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Theorem 8.6.2 [Erdős (1956)] There is a set \( S \) for which \( f\left( n\right) = \Theta \left( {\ln n}\right) \) . That is, there is a set \( S \) and constants \( {c}_{1},{c}_{2} \) so that for all sufficiently large \( n \)\n\n\[ \n{c}_{1}\ln n \leq f\left( n\right) \leq {c}_{2}\ln n \n\] | Proof. Define \( S \) randomly by\n\n\[ \n\Pr \left\lbrack {x \in S}\right\rbrack = {p}_{x} = \min \left\lbrack {{10}\sqrt{\frac{\ln x}{x}},1}\right\rbrack . \n\]\n\nFix \( n \) . Now \( f\left( n\right) \) is a random variable with mean\n\n\[ \n\mu = E\left\lbrack {f\left( n\right) }\right\rbrack = \frac{1}{2}\mathop{... | Yes |
Theorem 8.6.3 [Erdős and Tetali (1990) ] There is a set \( S \) for which \( g\left( n\right) = \) \( \Theta \left( {\ln n}\right) \) . That is, there is a set \( S \) and constants \( {c}_{1},{c}_{2} \) so that for all sufficiently large \( n \)\n\n\[{c}_{1}\ln n \leq g\left( n\right) \leq {c}_{2}\ln n\] | Proof. Define \( S \) randomly by\n\n\[ \Pr \left\lbrack {x \in S}\right\rbrack = {p}_{x} = \min \left\lbrack {{10}{\left( \frac{\ln x}{{x}^{2}}\right) }^{1/3},\frac{1}{2}}\right\rbrack . \]\n\nFix \( n \) . Now \( g\left( n\right) \) is a random variable and\n\n\[ \mu = E\left\lbrack {g\left( n\right) }\right\rbrack =... | Yes |
For all \( k \) there exists \( \epsilon > 0 \) so that for all sufficiently large \( n \) there exist graphs \( G \) on \( n \) vertices with \( \chi \left( G\right) > k \) and yet \( \chi \left( {\left. G\right| }_{S}\right) \leq 3 \) for every set \( S \) of vertices of size at most \( {\epsilon n} \) . | Proof. For a given \( k \) let \( c,\epsilon > 0 \) satisfy (with foresight)\n\n\[ c > 2{k}^{2}H\left( {1/k}\right) \ln 2 \]\n\n\[ \epsilon < {e}^{-5}{3}^{3}{c}^{-3} \]\n\nwhere \( H\left( x\right) = - x{\log }_{2}x - \left( {1 - x}\right) {\log }_{2}\left( {1 - x}\right) \) is the entropy function. Set \( p = c/n \) a... | Yes |
Lemma 9.1.2 For any two subsets \( A \) and \( B \) of \( {GF}\left( p\right) \) ;\n\n\( \left| {\mathop{\sum }\limits_{{i \in A}}\mathop{\sum }\limits_{{j \in B}}{d}_{ij}}\right| \leq {\left| A\right| }^{1/2}{\left| B\right| }^{1/2}{p}^{1/2}. \) | Proof. By the Cauchy-Schwarz Inequality and by the fact above:\n\n\[ \n{\left( \mathop{\sum }\limits_{{i \in A}}\mathop{\sum }\limits_{{j \in B}}{d}_{ij}\right) }^{2} \leq \left| A\right| \left( {\mathop{\sum }\limits_{{i \in A}}{\left( \mathop{\sum }\limits_{{j \in B}}{d}_{ij}\right) }^{2}}\right) \n\]\n\n\[ \n\leq \l... | Yes |
Theorem 9.2.1 For every partition of the set of vertices \( V \) into two disjoint subsets \( B \) and \( C \) :\n\n\[ e\left( {B, C}\right) \geq \frac{\left( {d - \lambda }\right) \left| B\right| \left| C\right| }{n}. \] | Proof. Put \( \left| V\right| = n, b = \left| B\right|, c = \left| C\right| = n - b \) . Let \( D = {dI} \) be the \( n \) by \( n \) scalar matrix with the degree of regularity of \( G \) on its diagonal. Observe that for any real vector \( x \) of length \( n \) (considered as a function \( x : V \mapsto R \) ) we ha... | Yes |
Corollary 9.2.2 If \( \lambda \) is the second largest eigenvalue of a \( d \) -regular graph \( G \) with \( n \) vertices, then \( G \) is an \( \left( {n, d, c}\right) \) -expander for \( c = \frac{d - \lambda }{2d} \) . | Proof. Let \( W \) be a set of \( w \leq n/2 \) vertices of \( G \) . By Theorem 9.2.1 there are at least \( \frac{\left( {d - \lambda }\right) w\left( {n - w}\right) }{n} \geq \frac{\left( {d - \bar{\lambda }}\right) w}{2} \) edges from \( W \) to its complement. Since no vertex in the complement is adjacent to more t... | Yes |
Theorem 9.2.4 Let \( G = \left( {V, E}\right) \) be a d-regular graph on \( n \) vertices, and suppose the absolute value of each of its eigenvalues but the first one is at most \( \lambda \) . For a vertex \( v \in V \) and a subset \( B \) of \( V \) denote by \( N\left( v\right) \) the set of all neighbours of \( v ... | Proof. Let \( A \) be the adjacency matrix of \( G \) and define a vector \( f : V \mapsto R \) by \( f\left( v\right) = 1 - b \) for \( v \in B \) and \( f\left( v\right) = - b \) for \( v \notin B \) . Clearly \( \mathop{\sum }\limits_{{v \in V}}f\left( v\right) = 0 \), i.e., \( f \) is orthogonal to the eigenvector ... | Yes |
Corollary 9.2.5 Let \( G = \\left( {V, E}\\right), d, n \) and \( \\lambda \) be as in Theorem 9.2.4. Then for every two sets of vertices \( B \) and \( C \) of \( G \), where \( \\left| B\\right| = {bn} \) and \( \\left| C\\right| = {cn} \) we have:\n\n\[ \n\\left| {e\\left( {B, C}\\right) - {cbdn}}\\right| \\leq \\la... | Proof. By Theorem 9.2.4\n\n\[ \n\\mathop{\\sum }\\limits_{{v \\in C}}{\\left( \\left| {N}_{B}\\left( v\\right) \\right| - bd\\right) }^{2} \\leq \\mathop{\\sum }\\limits_{{v \\in V}}{\\left( \\left| {N}_{B}\\left( v\\right) \\right| - bd\\right) }^{2} \\leq {\\lambda }^{2}b\\left( {1 - b}\\right) n. \n\]\n\nThus, by th... | Yes |
Theorem 9.2.7 Let \( G = \left( {V, E}\right) \) be a \( d \) -regular graph on \( n \) vertices, and suppose that each of its eigenvalues but the first one is at most \( \lambda \) . Let \( C \) be a set of cn vertices of \( G \) . Then, for every \( l \), the number of walks of length \( l \) in \( G \) that avoid \(... | Proof. Let \( A \) be the adjacency matrix of \( G \) and let \( {A}^{\prime } \) be the adjacency matrix of its induced subgraph on the complement of \( C \) . We claim that the maximum eigenvalue of \( {A}^{\prime } \) is at most \( \left( {1 - c}\right) d + {c\lambda } \) . To prove this claim we must show that for ... | Yes |
Corollary 9.2.8 Let \( G = \left( {V, E}\right), d, n,\lambda, C \) and \( c \) be as in Theorem 9.2.7 and suppose\n\n\[ \left( {1 - c}\right) d + {c\lambda } \leq \frac{d}{\sqrt{2}} \]\n\nThen, for every \( l \), the probability that a randomly chosen walk of length \( l \) in \( G \) avoids \( C \) is at most \( {2}^... | Proof. The number of walks of length \( l \) in \( G \) that avoid \( C \) is at most \( \left( {1 - c}\right) n((1 - \) \( c)d + {c\lambda }{)}^{l} \leq n{d}^{l}{2}^{-l/2} \), by Theorem 9.2.7. Since the total number of walks is \( n{d}^{l} \) , the desired result follows. - | Yes |
Theorem 1 Let \( G = \left( {V, E}\right) \) be a vertex-transitive graph. For an integer \( k \) and for two (not neccessarily distinct) vertices \( u, v \) of \( G \), let \( {P}^{k}\left( {u, v}\right) \) denote the probability that a random walk of length \( k \) starting at \( u \) ends at \( v \) . Then, for ever... | Proof. We need the following simple inequality, sometimes attributed to Chebyschev.\n\nClaim 2 For every sequence \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) of \( n \) reals and for any permutation \( \pi \) of \( \{ 1,\ldots, n\} \) :\n\n\[ \n\mathop{\sum }\limits_{{i = 1}}^{n}{a}_{i}{a}_{\pi \left( i\right) } \leq... | Yes |
Theorem 10.1.1 Let \( H \) be a strictly balanced graph with \( v \) vertices, \( e \) edges and \( a \) automorphisms. Let \( c > 0 \) be arbitrary. Let \( A \) be the property that \( G \) contains no copy of \( H \) . Then with \( p = c{n}^{-v/e} \) , | Proof. Let \( {A}_{\alpha },1 \leq \alpha \leq \left( \begin{array}{l} n \\ v \end{array}\right) v!/a \), range over the edge sets of possible copies of \( H \) and let \( {B}_{\alpha } \) be the event \( G\left( {n, p}\right) \supseteq {A}_{\alpha } \) . We apply Janson’s Inequality. As\n\n\[ \mathop{\lim }\limits_{{n... | Yes |
Theorem 10.3.1 [Bollobás (1988) ] Almost always\n\n\\[ \chi \\left( G\\right) ) \\sim \\frac{n}{2{\\log }_{2}n}. \\] | Proof. Let \\( \\alpha \\left( G\\right) = \\omega \\left( \\bar{G}\\right) \\) denote, as usual, the independence number of \\( G \\) . The complement of \\( G \\) has the same distribution \\( G\\left( {n,1/2}\\right) \\) . Hence \\( \\alpha \\left( G\\right) \\leq \\left( {2 + o\\left( 1\\right) }\\right) {\\log }_{... | Yes |
Theorem 10.7.2 For any irrational \( \alpha ,0 < \alpha < 1 \), setting \( p = p\left( n\right) = {n}^{-\alpha } \), and for any First Order A, | Both proofs are only outlined. | No |
Lemma 10.7.3 For every First Order \( A \) there is a \( t = t\left( A\right) \) so that if \( G, H \) are any graphs with \( G \vDash A \) and \( H \vDash \neg A \) then Spoiler wins \( {EHR}\left\lbrack {G, H, t}\right\rbrack \) . | A detailed proof would require a formal analysis of the First Order language so we give only an example. Let \( A \) be the property \( {\forall }_{x}{\exists }_{y}\left\lbrack {x \sim y}\right\rbrack \) of not containing an isolated point and set \( t = 2 \) . Spoiler begins by selecting an isolated point \( {y}_{1} \... | No |
Theorem 10.7.4 A function \( p = p\left( n\right) \) satisfies the Zero-One Law if and only if for every \( t \), letting \( G\left( {n, p\left( n\right) }\right), H\left( {m, p\left( m\right) }\right) \) be independently chosen random graphs on disjoint vertex sets\n\n\[ \mathop{\lim }\limits_{{m, n \rightarrow \infty... | Proof. We prove only the \ | No |
Theorem 10.7.5 For any fixed \( p,0 < p < 1 \), and any \( s, G\left( {n, p}\right) \) almost always has the full level \( s \) extension property. | Proof. For every distinct \( {u}_{1},\ldots ,{u}_{a},{v}_{1},\ldots ,{v}_{b}, x \in G \) with \( a + b \leq s \) we define \( {E}_{{u}_{1},\ldots ,{u}_{a},{v}_{1},\ldots ,{v}_{b}, x} \) to be the event that \( \left\{ {x,{u}_{i}}\right\} \in E\left( G\right) ,1 \leq i \leq a \) and \( \left\{ {x,{v}_{j}}\right\} \notin... | Yes |
Lemma 10.7.6 [Generic Extension] Let \( \left( {R, H}\right) \), as given above, be safe. Let \( t \geq 0 \) be an arbitrary, but fixed, integer. Then in \( G \sim G\left( {n,{n}^{-\alpha }}\right) \) almost surely for all \( {x}_{1},\ldots ,{x}_{r} \) there exist \( {y}_{1},\ldots ,{y}_{v} \) such that\n\n1. \( {y}_{1... | Proof. From Exercise 5 almost surely every \( {x}_{1},\ldots ,{x}_{r} \) has \( \Theta \left( {{n}^{v}{p}^{e}}\right) \left( {R, H}\right) \) extensions \( {y}_{1},\ldots ,{y}_{v} \) . Our rough notion will be that the number of these \( {y}_{1},\ldots ,{y}_{v} \) that fail to be generic, in any of the bounded number o... | No |
Lemma 10.7.7 [Finite Closure] Let \( \alpha, r > 0 \) be fixed. Set \( \varepsilon \) equal to the minimal value of \( \frac{{e\alpha } - v}{v} \) over all integers \( v, e \) with \( 1 \leq v \leq t \) and \( {e\alpha } - v > 0 \) . Let \( K \) be such that \( r - {K\varepsilon } < 0 \) . Then in \( G\left( {n,{n}^{-\... | Proof. If not there would be a rigid \( t \) -chain \( X = {X}_{0} \subset {X}_{1} \subset \ldots \subset {X}_{L} = Y \) with \( K + r < \left| Y\right| < K + r + t \) . Letting \( \left( {{X}_{i - 1},{X}_{i}}\right) \) have type \( \left( {{v}_{i},{e}_{i}}\right) \) the restriction of \( G \) to \( Y \) would have \( ... | Yes |
Theorem 11.2.2 Let \( f = f\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) be a function and let \( C \) be a \( C\left( {\infty, s, d, t}\right) \) - circuit computing \( f \), where \( s \cdot {\left( \frac{1}{2}\right) }^{t} \leq {0.5} \) . Then either \( f \) or its complement \( \bar{f} \) has a minterm of size at most... | Proof. Let us apply to \( C \), repeatedly, \( d - 2 \) times a random \( 1/\left( {10t}\right) \) - restriction. Each of these random restrictions, when applied to any bottom subcircuit of depth 2, transforms it by Lemma 11.2.1 with probability at least \( 1 - {\left( \frac{1}{2}\right) }^{t} \) from a \( t \) -Or-And... | Yes |
Corollary 11.2.3 For any \( d \geq 2 \), there is no \( C\left( {\infty ,\frac{1}{2} \cdot {2}^{\frac{1}{10}{n}^{1/\left( {d - 1}\right) }}, d,\frac{1}{10}{n}^{1/\left( {d - 1}\right) }}\right) \) -circuit that computes the parity function \( f\left( {{x}_{1},\ldots ,{x}_{n}}\right) = {x}_{1} \oplus \cdots \oplus {x}_{... | Proof. Assuming there is such a circuit we obtain, by Theorem 11.2.2, that the value of \( f \) can be fixed by assigning values to at most \( n - \frac{1}{2}{n}^{1/\left( {d - 1}\right) } + \frac{1}{10}{n}^{1/\left( {d - 1}\right) } < n \) variables. This is false, and hence there is no such circuit. | Yes |
Lemma 11.3.2 There is no polynomial \( P\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) over \( {GF}\left( 3\right) \) of degree at most \( \sqrt{n} \) which is equal to the parity of \( {x}_{1},\ldots ,{x}_{n} \) for a set \( S \) of at least \( {0.9} \cdot {2}^{n} \) distinct binary vectors \( \left( {{x}_{1},\ldots ,{x}_... | Proof. Suppose this is false, and suppose \( S \subset \{ 0,1{\} }^{n},\;\left| S\right| \geq {0.9} \cdot {2}^{n} \) and \( P\left( {{x}_{1},\ldots ,{x}_{n}}\right) = {x}_{1} \oplus \cdots \oplus {x}_{n} \) for all \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \in S \) . Define a polynomial \( Q = Q\left( {{y}_{1},\ldots ... | Yes |
Corollary 11.3.3 There is no circuit of depth \( d \) and size \( s \leq \frac{1}{10}{2}^{\frac{1}{2}{n}^{1/{2d}}} \) computing the parity of \( {x}_{1},{x}_{2},\ldots ,{x}_{n} \) using Not, And, Or and \( {\operatorname{Mod}}_{3} \) gates. | Proof. Suppose this is false and let \( C \) be such a circuit. Put \( \ell = \frac{1}{2} \cdot {n}^{1/{2d}} \) . By Lemma 11.3.1 there is a polynomial \( P = P\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) over \( {GF}\left( 3\right) \), whose degree is at most \( {\left( 2\ell \right) }^{d} = \sqrt{n} \), which is equal ... | Yes |
Theorem 11.4.1 The monotone circuit complexity of \( T \) is at least \( \Omega \left( {{m}^{3}/{\log }^{4}m}\right) \) . | Before we present the proof of this Theorem we introduce some notation and prove a simple lemma. For any Boolean function \( f = f\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) define \( A\left( f\right) = \) \( \left\{ {\left( {{x}_{1},\ldots ,{x}_{n}}\right) \in \{ 0,1{\} }^{n} : f\left( {{x}_{1},\ldots ,{x}_{n}}\right) ... | Yes |
Lemma 11.4.2 For all members \( {M}_{i} \) of \( M\left( C\right) \)\n\n\[ \n{A}_{i} - \left( {\mathop{\bigcup }\limits_{{j \leq i}}{\delta }_{ \sqcap }^{j}}\right) \subseteq {M}_{i} \subseteq {A}_{i} \cup \underset{j \leq i}{ \cup }{\delta }_{ \sqcup }^{j}.\n\]\n\n(11.10) | Proof. We apply induction on \( i \) . For \( i < 0\;{M}_{i} = {A}_{i} \) and thus (11.10) holds. Assuming (11.10) holds for all \( {M}_{j} \) with \( j < i \) we prove it for \( i \) . If \( {A}_{i} = {A}_{\ell } \cup {A}_{k} \) , then, by the induction hypothesis\n\n\[ \n{M}_{i} = {M}_{\ell } \cup {M}_{k} \cup {\delt... | Yes |
Lemma 11.5.1 Let \( f = f\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) be a non-atom Boolean function of \( n \) variables. Then there is an \( i,1 \leq i \leq n \) and an \( \varepsilon \in \{ 0,1\} \) such that for the function \( g = f\left( {{x}_{1},\ldots ,{x}_{i - 1},\varepsilon ,{x}_{i + 1},\ldots ,{x}_{n}}\right) ... | Proof. Fix a formula \( F \) computing \( f \) with \( l = L\left( f\right) \) And and \( {Or} \) gates. \( F \) can be represented by a binary tree each of whose \( l + 1 \) leaves is labeled by an atom \( {x}_{i} \) or \( {\bar{x}}_{i} \) . Let us choose, randomly, a variable \( {x}_{i},1 \leq i \leq n \) according t... | Yes |
Corollary 11.5.2 If \( f = f\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) and \( L\left( f\right) \leq {\left( \frac{n}{k}\right) }^{3/2} - 1 \), then one can assign values to \( n - k \) variables so that the resulting function \( g \) is an atom. | Proof. Repeated application of Lemma 11.5.1 \( n - k \) times yields a \( g \) with\n\n\[ \left( {L\left( g\right) + 1}\right) \leq \mathop{\prod }\limits_{{i = k + 1}}^{n}{\left( 1 - \frac{1}{i}\right) }^{3/2}\left( {L\left( f\right) + 1}\right) = {\left( k/n\right) }^{3/2}\left( {L\left( f\right) + 1}\right) \leq 1.\... | Yes |
Theorem 1 Let \( \mathcal{F} \) be an antichain. Then \[ \mathop{\sum }\limits_{{A \in \mathcal{F}}}\frac{1}{\left( \begin{matrix} n \\ \left| A\right| \end{matrix}\right) } \leq 1 \] | Proof. Let \( \sigma \) be a uniformly chosen permutation of \( \{ 1,\ldots, n\} \) and set \[ {\mathcal{C}}_{\sigma } = \{ \{ \sigma \left( j\right) : 1 \leq j \leq i\} : 0 \leq i \leq n\} \] (The cases \( i = 0, n \) give \( \varnothing ,\{ 1,\ldots, n\} \in \mathcal{C} \) respectively.) Define a random variable \[ X... | Yes |
Corollary 2 [Sperner’s Theorem] Let \( \\mathcal{F} \) be an antichain. Then\n\n\[ \n\\left| \\mathcal{F}\\right| \\leq \\left( \\begin{matrix} n \\\\ \\lfloor n/2\\rfloor \\end{matrix}\\right) \n\] | Proof. The function \( \\left( \\begin{array}{l} n \\\\ x \\end{array}\\right) \) is maximized at \( x = \\lfloor n/2\\rfloor \) so that\n\n\[ \n1 \\geq \\mathop{\\sum }\\limits_{{A \\in \\mathcal{F}}}\\frac{1}{\\left( \\begin{matrix} n \\\\ \\left| A\\right| \\end{matrix}\\right) } \\geq \\frac{\\left| \\mathcal{F}\\r... | Yes |
Theorem 12.1.1 Let \( \mathcal{A} \) be a family of \( n \) subsets of an \( m \) -set \( \Omega \) . Then\n\n\[ \operatorname{disc}\left( \mathcal{A}\right) \leq \sqrt{{2m}\ln \left( {2n}\right) }.\] | Proof. Let \( \chi : \Omega \rightarrow \{ - 1, + 1\} \) be random. For \( A \subset \Omega \) let \( {X}_{A} \) be the indicator random variable for \( \left| {\chi \left( A\right) }\right| > \alpha \) where we set \( \alpha = \sqrt{{2m}\ln \left( {2n}\right) } \) . If \( \left| A\right| = a \) then \( \chi \left( A\r... | Yes |
Theorem 12.2.1 [Spencer (1985a) ] Let \( \mathcal{A} \) be a family of \( n \) subsets of an \( n \) -element set \( \Omega \) . Then\n\n\[ \operatorname{disc}\left( \mathcal{A}\right) < 6\sqrt{n} \] | With \( \chi : \Omega \rightarrow \{ - 1, + 1\} \) random, \( A \in \mathcal{A},\chi \left( A\right) \) has zero mean and variance at most \( \sqrt{n} \) . If \( \left| {\chi \left( A\right) }\right| > 6\sqrt{n} \) then \( \chi \left( A\right) \) is at least six standard deviations off the mean. The probability of this... | No |
Lemma 12.2.3 Let \( \mathcal{A} \) be a family of \( n \) subsets of an \( r \) -set \( \Omega \) with \( r \leq {10}^{-9}n \) . Then there is a partial coloring \( \chi \) of \( \Omega \) with at most \( {10}^{-{40}}r \) points uncolored so that | ## Proof.\n\nWe outline the argument which leaves room to spare. Let \( {A}_{1},\ldots ,{A}_{n} \) denote the sets of \( \mathcal{A} \) . Let \( \chi : \Omega \rightarrow \{ - 1, + 1\} \) be random. For \( 1 \leq i \leq n \) define\n\n\[ \n{b}_{i} = \text{ nearest integer to }\frac{\chi \left( {A}_{i}\right) }{{20}\sqr... | Yes |
Theorem 12.3.1 Let \( \mathcal{A} \) be a family of \( n \) sets on \( m \) points with \( m \geq n \) . Suppose that lindisc \( \left( {\left. \mathcal{A}\right| }_{X}\right) \leq K \) for every subset \( X \) of at most \( n \) points. Then lindisc \( \left( \mathcal{A}\right) \leq K \) . | Proof. Let \( {p}_{1},\ldots ,{p}_{m} \in \left\lbrack {0,1}\right\rbrack \) be given. We define a reduction process. Call index \( j \) fixed if \( {p}_{j} \in \{ 0,1\} \), otherwise call it floating, and let \( F \) denote the set of floating indices. If \( \left| F\right| \leq n \) then halt. Otherwise, let \( {y}_{... | Yes |
Corollary 12.3.3 Let \( \mathcal{A} \) be a family of \( n \) sets on \( m \) points. Suppose \( \operatorname{disc}\left( {\left. \mathcal{A}\right| }_{X}\right) \leq K \) for every subset \( X \) with at most \( n \) points. Then \( \operatorname{disc}\left( \mathcal{A}\right) \leq {2K} \) . | Proof. For every \( X \subseteq \Omega \) with \( \left| X\right| \leq n \), herdisc \( \left( {\left. \mathcal{A}\right| }_{X}\right) \leq K \) so by Theorem 12.3.2 lindisc \( \left( {\left. \mathcal{A}\right| }_{X}\right) \leq K \) . By Theorem 12.3.1 lindisc \( \left( \mathcal{A}\right) \leq \bar{K} \) . But\n\n\[ \... | Yes |
For any family \( \mathcal{A} \) of \( n \) sets of arbitrary size | Apply Theorem 12.2.1 and Corollary 12.3.3. | No |
If a Hadamard matrix exists of order \( n > 1 \) then there exists a family \( \mathcal{A} \) consisting of \( n \) subsets of an \( n \) -set with | \[ \operatorname{disc}\left( \mathcal{A}\right) \geq \sqrt{n}/2 \] | No |
Theorem 12.5.1 Let \( \mathcal{A} \) be a finite family of finite sets, no restriction on either the number of sets nor on the cardinality of the sets, with \( \deg \left( \mathcal{A}\right) \leq t \) . Then\n\n\[ \operatorname{disc}\left( \mathcal{A}\right) \leq {2t} - 1 \] | Proof. For convenience write \( \mathcal{A} = \left\{ {{A}_{1},\ldots ,{A}_{m}}\right\} \) with all \( {A}_{i} \subseteq \Omega = \{ 1,\ldots, n\} \) . To each \( j \in \Omega \) there is assigned a value \( {\mathbf{x}}_{\mathbf{j}} \) which will change as the proof progresses. Initially all \( {x}_{j} = 0 \) . At the... | No |
Theorem 13.1.1 For every \( d \geq 1 \) there is a set of at least \( \left\lfloor {\frac{1}{2}{\left( \frac{2}{\sqrt{3}}\right) }^{d}}\right\rfloor \) points in the \( d \) -dimensional Euclidean space \( {R}^{d} \), such that all angles determined by three points from the set are strictly less than \( \pi /2 \) . | Proof.[Theorem 13.1.1] We select the points of a set \( X \) in \( {R}^{d} \) from the vertices of the \( d \) -dimensional cube. As usual, we view the vertices of the cube, which are 0,1-vectors of length \( d \), as the characteristic vectors of subsets of a \( d \) -element set; i.e., each 0,1-vector \( a \) of leng... | Yes |
Let \( {I}_{1},{I}_{2},\ldots ,{I}_{n} \) be parallel unit intervals in the plane, where\n\n\[ \n{I}_{i} = \{ \left( {x, y}\right) : x = i,0 \leq y \leq 1\}\n\]\n\nFor each \( i \) let us choose a point \( {p}_{i} \) randomly and independently from \( {I}_{i} \) according to a uniform distribution. Let \( X \) be the s... | We first estimate the probability that the triangle determined by the points \( {p}_{i},{p}_{i + a} \) and \( {p}_{i + k} \) is empty, for some fixed \( i, a \) and \( k = a + b \geq 3 \) . Let \( A = \left( {i, x}\right) \) , \( B = \left( {i + a, y}\right) \) and \( C = \left( {i + k, z}\right) \) be the points \( {p... | Yes |
Theorem 13.3.1 For all \( m \geq n \) ,\n\n\[ d\left( {m, n}\right) \leq \left( {n + 1}\right) /2 + \sqrt{\frac{n - 1}{2}\log m} \] | For the proof, we need a definition and two lemmas. For a vector \( \mathbf{a} = \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) of +1,-1-entries, the number of sign-changes in \( \mathbf{a} \) is the number of indices \( i \) , \( 1 \leq i \leq n - 1 \) such that \( {a}_{i} = - {a}_{i + 1} \) . For a matrix \( A \) of \( +... | No |
Lemma 13.3.2 For any matrix \( A \) of \( + 1, - 1 \) -entries, \( d\left( A\right) \leq s\left( A\right) + 1 \) . | Proof. Let \( A = \left( {a}_{i, j}\right) \) be an \( m \) by \( n \) matrix of \( + 1, - 1 \) entries and suppose \( s = s\left( A\right) \) . Let \( {t}_{1} < {t}_{2} < \ldots < {t}_{n} \) be arbitrary reals, and define \( n \) points \( {P}_{1},{P}_{2},\ldots ,{P}_{n} \) in \( {R}^{s + 1} \) by: \( {P}_{j} = \left(... | Yes |
Lemma 13.3.3 For every \( m \) by \( n \) matrix \( A \) of \( + 1, - 1 \) -entries there is a matrix \( B \) obtained from \( A \) by multiplying some of the columns of \( A \) by -1, such that \( s\left( B\right) \leq \left( {n - 1}\right) /2 + \sqrt{\frac{n - 1}{2}\log m} \) . | Proof. For each column of \( A \), randomly and independently, choose a number \( \epsilon \in \{ + 1, - 1\} \), where each of the two choices is equally likely, and multiply this column by \( \epsilon \) . Let \( B \) be the random sign-matrix obtained in this way. Consider an arbitrary fixed row of \( B \) . One can ... | Yes |
Lemma 13.4.1 If \( \\left( {X, R}\\right) \) is a range space of VC-dimension \( d \) with \( \\left| X\\right| = n \) points then \( \\left| R\\right| \\leq g\\left( {d, n}\\right) \) . | Proof. We apply induction on \( n + d \) . The assertion is trivially true for \( d = 0 \) and \( n = 0 \) . Assuming it holds for \( n \) and \( d - 1 \) and for \( n - 1 \) and \( d - 1 \) we prove it for \( n \) and \( d \) . Let \( S = \\left( {X, R}\\right) \) be a range space of VC-dimension \( d \) on \( n \) po... | Yes |
Corollary 13.4.3 Let \( \left( {X, R}\right) \) be a range space of VC-dimension \( d \geq 2 \), and let \( \left( {X,{R}_{h}}\right) \) be the range space on \( X \) in which \( {R}_{h} = \left\{ {\left( {{r}_{1} \cap \ldots \cap {r}_{h}}\right) : {r}_{1},\ldots {r}_{h} \in R}\right\} \) . Then \( {VC}\left( {X,{R}_{h... | Proof. Let \( A \) be an arbitrary subset of cardinality \( n \) of \( X \) . By Corollary 13.4.2 \( \left| {{P}_{R}\left( A\right) }\right| \leq g\left( {d, n}\right) \leq {n}^{d} \) . Since each member of \( {P}_{{R}_{h}}\left( A\right) \) is an intersection of \( h \) members of \( {P}_{R}\left( A\right) \) it follo... | Yes |
Theorem 13.4.4 There is a positive constant \( c \) such that if \( \left( {X, R}\right) \) is any range-space of VC-dimension at most \( d, A \subset X \) is a finite subset and \( \epsilon ,\delta > 0 \), then a random subset \( B \) of cardinality \( s \) of \( A \) where \( s \) is at least the minimum between \( \... | \[ \frac{c}{{\epsilon }^{2}}\left( {\operatorname{dlog}\frac{d}{\epsilon } + \log \frac{1}{\delta }}\right) \] is an \( \epsilon \) -sample for \( A \) with probability at least \( 1 - \delta \) . | Yes |
Theorem 13.4.5 Let \( \\left( {X, R}\\right) \) be a range space of VC-dimension \( d \), let \( A \) be a finite subset of \( X \) and supppose \( 0 < \\epsilon ,\\delta < 1 \) . Let \( N \) be a set obtained by \( m \) random independent draws from \( A \), where\n\n\[ m \\geq \\max \\left( {\\frac{4}{\\epsilon }\\lo... | Proof.[Theorem 13.4.5] Let \( \\left( {X, R}\\right) \) be a range space with VC-dimension \( d \), and let \( A \) be a subset of \( X \) of cardinality \( \\left| A\\right| = n \) . Suppose \( m \) satisfies (13.2), and let \( N = \\left( {{x}_{1},\\ldots ,{x}_{m}}\\right) \) be obtained by \( m \) independent random... | Yes |
Lemma 13.5.2 Let \( \left( {A,\mathcal{F}}\right), n, h, t \) and \( c \) be as above, let \( B \) be a finite subset of \( p > 1 \) points of \( A \), and let \( \mathcal{G} \) be a collection of \( m \) (not necessarily distinct) members of \( \mathcal{F} \). Then there are two distinct points \( x, y \) in \( B \), ... | Proof. We may and will assume that \( p \) is larger than \( c + 1 \). Let \( g \) be the largest integer such that \( c{g}^{t} \leq p - 1 \), that is, \( g = \left\lfloor {\left( \frac{p - 1}{c}\right) }^{1/t}\right\rfloor \). Let \( L \) be a random collection of \( g \) members of \( \mathcal{G} \), each picked, ran... | Yes |
Theorem 13.5.3 Let \( \left( {A,\mathcal{F}}\right) \) be a finite range space, where \( \left| A\right| = n \), and suppose that its dual shatter function \( h \) satisfies \( h\left( g\right) \leq c{g}^{t} \) for some fixed \( c, t > 0 \) . Then, there is a \( {C}^{\prime } = {C}^{\prime }\left( {c, t}\right) \) such... | Proof. Without loss of generality, assume that the number of points of \( A \) is even (otherwise, simply omit a point). By Theorem 13.5.1 there is a Hamilton path \( {x}_{1}{x}_{2}\ldots {x}_{n} \) on these points such that each member of \( \mathcal{F} \) is stabbed by at most \( C{n}^{1 - 1/t}\log n \) edges of the ... | No |
Theorem 1 Let \( C \) be bounded, convex and centrally symmetric about the origin. Then\n\n\[ \delta \left( C\right) \geq {2}^{-\left( {n - 1}\right) } \] | Proof. Fix \( \epsilon > 0 \) . Normalize so \( \mu = \mu \left( C\right) = 2 - \epsilon \) . For any real \( z \) let \( {C}_{z} \) denote the \ | No |
Theorem 14.1.1 [Shannon’s Theorem] Let \( p \in \left( {0,{.5}}\right) \) be fixed. For \( \epsilon > 0 \) arbitrarily small there exists a Coding Scheme with rate of transmission greater than \( 1 - H\left( p\right) - \epsilon \) and probability of incorrect transmission less than \( \epsilon \) . | Proof. Let \( \delta > 0 \) be such that \( p + \delta < {.5} \) and \( H\left( {p + \delta }\right) < H\left( p\right) + \epsilon /2 \) . For \( n \) large set \( m = n\left( {1 - H\left( p\right) - \epsilon }\right) \), guaranteeing the rate of transmission. Let \( f \) : \( \{ 0,1{\} }^{m} \rightarrow \{ 0,1{\} }^{n... | Yes |
Theorem 14.1.2 Let \( p \in \left( {0,{.5}}\right) \) be fixed. For \( \epsilon > 0 \) arbitrarily small there exists a Group Code with rate of transmission greater than \( 1 - H\left( p\right) - \epsilon \) and probability of incorrect transmission less than \( \epsilon \) . | Proof. For \( 1 \leq i \leq m \) let \( {u}_{i} \in \{ 0,1{\} }^{m} \) be that vector with a one in position \( i \) , all other entries zero. Let \( f\left( {u}_{1}\right) ,\ldots, f\left( {u}_{m}\right) \) be chosen randomly and independently and then extend \( f \) by setting\n\n\[ f\left( {{\epsilon }_{1}{u}_{1} + ... | Yes |
Corollary 14.2.2 If\n\n\[ \n n > \frac{{2}^{q}}{\mathop{\sum }\limits_{{i = 0}}^{k}\left( \begin{array}{l} q \\ i \end{array}\right) } \n\]\n\nthen Carole wins the \( \left( {n, q, k}\right) \) -Liar Game. | Proof.[Theorem 14.2.1] Fix a strategy for Paul. Now Carole plays randomly! That is, at each round, after Paul has selected a set \( S \) of chips Carole flips a coin - if it comes up heads she moves every chip not in \( S \) one position to the left and if it comes up tails she moves every chip in \( S \) one position ... | Yes |
Theorem 14.3.1 If \( \mathop{\sum }\limits_{i}{x}_{i}{2}^{-i} < 1 \) then Carole wins the \( \left( {{x}_{1},\ldots ,{x}_{k}}\right) \) -Tenure Game. | Proof. Fix a strategy for Paul. Now Carole plays randomly! That is, at each round, after Paul has selected a set \( S \) of chips Carole flips a coin - if it comes up heads she moves every chip not in \( S \) one position to the left and if it comes up tails she moves every chip in \( S \) one position to the left. For... | Yes |
Lemma 14.3.2 If a set of chips has weight at least one it may be split into two parts each of weight at least one half. | Proof. There must be two chips at some position \( i \), otherwise the weight is less than one. If there are two chips at position 1 simply split them. If there are two chips at position \( i > 1 \) glue them together, and consider them as one superchip at position \( i - 1 \) . Then the proof follows by induction on t... | No |
Theorem 14.3.3 If \( \sum {x}_{i}{2}^{-i} \geq 1 \) then Paul wins the \( \left( {{x}_{1},\ldots ,{x}_{k}}\right) \) -Tenure Game. | Proof. The initial weight is at least one. Applying the Lemma Paul splits the chips into two parts each of weight at least one half and sets \( S \) equal one of the parts. Carole moves all chips in one part one position to the left, doubling their weight, leaving a new position of weight at least one. Thus the weight ... | Yes |
Theorem 14.4.1 If \( \Pr \left\lbrack {\left| {S}_{n}\right| > \alpha }\right\rbrack < {n}^{-1} \) then \( \operatorname{VAL}\left( n\right) \leq \alpha \) . | Proof. Consider the game a win for Pusher if the final \( {\left| P\right| }_{\infty } > \alpha \) . Suppose Chooser announces that she will flip a fair coin each round to determine whether to reset \( P \) as \( P + v \) or \( P - v \) . Let \( {x}_{i} \) be the \( i \) -th coordinate for the final value of the positi... | Yes |
Proposition 14.6.2 Let \( X = \left( {{X}_{1},\ldots ,{X}_{n}}\right) \) be a random variable taking values in the set \( S = {S}_{1} \times {S}_{2} \times \ldots \times {S}_{n} \), where each of the coordinates \( {X}_{i} \) of \( X \) is a random variable taking values in \( {S}_{i} \) . Then\n\n\[ H\left( X\right) \... | Proof. This follows by induction from Lemma 14.6.1, part 3. ∎ | No |
Corollary 14.6.3 Let \( \mathcal{F} \) be a family of subsets of \( \{ 1,2,\ldots, n\} \) and let \( {p}_{i} \) denote the fraction of sets in \( \mathcal{F} \) that contain \( i \) . Then\n\n\[ \left| \mathcal{F}\right| \leq {2}^{\mathop{\sum }\limits_{{i = 1}}^{n}H\left( {p}_{i}\right) } \]\n\nwhere \( H\left( y\righ... | Proof. Associate each set \( F \in \mathcal{F} \) with its characteristic vector \( v\left( F\right) \), which is a binary vector of length \( n \) . Let \( X = \left( {{X}_{1},\ldots ,{X}_{n}}\right) \) be the random variable taking values in \( \{ 0,1{\} }^{n} \), where \( P\left( {X = v\left( F\right) }\right) = 1/\... | Yes |
Proposition 14.6.4 Let \( X = \left( {{X}_{1},\ldots ,{X}_{n}}\right) \) and \( S \) be as above. If \( \mathcal{G} \) is a family of subsets of \( \{ 1,\ldots, n\} \) and each \( i \in \{ 1,\ldots, n\} \) belongs to at least \( k \) members of \( \mathcal{G} \) then\n\n\[ \n{kH}\left( X\right) \leq \mathop{\sum }\limi... | Proof. We apply induction on \( k \) . For \( k = 1 \), replace each set \( G \in \mathcal{G} \) by a subset of it to obtain a family \( {\mathcal{G}}^{\prime } \) whose members form a partition of \( \{ 1,\ldots, n\} \) . By Lemma 14.6.1, part 2, \( \mathop{\sum }\limits_{{G \in \mathcal{G}}}H\left( {X\left( G\right) ... | Yes |
Corollary 14.6.5 Let \( \\mathcal{F} \) be a family of vectors in \( {S}_{1} \\times {S}_{2}\\ldots \\times {S}_{n} \) . Let \( \\mathcal{G} = \) \( \\left\\{ {{G}_{1},{G}_{2},\\ldots {G}_{m}}\\right\\} \) be a collection of subsets of \( N = \\{ 1,2,\\ldots, n\\} \), and suppose that each element \( i \\in N \) belong... | Proof. Let \( X = \\left( {{X}_{1},\\ldots ,{X}_{n}}\\right) \) be the random variable taking values in \( \\mathcal{F} \), where \( P\\left( {X = F}\\right) = \\frac{1}{\\left| \\mathcal{F}\\right| } \) for all \( F \\in \\mathcal{F} \) . By Proposition 14.6.4\n\n\[ \n{kH}\\left( X\\right) \\leq \\mathop{\\sum }\\limi... | Yes |
Corollary 14.6.7 [Chung et al. (1986) ] Let \( N \) be a finite set, and let \( \mathcal{F} \) be a family of subsets of \( N \) . Let \( \mathcal{G} = \left\{ {{G}_{1},\ldots {G}_{m}}\right\} \) be a collection of subsets of \( N \), and suppose that each element of \( S \) belongs to at least \( k \) members of \( \m... | \[ {\left| \mathcal{F}\right| }^{k} \leq \mathop{\prod }\limits_{{i = 1}}^{m}\left| {\mathcal{F}}_{i}\right| \] | Yes |
Corollary 14.6.8 Let \( \mathcal{F} \) be a family of graphs on the labeled set of vertices \( \{ 1,2,\ldots t\} \), and suppose that for any two members of \( \mathcal{F} \) there is a triangle contained in both of them. Then\n\n\[ \left| \mathcal{F}\right| < \frac{1}{4}{2}^{\left( \begin{array}{l} t \\ 2 \end{array}\... | Proof. Let \( N \) be the set of all \( \left( \begin{array}{l} t \\ 2 \end{array}\right) \) unordered pairs of vertices in \( T = \{ 1,2\ldots, t\} \) , and consider \( \mathcal{F} \) as a family of subsets of \( N \) . Let \( \mathcal{G} \) be the family of all subsets of \( N \) consisting of the edge-sets of unions... | Yes |
Theorem 15.2.1 Suppose \( n = {2}^{k} - 1 \) and \( d = {2t} + 1 \) . Then there exists a symmetric probability space \( \Omega \) of size \( 2{\left( n + 1\right) }^{t} \) and \( d \) -wise independent random variables \( {y}_{1},\ldots ,{y}_{n} \) over \( \Omega \) each of which takes the values 0 and 1 with probabil... | The space and the variables are explicitly constructed, given a representation of the field \( F = {GF}\left( {2}^{k}\right) \) as a \( k \) -dimensional algebra over \( {GF}\left( 2\right) \) . Proof. Let \( {x}_{1},\ldots ,{x}_{n} \) be the \( n \) nonzero elements of \( F \), represented as column-vectors of length ... | Yes |
Lemma 15.2.2 Any set of \( d = {2t} + 1 \) columns of \( H \) is linearly independent over \( {GF}\left( 2\right) \) . | Proof. Let \( J \subset \{ 1,2,\ldots, n\} \) be a subset of cardinality \( \left| J\right| = {2t} + 1 \) of the set of indices of the columns of \( H \) . Suppose that \( \mathop{\sum }\limits_{{j \in J}}{z}_{j}{H}_{j} = 0 \), where \( {H}_{j} \) denotes the \( j \) -th column of \( H \) and \( {z}_{j} \in {GF}\left( ... | Yes |
Proposition 15.2.3 If the random variables \( {y}_{1},\ldots ,{y}_{n} \) over the sample space \( \Omega \) are \( d \) -wise independent and none of them is almost constant then \( \left| \Omega \right| \geq m\left( {n, d}\right) \) . | Proof. Clearly we may assume that the expected value of each \( {y}_{j} \) is 0 (since otherwise we can replace \( {y}_{j} \) by \( {y}_{j} - E\left( {y}_{j}\right) \) ). For each subset \( S \) of \( \{ 1,\ldots, n\} \), define \( {\alpha }_{S} = \mathop{\prod }\limits_{{j \in S}}{y}_{j} \) . Observe that since no \( ... | Yes |
Theorem 1 The crossing number of any simple graph \( G = \left( {V, E}\right) \) with \( \left| E\right| \geq 4\left| V\right| \) is at least \( \frac{{\left| E\right| }^{3}}{{64}{\left| V\right| }^{2}} \) . | Proof. By Euler’s formula any simple planar graph with \( n \) vertices has at most \( {3n} - 6 \) edges, implying that the crossing number of any simple graph with \( n \) vertices and \( m \) edges is at least \( m - \left( {{3n} - 6}\right) > m - {3n} \) . Let \( G = \left( {V, E}\right) \) be a graph with \( \left|... | Yes |
Theorem 2 Let \( P \) be a set of \( n \) distinct points in the plane, and let \( L \) be a set of \( m \) distinct lines. Then, the number of incidences between the members of \( P \) and those of \( L \) (that is, the number of pairs \( \left( {p, l}\right) \) with \( p \in P, l \in L \) and \( p \in l \) ) is at mo... | Proof. Denote the number of incidences by \( I \) . Let \( G = \left( {V, E}\right) \) be the graph whose vertices are all members of \( P \), where two are adjacent if and only if they are consecutive points of \( P \) on some line in \( L \) . Clearly, \( \left| V\right| = n \) and \( \left| E\right| = I - m \) . Not... | Yes |
Theorem 3 For any three sets \( A, B \) and \( C \) of \( s \) real numbers each,\n\n\[ \left| {A \cdot B + C}\right| = \left| {\{ {ab} + c : a \in A, b \in B, c \in C\} }\right| \geq \Omega \left( {s}^{3/2}\right) . | Proof. Put \( R = A \cdot B + C,\left| R\right| = r \) and define\n\n\[ P = \{ \left( {a, t}\right) : a \in A, t \in R\} ,\;L = \{ y = {bx} + c : b \in B, c \in C\} .\n\nThus \( P \) is a set of \( n = {sr} \) points in the plane, \( L \) is a set of \( m = {s}^{2} \) lines in the plane, and each line \( y = {bx} + c \... | Yes |
Proposition 1 Let \( G = \left( {V, E}\right) \) be a triangle-free graph on \( n \) vertices with maximum degree at most \( d \geq 1 \) . Then\n\n\[ \alpha \left( G\right) \geq \frac{n\log d}{8d} \]\n\nwhere the logarithm here and in what follows is in base 2. | Proof. If, say, \( d < {16} \) the result follows from the trivial bound \( \alpha \left( G\right) \geq n/\left( {d + 1}\right) \) and hence we may and will assume that \( d \geq {16} \) . Let \( W \) be a random independent set of vertices in \( G \), chosen uniformly among all independent sets in \( G \) . For each v... | Yes |
Theorem 2 [Ajtai et al. (1980) ] There exists an absolute constant \( b \) such that \( r\left( {3, k}\right) \leq b{k}^{2}/\log k \) for every \( k > 1 \) . | Proof. Let \( G = \left( {V, E}\right) \) be a triangle-free graph on \( 8{k}^{2}/\log k \) vertices. If \( G \) has a vertex of degree at least \( k \) then its neighborhood contains an independent set of size \( k \) . Otherwise, by proposition 1 above, \( G \) contains an independent set of size at least \( \frac{8{... | Yes |
Proposition 1.1 Let \( {\left( {X}_{n}\right) }_{n \geq 1} \) be a sequence of real random variables such that, for every \( n \geq 1,{X}_{n} \) follows the \( \mathcal{N}\left( {{m}_{n},{\sigma }_{n}^{2}}\right) \) -distribution. Suppose that \( {X}_{n} \) converges in \( {L}^{2} \) to X. Then:\n\n(i) The random varia... | (i) The convergence in \( {L}^{2} \) implies that \( {m}_{n} = E\left\lbrack {X}_{n}\right\rbrack \) converges to \( E\left\lbrack X\right\rbrack \) and \( {\sigma }_{n}^{2} = \) \( \operatorname{var}\left( {X}_{n}\right) \) converges to \( \operatorname{var}\left( X\right) \) as \( n \rightarrow \infty \) . Then, sett... | Yes |
Proposition 1.2 Under the preceding assumptions, the random variables \( {X}_{1},\ldots ,{X}_{d} \) are independent if and only if the covariance matrix \( {\left( \operatorname{cov}\left( {X}_{j},{X}_{k}\right) \right) }_{1 \leq j, k \leq d} \) is diagonal or equivalently if and only if \( {q}_{X} \) is of diagonal fo... | Proof If the random variables \( {X}_{1},\ldots ,{X}_{d} \) are independent, the covariance matrix \( {\left( \operatorname{cov}\left( {X}_{j},{X}_{k}\right) \right) }_{j, k = 1,\ldots d} \) is diagonal. Conversely, if this matrix is diagonal, we have for every \( u = \mathop{\sum }\limits_{{j = 1}}^{d}{u}_{j}{e}_{j} \... | Yes |
Proposition 1.7 If \( {\left( {X}_{t}\right) }_{t \in T} \) is a Gaussian process, the closed linear subspace of \( {L}^{2} \) spanned by the variables \( {X}_{t}, t \in T \), is a Gaussian space, which is called the Gaussian space generated by the process \( X \) . | Proof It suffices to observe that an \( {L}^{2} \) -limit of centered Gaussian variables is still centered Gaussian, by Proposition 1.1. | No |
Theorem 1.9 Let \( H \) be a centered Gaussian space and let \( {\left( {H}_{i}\right) }_{i \in I} \) be a collection of linear subspaces of \( H \) . Then the subspaces \( {H}_{i}, i \in I \), are (pairwise) orthogonal in \( {L}^{2} \) if and only the \( \sigma \) -fields \( \sigma \left( {H}_{i}\right), i \in I \), a... | Proof Suppose that the \( \sigma \) -fields \( \sigma \left( {H}_{i}\right) \) are independent. Then, if \( i \neq j \), if \( X \in {H}_{i} \) and \( Y \in {H}_{j} \) ,\n\n\[ E\left\lbrack {XY}\right\rbrack = E\left\lbrack X\right\rbrack E\left\lbrack Y\right\rbrack = 0, \]\n\nso that the linear spaces \( {H}_{i} \) a... | Yes |
Corollary 1.10 Let \( H \) be a (centered) Gaussian space and let \( K \) be a closed linear subspace of \( H \) . Let \( {p}_{K} \) denote the orthogonal projection onto \( K \) in the Hilbert space \( {L}^{2} \), and let \( X \in H \) .\n\n(i) We have\n\n\[ E\left\lbrack {X \mid \sigma \left( K\right) }\right\rbrack ... | (i) Let \( Y = X - {p}_{K}\left( X\right) \) . Then \( Y \) is orthogonal to \( K \) and, by Theorem \( {1.9}, Y \) is independent of \( \sigma \left( K\right) \) . Then,\n\n\[ E\left\lbrack {X \mid \sigma \left( K\right) }\right\rbrack = E\left\lbrack {{p}_{K}\left( X\right) \mid \sigma \left( K\right) }\right\rbrack ... | Yes |
Theorem 1.11 Let \( \\Gamma \) be a symmetric function of positive type on \( T \\times T \) . There exists, on an appropriate probability space \( \\left( {\\Omega ,\\mathcal{F}, P}\\right) \), a centered Gaussian process whose covariance function is \( \\Gamma \) . | Example Consider the case \( T = \\mathbb{R} \) and let \( \\mu \) be a finite measure on \( \\mathbb{R} \), which is also symmetric (i.e. \( \\mu \\left( {-A}\\right) = \\mu \\left( A\\right) \) ). Then set, for every \( s, t \\in \\mathbb{R} \) ,\n\n\\[ \n\\Gamma \\left( {s, t}\\right) = \\int {\\mathrm{e}}^{\\mathrm... | Yes |
Proposition 1.13 Let \( \\left( {E,\\mathcal{E}}\\right) \) be a measurable space, and let \( \\mu \) be a \( \\sigma \) -finite measure on \( \\left( {E,\\mathcal{E}}\\right) \). There exists, on an appropriate probability space \( \\left( {\\Omega ,\\mathcal{F}, P}\\right) \), a Gaussian white noise with intensity \(... | Proof We rely on elementary Hilbert space theory. Let \( \\left( {{f}_{i}, i \\in I}\\right) \) be a total orthonormal system in the Hilbert space \( {L}^{2}\\left( {E,\\mathcal{E},\\mu }\\right) \). For every \( f \\in {L}^{2}\\left( {E,\\mathcal{E},\\mu }\\right) \), \[ f = \\mathop{\\sum }\\limits_{{i \\in I}}{\\alp... | Yes |
Proposition 2.2 Pre-Brownian motion is a centered Gaussian process with covariance\n\n\[ K\left( {s, t}\right) = \min \{ s, t\} \overset{\text{ (not.) }}{ = }s \land t. \] | Proof By the definition of a Gaussian white noise, the variables \( {B}_{t} \) belong to a common Gaussian space, and therefore \( {\left( {B}_{t}\right) }_{t \geq 0} \) is a Gaussian process. Moreover, for every \( s, t \geq 0 \) ,\n\n\[ E\left\lbrack {{B}_{s}{B}_{t}}\right\rbrack = E\left\lbrack {G\left( \left\lbrack... | Yes |
Corollary 2.4 Let \( {\left( {B}_{t}\right) }_{t \geq 0} \) be a pre-Brownian motion. Then, for every choice of \( 0 = {t}_{0} < {t}_{1} < \cdots < {t}_{n} \), the law of the vector \( \left( {{B}_{{t}_{1}},{B}_{{t}_{2}},\ldots ,{B}_{{t}_{n}}}\right) \) has density | Proof The random variables \( {B}_{{t}_{1}},{B}_{{t}_{2}} - {B}_{{t}_{1}},\ldots ,{B}_{{t}_{n}} - {B}_{{t}_{n - 1}} \) are independent with respective distributions \( \mathcal{N}\left( {0,{t}_{1}}\right) ,\mathcal{N}\left( {0,{t}_{2} - {t}_{1}}\right) ,\ldots ,\mathcal{N}\left( {0,{t}_{n} - {t}_{n - 1}}\right) \). Hen... | Yes |
Proposition 2.5 Let \( B \) be a pre-Brownian motion. Then,\n\n(i) \( - B \) is also a pre-Brownian motion (symmetry property);\n\n(ii) for every \( \lambda > 0 \), the process \( {B}_{t}^{\lambda } = \frac{1}{\lambda }{B}_{{\lambda }^{2}t} \) is also a pre-Brownian motion (invariance under scaling);\n\n(iii) for every... | Proof (i) and (ii) are very easy. Let us prove (iii). With the notation of the proof of Proposition 2.3, the \( \sigma \) -field generated by \( {B}^{\left( s\right) } \) is \( \sigma \left( {\widetilde{H}}_{s}\right) \), which is independent of \( \sigma \left( {H}_{s}\right) = \sigma \left( {{B}_{r}, r \leq s}\right)... | No |
Theorem 2.9 (Kolmogorov’s lemma) Let \( X = {\left( {X}_{t}\right) }_{t \in I} \) be a random process indexed by a bounded interval \( I \) of \( \mathbb{R} \), and taking values in a complete metric space \( \left( {E, d}\right) \) . Assume that there exist three reals \( q,\varepsilon, C > 0 \) such that, for every \... | Proof To simplify the presentation, we take \( I = \left\lbrack {0,1}\right\rbrack \), but the proof would be the same for any bounded interval (closed or not). We fix \( \alpha \in \left( {0,\frac{\varepsilon }{q}}\right) \).\n\nThe assumption of the theorem implies that, for \( a > 0 \) and \( s, t \in I \) ,\n\n\[ P... | Yes |
Lemma 2.10 Let \( f \) be a mapping defined on \( D \) and with values in the metric space \( \\left( {E, d}\\right) \). Assume that there exists a real \( \\alpha > 0 \) and a constant \( K < \\infty \) such that, for every integer \( n \\geq 1 \) and every \( i \\in \\left\\{ {1,2,\\ldots ,{2}^{n} - 1}\\right\\} \), | Proof of Lemma 2.10 Fix \( s, t \\in D \) with \( s < t \). Let \( p \\geq 1 \) be the smallest integer such that \( {2}^{-p} \\leq t - s \), and let \( k \\geq 0 \) be the smallest integer such that \( k{2}^{-p} \\geq s \). Then, we may write | No |
Corollary 2.11 Let \( B = {\left( {B}_{t}\right) }_{t \geq 0} \) be a pre-Brownian motion. The process \( B \) has a modification whose sample paths are continuous, and even locally Hölder continuous with exponent \( \frac{1}{2} - \delta \) for every \( \delta \in \left( {0,\frac{1}{2}}\right) \) . | Proof If \( s < t \), the random variable \( {B}_{t} - {B}_{s} \) is distributed according to \( \mathcal{N}\left( {0, t - s}\right) \) , and thus \( {B}_{t} - {B}_{s} \) has the same law as \( \sqrt{t - s}U \), where \( U \) is \( \mathcal{N}\left( {0,1}\right) \) . Consequently, for every \( q > 0 \) ,\n\n\[ E\left\l... | Yes |
Theorem 2.13 (Blumenthal’s zero-one law) The \( \\sigma \) -field \( {\\mathcal{F}}_{0 + } \) is trivial, in the sense that \( P\\left( A\\right) = 0 \) or 1 for every \( A \\in {\\mathcal{F}}_{0 + } \) . | Proof Let \( 0 < {t}_{1} < {t}_{2} < \\cdots < {t}_{k} \) and let \( g : {\\mathbb{R}}^{k} \\rightarrow \\mathbb{R} \) be a bounded continuous function. Also fix \( A \\in {\\mathcal{F}}_{0 + } \). Then, by a continuity argument,\n\n\[ E\\left\\lbrack {{\\mathbf{1}}_{A}g\\left( {{B}_{{t}_{1}},\\ldots ,{B}_{{t}_{k}}}\\r... | Yes |
Corollary 2.15 Almost surely, the function \( t \mapsto {B}_{t} \) is not monotone on any nontrivial interval. | Proof Using assertion (i) of Proposition 2.14 and the simple Markov property, we immediately get that a.s. for every rational \( q \in {\mathbb{Q}}_{ + } \), for every \( \varepsilon > 0 \) ,\n\n\[ \mathop{\sup }\limits_{{q \leq t \leq q + \varepsilon }}{B}_{t} > {B}_{q},\;\mathop{\inf }\limits_{{q \leq t \leq q + \var... | Yes |
Proposition 2.16 Let \( 0 = {t}_{0}^{n} < {t}_{1}^{n} < \cdots < {t}_{{p}_{n}}^{n} = t \) be a sequence of subdivisions of \( \left\lbrack {0, t}\right\rbrack \) whose mesh tends to 0 (i.e. \( \mathop{\sup }\limits_{{1 \leq i \leq {p}_{n}}}\left( {{t}_{i}^{n} - {t}_{i - 1}^{n}}\right) \rightarrow 0 \) as \( n \rightarr... | Proof This is an immediate consequence of Proposition 1.14, writing \( {B}_{{t}_{i}^{n}} - {B}_{{t}_{i - 1}^{n}} = \) \( G\left( \left( {{t}_{i - 1}^{n},{t}_{i}^{n}}\right\rbrack \right) \), where \( G \) is the Gaussian white noise associated with \( B \) . | Yes |
Corollary 2.17 Almost surely, the function \( t \mapsto {B}_{t} \) has infinite variation on any non-trivial interval. | Proof From the simple Markov property, it suffices to consider the interval \( \left\lbrack {0, t}\right\rbrack \) for some fixed \( t > 0 \) . We use Proposition 2.16, and note that by extracting a subsequence we may assume that the convergence in this proposition holds a.s. We then observe that\n\n\[ \mathop{\sum }\l... | Yes |
Theorem 2.21 For every \( t > 0 \), set \( {S}_{t} = \mathop{\sup }\limits_{{s \leq t}}{B}_{s} \) . Then, if \( a \geq 0 \) and \( b \in ( - \infty, a\rbrack \) , we have\n\n\[ P\left( {{S}_{t} \geq a,{B}_{t} \leq b}\right) = P\left( {{B}_{t} \geq {2a} - b}\right) . \]\n\nIn particular, \( {S}_{t} \) has the same distr... | Proof We apply the strong Markov property at the stopping time\n\n\[ {T}_{a} = \inf \left\{ {t \geq 0 : {B}_{t} = a}\right\} . \]\n\nWe already saw (Proposition 2.14) that \( {T}_{a} < \infty \) a.s. Then, using the notation of Theorem 2.20, we have\n\n\[ P\left( {{S}_{t} \geq a,{B}_{t} \leq b}\right) = P\left( {{T}_{a... | Yes |
Corollary 2.22 For every \( a > 0,{T}_{a} \) has the same distribution as \( \frac{{a}^{2}}{{B}_{1}^{2}} \) and has density\n\n\[ f\left( t\right) = \frac{a}{\sqrt{{2\pi }{t}^{3}}}\exp \left( {-\frac{{a}^{2}}{2t}}\right) {\mathbf{1}}_{\{ t > 0\} }.\] | Proof Using Theorem 2.21 in the second equality, we have, for every \( t \geq 0 \), \n\n\[ P\left( {{T}_{a} \leq t}\right) = P\left( {{S}_{t} \geq a}\right) = P\left( {\left| {B}_{t}\right| \geq a}\right) = P\left( {{B}_{t}^{2} \geq {a}^{2}}\right) = P\left( {t{B}_{1}^{2} \geq {a}^{2}}\right) = P\left( {\frac{{a}^{2}}{... | Yes |
Proposition 3.4 Let \( {\left( {X}_{t}\right) }_{t \geq 0} \) be a random process with values in a metric space \( \left( {E, d}\right) \) (equipped with its Borel \( \sigma \) -field). Suppose that \( X \) is adapted and that the sample paths of \( X \) are right-continuous (i.e. for every \( \omega \in \Omega, t \map... | Proof We treat only the case of right-continuous sample paths, as the other case is similar. Fix \( t > 0 \) . For every \( n \geq 1 \) and \( s \in \left\lbrack {0, t}\right\rbrack \), define a random variable \( {X}_{s}^{n} \) by setting\n\n\[ \n{X}_{s}^{n} = {X}_{{kt}/n}\;\text{ if }s \in \lbrack \left( {k - 1}\righ... | Yes |
Proposition 3.6 Write \( {\mathcal{G}}_{t} = {\mathcal{F}}_{t + } \) for every \( t \in \left\lbrack {0,\infty }\right\rbrack \) . (i) A random variable \( T : \Omega \rightarrow \left\lbrack {0,\infty }\right\rbrack \) is a stopping time of the filtration \( \left( {\mathcal{G}}_{t}\right) \) if and only if \( \{ T < ... | Proof (i) Suppose that \( T \) is a stopping time of the filtration \( \left( {\mathcal{G}}_{t}\right) \) . Then, for every \( t > 0 \) ,\n\n\[ \{ T < t\} = \mathop{\bigcup }\limits_{{q \in {\mathbb{Q}}_{ + }, q < t}}\{ T \leq q\} \in {\mathcal{F}}_{t} \]\n\nbecause \( \{ T \leq q\} \in {\mathcal{G}}_{q} \subset {\math... | Yes |
Theorem 3.7 Let \( {\left( {X}_{t}\right) }_{t \geq 0} \) be a progressive process with values in a measurable space \( \left( {E,\mathcal{E}}\right) \), and let \( T \) be a stopping time. Then the function \( \omega \mapsto {X}_{T}\left( \omega \right) \mathrel{\text{:=}} \) \( {X}_{T\left( \omega \right) }\left( \om... | Proof We use property (j) above. Let \( t \geq 0 \) . The restriction to \( \{ T \leq t\} \) of the function \( \omega \mapsto {X}_{T}\left( \omega \right) \) is the composition of the two mappings\n\n\[ \n\{ T \leq t\} \ni \omega \mapsto \left( {\omega, T\left( \omega \right) \land t}\right) \n\]\n\n\[ \n{\mathcal{F}}... | Yes |
Proposition 3.8 Let \( T \) be a stopping time and let \( S \) be an \( {\mathcal{F}}_{T} \) -measurable random variable with values in \( \left\lbrack {0,\infty }\right\rbrack \), such that \( S \geq T \) . Then \( S \) is also a stopping time. | Proof For the first assertion, we write, for every \( t \geq 0 \), \[ \{ S \leq t\} = \{ S \leq t\} \cap \{ T \leq t\} \in {\mathcal{F}}_{t} \] since \( \{ S \leq t\} \) is \( {\mathcal{F}}_{T} \) -measurable. The second assertion follows since \( {T}_{n} \geq T \), and \( {T}_{n} \) is a function of \( T \), hence \( ... | No |
Proposition 3.9 Let \( {\left( {X}_{t}\right) }_{t \geq 0} \) be an adapted process with values in a metric space \( \left( {E, d}\right) \). (i) Assume that the sample paths of \( X \) are right-continuous, and let \( O \) be an open subset of \( E \). Then \[ {T}_{O} = \inf \left\{ {t \geq 0 : {X}_{t} \in O}\right\} ... | Proof (i) For every \( t > 0 \), \[ \left\{ {{T}_{O} < t}\right\} = \mathop{\bigcup }\limits_{{s \in \lbrack 0, t) \cap \mathbb{Q}}}\left\{ {{X}_{s} \in O}\right\} \in {\mathcal{F}}_{t} \] and we use Proposition 3.6 (i). (ii) For every \( t \geq 0 \), \[ \left\{ {{T}_{F} \leq t}\right\} = \left\{ {\mathop{\inf }\limits... | Yes |
Proposition 3.12 Let \( {\left( {X}_{t}\right) }_{t \geq 0} \) be an adapted process and let \( f : \mathbb{R} \rightarrow {\mathbb{R}}_{ + } \) be a convex function such that \( E\left\lbrack {f\left( {X}_{t}\right) }\right\rbrack < \infty \) for every \( t \geq 0 \). (i) If \( {\left( {X}_{t}\right) }_{t \geq 0} \) i... | Proof By Jensen’s inequality, we have, for \( s < t \), \[ E\left\lbrack {f\left( {X}_{t}\right) \mid {\mathcal{F}}_{s}}\right\rbrack \geq f\left( {E\left\lbrack {{X}_{t} \mid {\mathcal{F}}_{s}}\right\rbrack }\right) \geq f\left( {X}_{s}\right) . \] In the last inequality, we need the fact that \( f \) is nondecreasing... | Yes |
Proposition 3.13 Let \( {\left( {X}_{t}\right) }_{t \geq 0} \) be a submartingale or a supermartingale. Then, for every \( t > 0 \), \[ \mathop{\sup }\limits_{{0 \leq s \leq t}}E\left\lbrack \left| {X}_{s}\right| \right\rbrack < \infty \] | Proof It is enough to treat the case where \( {\left( {X}_{t}\right) }_{t \geq 0} \) is a submartingale. Since \( {\left( {X}_{t}\right) }^{ + } \) is also a submartingale, we have for every \( s \in \left\lbrack {0, t}\right\rbrack \), \[ E\left\lbrack {\left( {X}_{s}\right) }^{ + }\right\rbrack \leq E\left\lbrack {\l... | Yes |
Proposition 3.14 Let \( {\left( {M}_{t}\right) }_{t \geq 0} \) be a square integrable martingale (that is, \( {M}_{t} \in {L}^{2} \) for every \( t \geq 0 \) ). Let \( 0 \leq s < t \) and let \( s = {t}_{0} < {t}_{1} < \cdots < {t}_{p} = t \) be a subdivision of the interval \( \left\lbrack {s, t}\right\rbrack \) . The... | Proof For every \( i = 1,\ldots, p \) ,\n\n\[ E\left\lbrack {{\left( {M}_{{t}_{i}} - {M}_{{t}_{i - 1}}\right) }^{2} \mid {\mathcal{F}}_{s}}\right\rbrack = E\left\lbrack {E\left\lbrack {{\left( {M}_{{t}_{i}} - {M}_{{t}_{i - 1}}\right) }^{2} \mid {\mathcal{F}}_{{t}_{i - 1}}}\right\rbrack \mid {\mathcal{F}}_{s}}\right\rbr... | Yes |
Theorem 3.17 Let \( {\left( {X}_{t}\right) }_{t \geq 0} \) be a supermartingale, and let \( D \) be a countable dense subset of \( {\mathbb{R}}_{ + } \). (i) For almost every \( \omega \in \Omega \), the restriction of the function \( s \mapsto {X}_{s}\left( \omega \right) \) to the set \( D \) has a right-limit \[ {X}... | Proof (i) Fix \( T \in D \). By the remark following Proposition 3.15, we have \[ \mathop{\sup }\limits_{{s \in D \cap \left\lbrack {0, T}\right\rbrack }}\left| {X}_{s}\right| < \infty ,\;\text{ a.s. } \] As in the proof of Proposition 3.15, we can choose a sequence \( {\left( {D}_{m}\right) }_{m \geq 1} \) of finite s... | Yes |
Theorem 3.18 Assume that the filtration \( \left( {\mathcal{F}}_{t}\right) \) is right-continuous and complete. Let \( X = {\left( {X}_{t}\right) }_{t \geq 0} \) be a supermartingale, such that the function \( t \rightarrow E\left\lbrack {X}_{t}\right\rbrack \) is right-continuous. Then \( X \) has a modification with ... | Proof Let \( D \) be a countable dense subset of \( {\mathbb{R}}_{ + } \) as in Theorem 3.17. Let \( N \) be the negligible set defined in (3.3). We set, for every \( t \geq 0 \) ,\n\n\[ \n{Y}_{t}\left( \omega \right) = \left\{ \begin{array}{ll} {X}_{t + }\left( \omega \right) & \text{ if }\omega \notin N \\ 0 & \text{... | Yes |
Theorem 3.19 Let \( X \) be a supermartingale with right-continuous sample paths. Assume that the collection \( {\left( {X}_{t}\right) }_{t \geq 0} \) is bounded in \( {L}^{1} \). Then there exists a random variable \( {X}_{\infty } \in {L}^{1} \) such that\n\n\[ \mathop{\lim }\limits_{{t \rightarrow \infty }}{X}_{t} =... | Proof Let \( D \) be a countable dense subset of \( {\mathbb{R}}_{ + } \). From the proof of Theorem 3.17, we have, for every \( T \in D \) and \( a < b \),\n\n\[ E\left\lbrack {{M}_{ab}^{X}\left( {D \cap \left\lbrack {0, T}\right\rbrack }\right) }\right\rbrack \leq \frac{1}{b - a}E\left\lbrack {\left( {X}_{T} - a\righ... | Yes |
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