrix} \Delta {X}_{\left( 1\right) } & 0... | Derive the ML estimators of the parameters. (Hint: Use similar arguments as in the proof of Proposition 7.3.) | No |
Proposition 9.2 (Local Identification of the B-Model)\n\nLet \( \mathrm{B} \) be a nonsingular \( \left( {K \times K}\right) \) matrix. Then, for a given symmetric, positive definite \( \left( {K \times K}\right) \) matrix \( {\sum }_{u} \) and an \( \left( {N \times {K}^{2}}\right) \) matrix \( {C}_{\mathrm{B}} \), th... | Proof: Using the same kind of reasoning as in the proof of Proposition 9.1, the result of Proposition 9.2 follows by noting that\n\n\[ \frac{\partial \operatorname{vech}\left( {\mathrm{{BB}}}^{\prime }\right) }{\partial \operatorname{vec}{\left( \mathrm{B}\right) }^{\prime }} = {\mathbf{D}}_{K}^{ + }\left( {{I}_{{K}^{2... | Yes |
Proposition 9.4 (Local Identification of a SVECM)\n\nSuppose the reduced form model (9.2.1) with Beveridge-Nelson MA representation (9.2.2) is given. Let \( \mathrm{B} \) be a nonsingular \( \left( {K \times K}\right) \) matrix. Then, the set of equations\n\n\[ \n{\sum }_{u} = {\mathrm{{BB}}}^{\prime },\;{C}_{l}\operat... | Proof: The model underlying Proposition 9.4 is a B-model. Therefore the proposition can be shown using the same arguments as for Proposition 9.2. Details are omitted. | No |
Define \( C = {\mathrm{B}}^{-1}\mathrm{\;A} \) and write the concentrated log-likelihood (9.3.3) as\n\n\[ \ln {l}_{c}\left( C\right) = \text{ constant } + T\ln \left| C\right| - \frac{T}{2}\operatorname{tr}\left( {{C}^{\prime }C{\widetilde{\sum }}_{u}}\right) . \]\n\nUse the rules for matrix differentiation from Append... | Next show that\n\n\[ \frac{\partial \operatorname{vec}\left( {{\mathrm{B}}^{-1}\mathrm{\;A}}\right) }{\partial \operatorname{vec}{\left( \mathrm{A}\right) }^{\prime }} = {I}_{K} \otimes {\mathrm{B}}^{-1} \]\n\nand\n\n\[ \frac{\partial \operatorname{vec}\left( {{\mathrm{B}}^{-1}\mathrm{\;A}}\right) }{\partial \operatorn... | No |
Show that the dynamic multipliers have the asymptotic distributions given in Section 10.6. Show also that the \( n \) -th interim multipliers have an asymptotic normal distribution, | \[ \sqrt{T}\operatorname{vec}\left( {{\widehat{M}}_{n} - {M}_{n}}\right) \overset{d}{ \rightarrow }\mathcal{N}\left( {0,{\sum }_{\widehat{\mathbf{m}}}\left( n\right) }\right) ,\] where \[ {\sum }_{\widehat{\mathbf{m}}}\left( n\right) = \left( {{G}_{0} + \cdots + {G}_{n}}\right) {\sum }_{\widehat{\mathbf{\beta }}}{\left... | Yes |
Proposition 11.3 (Characterization of Noncausality)\n\nLet\n\n\[ \n{y}_{t} = \left\lbrack \begin{matrix} {z}_{t} \\ {x}_{t} \end{matrix}\right\rbrack \n\]\n\nbe a stable and invertible \( \operatorname{VARMA}\left( {p, q}\right) \) process as in (11.7.1) with possibly nonzero mean. Then \( {x}_{t} \) is not Granger-cau... | Remark 1 Obviously, the restrictions characterizing Granger-noncausality are not quite so easy here as in the \( \operatorname{VAR}\left( p\right) \) case. Consider, for instance, a bivariate \( \operatorname{VARMA}\left( {1,1}\right) \) process\n\n\[ \n\left\lbrack \begin{array}{l} {z}_{t} \\ {x}_{t} \end{array}\right... | No |
Evaluate the autocovariances \( {\Gamma }_{y}\left( h\right), h = 1,2,3 \), of the bivariate \( \operatorname{VARMA}\left( {2,1}\right) \) process\n\n\[ \n{y}_{t} = \left\lbrack \begin{array}{l} {.3} \\ {.5} \end{array}\right\rbrack + \left\lbrack \begin{array}{ll} {.5} & {.1} \\ {.4} & {.5} \end{array}\right\rbrack {y... | (Hint: The use of a computer will greatly simplify this problem.) | No |
Generalize Propositions 2.4 and 2.5 to the \( \operatorname{VARMA}\left( {p, q}\right) \) case. | (Hint: Show that for a \( K \) -dimensional VARMA \( \left( {p, q}\right) \) process,\n\n\[ \n{\phi }_{{jk}, i} = 0,\;\text{ for }i = 1,2,\ldots , \n\]\n\nis equivalent to\n\n\[ \n{\phi }_{{jk}, i} = 0,\;\text{ for }i = 1,2,\ldots, p\left( {K - 1}\right) + q; \n\]\n\nand\n\n\[ \n{\theta }_{{jk}, i} = 0,\;\text{ for }i ... | No |
Proposition 12.1 (Asymptotic Properties of ML Estimators)\n\nLet \( {y}_{t} \) be a \( K \) -dimensional, stationary Gaussian process with stable and invertible \( \operatorname{VARMA}\left( {p, q}\right) \) representation\n\n\[ \n{A}_{0}\left( {{y}_{t} - \mu }\right) = {A}_{1}\left( {{y}_{t - 1} - \mu }\right) + \cdot... | where\n\n\[ \n{\sum }_{\widetilde{\mu }} = A{\left( 1\right) }^{-1}M\left( 1\right) {\sum }_{u}M{\left( 1\right) }^{\prime }A{\left( 1\right) }^{\prime - 1}, \n\]\n\n\[ \n{\sum }_{\widetilde{\mathbf{\gamma }}} = {\mathcal{I}}_{a}{\left( \mathbf{\gamma }\right) }^{-1} = \operatorname{plim}{\left\lbrack \frac{1}{T}\matho... | Yes |
Suppose that \( \ln \left| {{\widetilde{\sum }}_{u}\left( {\mu ,\gamma }\right) }\right| \) given in (12.3.11) is to be minimized with respect to \( \mathbf{\gamma } \) . Show that the resulting normal equations are\n\n\[\n\frac{\partial \ln \left| {{\widetilde{\sum }}_{u}\left( {\mu ,\mathbf{\gamma }}\right) }\right| ... | Thus, the normal equations are equivalent to those obtained from the log-likelihood function. | No |
Proposition 15.2 (Asymptotic Properties of the White Noise Covariance Matrix Estimator) Let\n\n\[ \n{\widehat{u}}_{t}\left( n\right) \mathrel{\text{:=}} {y}_{t} - \mathop{\sum }\limits_{{i = 1}}^{n}{\widehat{\Pi }}_{i}\left( n\right) {y}_{t - i},\;t = 1,\ldots, T, \n\]\n\nbe the multivariate LS residuals from a \( \ope... | We know from Chapter 3, Propositions 3.2 and 3.4, that, for a Gaussian process, \( {T}^{-1}U{U}^{\prime } \) has an asymptotic normal distribution,\n\n\[ \n\sqrt{T}\operatorname{vech}\left( {{T}^{-1}U{U}^{\prime } - {\sum }_{u}}\right) \overset{d}{ \rightarrow }\mathcal{N}\left( {0,2{\mathbf{D}}_{K}^{ + }\left( {{\sum ... | Yes |
Proposition 15.3 (Asymptotic Distributions of Estimated Forecasts)\n\nUnder the conditions of Proposition 15.1, if \( {y}_{t} \) is a Gaussian process and if independent processes with identical stochastic structures are used for estimation and forecasting, respectively, then\n\n\[ \sqrt{\frac{T}{{n}_{T}}}\left\lbrack ... | Remark 1 The proposition implies that for large samples the forecast vector \( {\widetilde{\mathbf{y}}}_{T}\left( h\right) \) has approximate MSE matrix\n\n\[ {\mathbf{\sum }}_{\widetilde{\mathbf{y}}}\left( h\right) = \left( {1 + \frac{K{n}_{T}}{T}}\right) {\mathbf{\sum }}_{\mathbf{y}}\left( h\right) \]\n\n(15.3.2)\n\n... | Yes |
Proposition 15.6 (Asymptotic Distribution of VECM Estimators) Under the conditions of Proposition 15.5,\n\n\[ \left. {T\left( {{\widehat{\widehat{\mathbf{\beta }}}}_{\left( K - r\right) }^{\prime } - {\mathbf{\beta }}_{\left( K - r\right) }^{\prime }}\right) \overset{d}{ \rightarrow }{\left( {\int }_{0}^{1}{\mathbf{W}}... | where \( {\mathbf{W}}_{K - r}^{\# } \) and \( {\mathbf{W}}_{r}^{\# } \) are independent \( \left( {K - r}\right) \) - and \( r \) -dimensional Wiener processes, respectively, as in Proposition 7.2. Furthermore,\n\n\[ \frac{\sqrt{T - {n}_{T}}\mathbf{f}{\left( {n}_{T}\right) }^{\prime }\left\lbrack {\widehat{\mathbf{\gam... | Yes |
Proposition 15.7 (Asymptotic Distributions of Tests for the Cointegrating Rank)\n\nSuppose \( {y}_{t} \) is generated by an infinite order process as described in Section 15.5.1. Moreover, suppose that\n\n\[ \n{n}_{T} \rightarrow \infty \;\text{ and }\;{n}_{T}^{3}/T \rightarrow 0\;\text{ as }\;T \rightarrow \infty .\n\... | Notice that in this proposition we just have an upper bound for the rate at which the lag order \( {n}_{T} \) has to go to infinity. No lower bound for the rate of divergence is needed. In fact, Lütkepohl & Saikkonen (1999b) considered also processes with nonzero mean term and, in addition, they treated the case where ... | No |
Proposition 18.1 (Asymptotic Properties of the ML Estimator)\n\nWith all the assumptions stated in the foregoing, the ML estimator \( \widetilde{\mathbf{\delta }} \) of \( {\mathbf{\delta }}_{0} \) is consistent and asymptotically normally distributed,\n\n\[ \sqrt{T}\left( {\widetilde{\mathbf{\delta }} - {\mathbf{\delt... | Pagan (1980) gives a proof of this proposition based on Crowder (1976) (see also Schneider (1988)). Other sets of conditions are possible to accommodate the situation where the inputs \( {x}_{t} \) are stochastic. They may, in fact, contain lagged \( {y}_{t} \) ’s. Moreover, \( \mathbf{B} \) may have eigenvalues on the... | No |
Consider the \( K \) -dimensional Gaussian stable VAR(1) process \( {y}_{t} = A{y}_{t - 1} + {u}_{t} \) with \( {y}_{0} \sim \mathcal{N}\left( {0,0}\right) \) and \( {u}_{t} \sim \mathcal{N}\left( {0,{\sum }_{u}}\right) \) for \( t = 1,2,\ldots \) . Use the Kalman filter recursions to determine \( {y}_{t \mid t - 1} \)... | (a) Show that \( {y}_{t \mid t - 1} = A{y}_{t - 1} \) . | No |
Example 1.1(a) Consider a population consisting of individuals able to produce offspring of the same kind. The number of individuals initially present, denoted by \( {X}_{0} \), is called the size of the zeroth generation All offspring of the zeroth generation constitute the first generation and their number is denoted... | Since \( {X}_{n} = 0 \) implies that \( {X}_{n + 1} = 0 \), it follows that \( P\left\{ {{X}_{n} = 0}\right\} \) is increasing and thus \( \mathop{\lim }\limits_{{n \rightarrow \infty }}P\left\{ {{X}_{n} = 0}\right\} \) exists What does it represent? To answer this use Proposition 1.1.1 as follows:\n\n\[ \mathop{\lim }... | Yes |
Example 1.3(a) The Matching Problem. At a party \( n \) people put their hats in the center of a room where the hats are mixed together Each person then randomly selects one We are interested in the mean and variance of \( X \) -the number that select their own hat | To solve, we use the representation\n\n\[ \nX = {X}_{1} + {X}_{2} + \cdots + {X}_{n} \]\n\nwhere\n\n\[ \n{X}_{t} = \left\{ \begin{array}{ll} 1 & \text{ if the }i\text{ th person selects his or her own hat } \\ 0 & \text{ otherwise } \end{array}\right. \]\n\nNow, as the \( i \) th person is equally likely to select any ... | Yes |
A useful identity can be obtained by noting that\n\n(1.3.5)\n\n\[ {\left( 1 - 1\right) }^{N} = \left\{ \begin{array}{ll} 1 & \text{ if }N = 0 \\ 0 & \text{ if }N > 0. \end{array}\right. \] | But by the binomial theorem,\n\n(1.3.6)\n\n\[ {\left( 1 - 1\right) }^{N} = \mathop{\sum }\limits_{{i = 0}}^{N}\left( \begin{matrix} N \\ i \end{matrix}\right) {\left( -1\right) }^{i} \]\n\n\[ = \mathop{\sum }\limits_{{i = 0}}^{n}\left( \begin{matrix} N \\ i \end{matrix}\right) {\left( -1\right) }^{i}\;\text{ since }\le... | Yes |
Example 1.3(c) A graph is a set of elements, called nodes, and a set of (unordered) pairs of nodes, called edges For instance, Figure 1.3.1 illustrates a graph with the set of nodes \( N = \{ 1,2,3,4,5\} \) and the set of edges \( E = \{ \left( {1,2}\right) ,\left( {1,3}\right) ,\left( {1,5}\right) ,\left( {2,3}\right)... | Solution. Suppose that the graph contains \( m \) edges, and arbitrarily number them as \( 1,2,\ldots, m \) For any set of nodes \( B \), if we let \( C\left( B\right) \) denote the number of edges that have exactly one of their nodes in \( B \), then the problem is to show that \( \mathop{\max }\limits_{B}C\left( B\ri... | Yes |
Example 1.4(a) Let \( X \) and \( Y \) be independent normal random variables with respective means \( {\mu }_{1} \) and \( {\mu }_{2} \) and respective variances \( {\sigma }_{1}^{2} \) and \( {\sigma }_{2}^{2} \). The moment generating function of their sum is given by | \n\[{\psi }_{X + Y}\left( t\right) = E\left\lbrack {e}^{t\left( {X + Y}\right) }\right\rbrack\]\n\[= E\left\lbrack {e}^{\iota X}\right\rbrack E\left\lbrack {e}^{\iota Y}\right\rbrack \;\text{(by independence)}\]\n\[= {\psi }_{X}\left( t\right) {\psi }_{Y}\left( t\right)\]\n\[= \exp \{ \left( {{\mu }_{1} + {\mu }_{2}}\r... | Yes |
Example 1.4(b) The Multivariate Normal Distribution. Let \( {Z}_{1} \) , , \( {Z}_{n} \) be independent standard normal random variables. If for some constants \( {a}_{ij},1 \leq i \leq m,1 \leq j \leq n \), and \( {\mu }_{i},1 \leq i \leq m \) , \[ {X}_{1} = {a}_{11}{Z}_{1} + \cdots + {a}_{1n}{Z}_{n} + {\mu }_{1} \] \... | Let us now consider \[ \psi \left( {{t}_{1},\ldots ,{t}_{m}}\right) = E\left\lbrack {\exp \left\{ {{t}_{1}{X}_{1} + \cdots + {t}_{m}{X}_{m}}\right\} }\right\rbrack \] the joint moment generating function of \( {X}_{1},\ldots ,{X}_{m} \) . The first thing to note is that since \( \mathop{\sum }\limits_{{t = 1}}^{m}{t}_{... | Yes |
Example 1.5(A) The Sum of a Random Number of Random Variables. Let \( {X}_{1},{X}_{2},\ldots \) denote a sequence of independent and identically distributed random variables; and let \( N \) denote a nonnegative integer valued random variable that is independent of the sequence \( {X}_{1},{X}_{2},\ldots \) . We shall c... | \[ E\left\lbrack {\left. {\exp \left\{ {t\mathop{\sum }\limits_{1}^{N}{X}_{i}}\right\} }\right| \;N = n}\right\rbrack = E\left\lbrack {\left. {\exp \left\{ {t\mathop{\sum }\limits_{1}^{n}{X}_{t}}\right\} }\right| N = n}\right\rbrack = E\left\lbrack {\exp \left\{ {t\mathop{\sum }\limits_{1}^{n}{X}_{t}}\right\} }\right\r... | Yes |
A miner is trapped in a mine containing three doors. The first door leads to a tunnel that takes him to safety after two hours of travel The second door leads to a tunnel that returns him to the mine after three hours of travel. The third door leads to a tunnel that returns him to his mine after five hours. Assuming th... | Let \( Y \) denote the door initially chosen Then\n\n(15.2)\n\n\[ E\left\lbrack {e}^{\iota X}\right\rbrack = \frac{1}{3}\left( {E\left\lbrack {{e}^{\iota X} \mid Y = 1}\right\rbrack + E\left\lbrack {{e}^{\iota X} \mid Y = 2}\right\rbrack + E\left\lbrack {{e}^{\iota X} \mid Y = 3}\right\rbrack }\right) . \]\n\nNow given... | Yes |
Suppose in the matching problem, Example 1.3(a), that those choosing their own hats depart, while the others (those without a match) put their selected hats in the center of the room, mix them up, and then reselect. If this process continues until each individual has his or her own hat, find \( E\left\lbrack {R}_{n}\ri... | We will now show that \( E\left\lbrack {R}_{n}\right\rbrack = n \) . The proof will be by induction on \( n \), the number of individuals. As it is obvious for \( n = 1 \) assume that \( E\left\lbrack {R}_{k}\right\rbrack = k \) for \( k = 1,., n - 1 \) To compute \( E\left\lbrack {R}_{n}\right\rbrack \), start by cond... | Yes |
Example 1.5(b) Suppose that \( X \) and \( Y \) are independent random variables having respective distributions \( F \) and \( G \) Then the distribution of \( X + Y \) -which we denote by \( F * G \), and call the convolution of \( F \) and \( G \) -is given by\n\n\[ \left( {F * G}\right) \left( a\right) = P\{ X + Y ... | \[ = {\int }_{-\infty }^{\infty }P\{ X + Y \leq a \mid Y = y\} {dG}\left( y\right) \]\n\[ = {\int }_{-\infty }^{\infty }P\{ X + y \leq a \mid Y = y\} {dG}\left( y\right) \]\n\[ = {\int }_{-\infty }^{\infty }F\left( {a - y}\right) {dG}\left( y\right) \] | Yes |
Example 1.5(E) The Ballot Problem. In an election, candidate \( A \) receives \( n \) votes and candidate \( B \) receives \( m \) votes, where \( n > \) \( m \) . Assuming that all orderings are equally likely, show that the probability that \( A \) is always ahead in the count of votes is \( (n - \) \( m)/\left( {n +... | Solution. Let \( {P}_{n, m} \) denote the desired probability By conditioning on which candidate receives the last vote counted we have\n\n\[ \n{P}_{n, m} = P\{ A\text{ always ahead } \mid A\text{ receives last vote }\} \frac{n}{n + m} \n\]\n\n\[ \n+ P\{ A\text{ always ahead } \mid B\text{ receives last vote }\} \frac{... | Yes |
Example 1.5(f) The Matching Problem Revisited. Let us reconsider Example 1 3(a) in which \( n \) individuals mix their hats up and then randomly make a selection We shall compute the probability of exactly \( k \) matches | First let \( E \) denote the event that no matches occur, and to make explicit the dependence on \( n \) write \( {P}_{n} = P\left( E\right) \) Upon conditioning on whether or not the first individual selects his or her own hat-call these events \( M \) and \( {M}^{c} \) -we obtain\n\n\[ \n{P}_{n} = P\left( E\right) = ... | No |
Consider an \( n \) component system that is subject to randomly occurring shocks Suppose that each shock has a value that, independent of all else, is chosen from a distribution \( G \) If a shock of value \( x \) occurs, then each component that was working at the moment the shock arrived will, independently, instant... | To compute \( P\{ N > k\} \) let \( {E}_{i}, i = 1,\;, n \), denote the event that component \( i \) has survived the first \( k \) shocks. Then\n\n\[ P\{ N > k\} = P\left( {\mathop{\bigcup }\limits_{1}^{n}{E}_{1}}\right) \]\n\n\[ = \mathop{\sum }\limits_{t}P\left( {E}_{t}\right) - \mathop{\sum }\limits_{{t < l}}P\left... | Yes |
Example 1.5(1) Classifying a Poisson Number of Events. Suppose that we are observing events, and that \( N \), the total number that occur, is a Poisson random variable with mean \( \lambda \) . Suppose also that each event that occurs is, independent of other events, classified as a type \( j \) event with probability... | For any nonnegative integers \( {n}_{l}, j = 1,\;, k \), let \( n = \mathop{\sum }\limits_{{j = 1}}^{k}{n}_{j} \) . Then, since \( N = \mathop{\sum }\limits_{l}{N}_{l} \), we have that\n\n\[ P\left\{ {{N}_{\jmath } = {n}_{\jmath }, j = 1,\;., k}\right\} \]\n\n\[ = P\left\{ {{N}_{j} = {n}_{j}, j = 1,\ldots, k \mid N = n... | Yes |
Example 1.6(a) Consider a post office having two clerks, and suppose that when \( A \) enters the system he discovers that \( B \) is being served by one of the clerks and \( C \) by the other. Suppose also that \( A \) is told that his service will begin as soon as either \( B \) or \( C \) leaves. If the amount of ti... | The answer is obtained by reasoning as follows: Consider the time at which \( A \) first finds a free clerk. At this point either \( B \) or \( C \) would have just left and the other one would still be in service. However, by the lack of memory of the exponential, it follows that the amount of additional time that thi... | Yes |
Example 1.6(b) Let \( {X}_{1},{X}_{2},\ldots \) be independent and identically distributed continuous random variables with distribution \( F \) . We say that a record occurs at time \( n, n > 0 \), and has value \( {X}_{n} \) if \( {X}_{n} > \max \left( {{X}_{1},\ldots ,{X}_{n - 1}}\right) \), where \( {X}_{0} = - \in... | As a preliminary to computing the distribution of \( {\tau }_{t} \), let us note that the record times of the sequence \( {X}_{1},{X}_{2}, \) . will be the same as for the sequence \( F\left( {X}_{1}\right), F\left( {X}_{2}\right) ,\; \), and since \( F\left( X\right) \) has a uniform \( \left( {0,1}\right) \) distribu... | Yes |
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