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Theorem 59.8.6 Let \( \mu \) and \( \nu \) be probability measures on the Borel sets of \( {\mathbb{R}}^{p} \) and suppose \( {\phi }_{\mu }\left( \mathbf{t}\right) = {\phi }_{\nu }\left( \mathbf{t}\right) \) . Then \( \mu = \nu \) . | Proof: The proof is identical to the above. Just replace \( {\lambda }_{\mathbf{X}} \) with \( \mu \) and \( {\lambda }_{\mathbf{Y}} \) with \( \nu \) . \( \blacksquare \) | Yes |
Proposition 59.9.3 Equations 59.9.14 and 59.9.13 hold with \( {\mathcal{X}}_{E} \) replaced by any nonnegative Borel measurable function and for any bounded continuous function or for any function in \( {L}^{1} \) . | Proof: The two equations hold for simple functions in place of \( {\mathcal{X}}_{E} \) and so an application of the monotone convergence theorem applied to an increasing sequence of simple functions converging pointwise to a given nonnegative Borel measurable function yields the conclusion of the proposition in the cas... | Yes |
Lemma 59.9.4 Let \( {\mathbf{X}}_{1},\cdots ,{\mathbf{X}}_{n} \) be random vectors with values in \( {\mathbb{R}}^{{p}_{1}},\cdots ,{\mathbb{R}}^{{p}_{n}} \) respectively and let \( \mathbf{g} : {\mathbb{R}}^{{p}_{1}} \times \cdots \times {\mathbb{R}}^{{p}_{n}} \rightarrow {\mathbb{R}}^{k} \) be Borel measurable. Then ... | Proof: First let \( E \) be a Borel set in \( {\mathbb{R}}^{k} \) . From the definition,\n\n\[{\lambda }_{\mathbf{g}\left( {{\mathbf{X}}_{1},\cdots ,{\mathbf{X}}_{n}}\right) }\left( E\right) = P\left( {\mathbf{g}\left( {{\mathbf{X}}_{1},\cdots ,{\mathbf{X}}_{n}}\right) \in E}\right)\]\n\n\[= P\left( {\left( {{\mathbf{X... | Yes |
Proposition 59.9.5 Let \( {\nu }_{1},\cdots ,{\nu }_{n} \) be Radon probability measures defined on \( {\mathbb{R}}^{p} \) . Then there exists a probability space and independent random vectors \( \left\{ {{\mathbf{X}}_{1},\cdots ,{\mathbf{X}}_{n}}\right\} \) defined on this probability space such that \( {\lambda }_{{... | Proof: Let \( \left( {\Omega ,\mathcal{S}, P}\right) \equiv \left( {{\left( {\mathbb{R}}^{p}\right) }^{n},{\mathcal{S}}_{1} \times \cdots \times {\mathcal{S}}_{n},{\nu }_{1} \times \cdots \times {\nu }_{n}}\right) \) where this is just the product \( \sigma \) algebra and product measure which satisfies the following f... | Yes |
Lemma 59.9.6 If \( {\left\{ {X}_{i}\right\} }_{i = 1}^{n} \) are independent random variables having values in \( \mathbb{R} \) , \n\n\[ \nE\left( {\mathop{\prod }\limits_{{i = 1}}^{n}{X}_{i}}\right) = \mathop{\prod }\limits_{{i = 1}}^{n}E\left( {X}_{i}\right) \n\] | Proof: By Lemma 59.9.4 and denoting by \( P \) the product, \( \mathop{\prod }\limits_{{i = 1}}^{n}{X}_{i} \) , \n\n\[ \nE\left( {\mathop{\prod }\limits_{{i = 1}}^{n}{X}_{i}}\right) = {\int }_{\mathbb{R}}{zd}{\lambda }_{P}\left( z\right) = {\int }_{{\mathbb{R}}^{n}}\mathop{\prod }\limits_{{i = 1}}^{n}{x}_{i}d{\lambda }... | Yes |
Proposition 59.10.2 Suppose \( {\int }_{{\mathbb{F}}^{{p}_{1}} \times {\mathbb{F}}^{{p}_{2}}}\left| \mathbf{x}\right| d{\lambda }_{\left( \mathbf{X},\mathbf{Y}\right) }\left( x\right) < \infty \) . Then \( E\left( {\mathbf{X} \mid \mathbf{y}}\right) \) exists for \( {\lambda }_{\mathbf{Y}} \) a.e. \( \mathbf{y} \) and\... | Proof: \( \infty > {\int }_{{\mathbb{F}}^{{p}_{1}} \times {\mathbb{F}}^{{p}_{2}}}\left| \mathbf{x}\right| d{\lambda }_{\left( \mathbf{X},\mathbf{Y}\right) } = {\int }_{{\mathbb{F}}^{{p}_{2}}}{\int }_{{\mathbb{F}}^{{p}_{1}}}\left| \mathbf{x}\right| d{\lambda }_{\mathbf{X} \mid \mathbf{y}}\left( x\right) d{\lambda }_{\ma... | Yes |
Proposition 59.10.5 Let \( {\left\{ {X}_{i}\right\} }_{i = 1}^{n} \) be a finite sequence of real random variables defined on \( \Omega \) where \( \left( {\Omega ,\mathcal{S}, P}\right) \) is a probability space. Let \( {U}_{\left\lbrack a, b\right\rbrack }\left( \omega \right) \) denote the number of upcrossings of \... | Proof: Let \( {X}_{0}\left( \omega \right) \equiv a + 1 \), let \( {Y}_{0}\left( \omega \right) \equiv 0 \), and let \( {Y}_{k}\left( \omega \right) \) remain 0 for \( k = 0,\cdots, l \) until \( {X}_{l}\left( \omega \right) \leq a \) . When this happens (if ever), \( {Y}_{l + 1}\left( \omega \right) \equiv 1 \) . Then... | Yes |
Lemma 59.10.7 (upcrossing lemma) Let \( {\left\{ {X}_{i}\right\} }_{i = 1}^{n} \) be a submartingale and suppose\n\n\[ E\left( \left| {X}_{n}\right| \right) < \infty .\n\]\n\nThen\n\[ E\left( {U}_{\left\lbrack a, b\right\rbrack }\right) \leq \frac{E\left( \left| {X}_{n}\right| \right) + \left| a\right| }{b - a}. \] | Proof: Let \( \phi \left( x\right) \equiv a + {\left( x - a\right) }^{ + } \) . Thus \( \phi \) is a convex and increasing function.\n\n\[ \phi \left( {X}_{k + r}\right) - \phi \left( {X}_{k}\right) = \mathop{\sum }\limits_{{i = k + 1}}^{{k + r}}\phi \left( {X}_{i}\right) - \phi \left( {X}_{i - 1}\right) \]\n\n\[ = \ma... | Yes |
Theorem 59.10.8 (submartingale convergence theorem) Let \( {\left\{ {X}_{i}\right\} }_{i = 1}^{\infty } \) be a sub-martingale with \( K \equiv \sup \left\{ {E\left( \left| {X}_{n}\right| \right) : n \geq 1}\right\} < \infty \) . Then there exists a random variable, \( {X}_{\infty } \), such that \( E\left( \left| {X}_... | Proof: Let \( a, b \in \mathbb{Q} \) and let \( a < b \) . Let \( {U}_{\left\lbrack a, b\right\rbrack }^{n}\left( \omega \right) \) be the number of upcrossings of \( {\left\{ {X}_{i}\left( \omega \right) \right\} }_{i = 1}^{n} \) . Then let\n\n\[ {U}_{\left\lbrack a, b\right\rbrack }\left( \omega \right) \equiv \matho... | Yes |
Lemma 59.11.2 Let \( \\mathbf{Y} \) be a random vector with values in \( {\\mathbb{R}}^{p} \) and let \( f \) be bounded and measurable with respect to the Radon measure \( {\\lambda }_{\\mathbf{Y}} \), and satisfy\n\n\[ \n\\int f\\left( \\mathbf{y}\\right) {e}^{i\\mathbf{t} \\cdot \\mathbf{y}}d{\\lambda }_{\\mathbf{Y}... | Proof: You could write the following for \( \\phi \\in \\mathcal{G} \)\n\n\[ \n\\int \\phi \\left( \\mathbf{t}\\right) \\int f\\left( \\mathbf{y}\\right) {e}^{i\\mathbf{t} \\cdot \\mathbf{y}}d{\\lambda }_{\\mathbf{Y}}{dt} = 0 = \\int f\\left( \\mathbf{y}\\right) \\left( {\\int \\phi \\left( \\mathbf{t}\\right) {e}^{i\\... | Yes |
Lemma 59.11.3 Suppose \( \mathbf{X},{\mathbf{Y}}_{1},{\mathbf{Y}}_{2},\cdots ,{\mathbf{Y}}_{k} \) are random vectors \( \mathbf{X} \) having values in \( {\mathbb{R}}^{n} \) and \( {\mathbf{Y}}_{j} \) having values in \( {\mathbb{R}}^{{p}_{j}} \) and\n\n\[ \mathbf{X},{\mathbf{Y}}_{j} \in {L}^{1}\left( \Omega \right) \]... | Proof: For the sake of brevity, denote by \( \mathbf{Y} \) the vector \( \left( {{\mathbf{Y}}_{1},\cdots ,{\mathbf{Y}}_{k}}\right) \) and by \( \mathbf{y} \) the vector \( \left( {{\mathbf{y}}_{1},\cdots ,{\mathbf{y}}_{k}}\right) \) and let \( \mathop{\prod }\limits_{{j = 1}}^{k}{\mathbb{R}}^{{p}_{j}} \equiv {\mathbb{R... | Yes |
Lemma 59.11.4 There exists a unique function \( \mathbf{Z}\left( \omega \right) \) which satisfies\n\n\[ \n{\int }_{F}\mathbf{X}\left( \omega \right) {dP} = {\int }_{F}\mathbf{Z}\left( \omega \right) {dP} \n\]\n\nfor all \( F \in \sigma \left( {{\mathbf{Y}}_{1},\cdots ,{\mathbf{Y}}_{k}}\right) \) such that \( \mathbf{Z... | Proof: It is like the above. Letting \( E \) be a Borel set in \( {\mathbb{R}}^{p} \) ,\n\n\[ \n{\int }_{{\mathbf{Y}}^{-1}\left( E\right) }\mathbf{X}{dP} = {\int }_{{\mathbb{R}}^{n} \times {\mathbb{R}}^{P}}{\mathcal{X}}_{{\mathbb{R}}^{n} \times E}\left( {\mathbf{x},\mathbf{y}}\right) \mathbf{x}d{\lambda }_{\left( \math... | Yes |
Corollary 59.12.5 Let \( \mathcal{K} \) be a \( \pi \) system of subsets of \( \Omega \) and suppose two probability measures, \( \mu \) and \( \nu \) defined on \( \sigma \left( \mathcal{K}\right) \) are equal on \( \mathcal{K} \) . Then \( \mu = \nu \) . | Proof: This follows from the Lemma 12.12.3 on Page 347. Let\n\n\[ \mathcal{G} \equiv \{ E \in \sigma \left( \mathcal{K}\right) : \mu \left( E\right) = \nu \left( E\right) \}\]\n\nThen \( \mathcal{K} \subseteq \mathcal{G} \), since \( \mu \) and \( \nu \) are both probability measures, it follows that if \( E \in \mathc... | Yes |
Lemma 59.12.6 If \( E \) is a separable Banach space with \( {B}^{\prime } \) the closed unit ball in \( {E}^{\prime } \), then there exists a sequence \( {\left\{ {f}_{n}\right\} }_{n = 1}^{\infty } \equiv {D}^{\prime } \subseteq {B}^{\prime } \) with the property that for every \( x \in E \) , \[ \parallel x\parallel... | Letting \( {\left\{ {f}_{n}\right\} }_{n = 1}^{\infty } = {D}^{\prime } \) be the sequence of Lemma 59.12.6 it follows that \[ \{ x \in E : \parallel x - a\parallel \leq \delta \} = \left\{ {x \in E : \mathop{\sup }\limits_{{f \in {D}^{\prime }}}\left| {f\left( {x - a}\right) }\right| \leq \delta }\right\} = \left\{ {x... | No |
Theorem 59.12.9 Let \( \mu \) and \( \nu \) be two probability measures on \( \mathcal{B}\left( E\right) \) where \( E \) is a separable real Banach space. Suppose\n\n\[{\phi }_{\mu }\left( {x}^{ * }\right) = {\phi }_{\nu }\left( {x}^{ * }\right)\]\n\nfor all \( {x}^{ * } \in {E}^{\prime } \) . Then \( \mu = \nu \) . | Proof: It suffices to verify that \( \mu \left( A\right) = \nu \left( A\right) \) for all \( A \in \mathcal{K} \) where \( \mathcal{K} \) is the set of cylindrical sets. Fix \( {\mathbf{g}}_{n} \in {\left( {E}^{\prime }\right) }^{n} \) . Thus the two measures are equal if for all such \( {\mathbf{g}}_{n}, n \in \mathbb... | Yes |
Theorem 59.13.3 Let \( {\left\{ {X}_{k}\right\} }_{k = 1}^{n} \) be random variables such that \( {X}_{k} \) has values in \( {E}_{k} \), a real separable Banach space. Then the random variables are independent if and only if\n\n\[ E\left( {e}^{iP}\right) = \mathop{\prod }\limits_{{j = 1}}^{n}E\left( {e}^{i{t}_{j}^{ * ... | Proof: If the random variables are independent, then so are the random variables, \( {t}_{j}^{ * }\left( {X}_{j}\right) \) and so the equation follows.\n\nThe interesting case is when the equation holds.\n\nIt suffices to consider only the case where each \( {E}_{k} = E \) . This is because you can consider each \( {X}... | Yes |
Corollary 59.13.4 Let \( {\left\{ {X}_{k}\right\} }_{k = 1}^{n} \) be random variables such that \( {X}_{k} \) has values in \( {E}_{k} \), a real separable Banach space. Then the random variables are independent if and only if\n\n\[ E\left( {e}^{iP}\right) = \mathop{\prod }\limits_{{j = 1}}^{n}E\left( {e}^{i{t}_{j}^{ ... | Proof: The easy direction follows from Theorem 59.13.3. Suppose then the above equation holds for all \( {t}_{j}^{ * } \in {M}_{j} \) . Then let \( {t}_{j}^{ * } \in {E}^{\prime } \) and let \( \left\{ {t}_{nj}^{ * }\right\} \) be a sequence in \( {M}_{j} \) such that\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty ... | Yes |
Lemma 59.14.1 Suppose \( K \) is a compact subset of \( U \) an open set in \( E \) a metric space. Then there exists \( \delta > 0 \) such that\n\n\[ \operatorname{dist}\left( {x, K}\right) + \operatorname{dist}\left( {x,{U}^{C}}\right) \geq \delta \text{ for all }x \in E. \] | Proof: For each \( x \in K \), there exists a ball, \( B\left( {x,{\delta }_{x}}\right) \) such that \( B\left( {x,3{\delta }_{x}}\right) \subseteq U \) . Finitely many of these balls cover \( K \) because \( K \) is compact, say \( {\left\{ B\left( {x}_{i},{\delta }_{{x}_{i}}\right) \right\} }_{i = 1}^{m} \) . Let\n\n... | Yes |
Corollary 59.14.2 Suppose \( K \) is a compact subset of \( U \), an open set in \( E \) a metric space. Then there exists a uniformly continuous function \( f \) defined on all of \( E \) , having values in \( \left\lbrack {0,1}\right\rbrack \) such that \( f\left( x\right) = 0 \) if \( x \notin U \) and \( f\left( x\... | Proof: Consider\n\n\[ f\left( x\right) \equiv \frac{\operatorname{dist}\left( {x,{U}^{C}}\right) }{\operatorname{dist}\left( {x,{U}^{C}}\right) + \operatorname{dist}\left( {x, K}\right) }.\]\n\nThen some algebra yields\n\n\[ \left| {f\left( x\right) - f\left( {x}^{\prime }\right) }\right| \leq \]\n\n\[ \frac{1}{\delta ... | Yes |
Theorem 59.14.3 Let \( \mu \) be a finite measure on \( \mathcal{B}\left( E\right) \) where \( E \) is a separable Banach space and let \( f \in {L}^{p}\left( {E;\mu }\right) \) . Then for any \( \varepsilon > 0 \), there exists a uniformly continuous, bounded \( g \) defined on \( E \) such that \[ \parallel f - g{\pa... | Proof: As usual in such situations, it suffices to consider only \( f \geq 0 \) . Then by Theorem 11.3.9 on Page 257 and an application of the monotone convergence theorem, there exists a simple measurable function, \[ s\left( x\right) \equiv \mathop{\sum }\limits_{{k = 1}}^{m}{c}_{k}{\mathcal{X}}_{{A}_{k}}\left( x\rig... | Yes |
Lemma 59.14.6 For A a Borel set in E, a separable Banach space, define\n\n\[ \n{S}_{A} \equiv \{ \left( {x, y}\right) \in E \times E : x + y \in A\} \n\]\n\nThen \( {S}_{A} \in \mathcal{B}\left( E\right) \times \mathcal{B}\left( E\right) \), the \( \sigma \) algebra of product measurable sets, the smallest \( \sigma \)... | Proof: Let \( \mathcal{K} \) denote the open sets in \( E \) . Then \( \mathcal{K} \) is a \( \pi \) system. Let\n\n\[ \n\mathcal{G} \equiv \{ A \in \sigma \left( \mathcal{K}\right) = \mathcal{B}\left( E\right) : {S}_{A} \in \mathcal{B}\left( E\right) \times \mathcal{B}\left( E\right) \} .\n\]\n\nThen \( \mathcal{K} \s... | Yes |
Theorem 59.14.7 Let \( \mu ,\nu \), and \( \lambda \) be finite measures on \( \mathcal{B}\left( E\right) \) for \( E \) a separable Banach space. Then\n\n\[ \mu * \nu = \nu * \mu \]\n\n\( \left( {59.14.23}\right) \)\n\n\[ \left( {\mu * \nu }\right) * \lambda = \mu * \left( {\nu * \lambda }\right) \]\n\n(59.14.24) | Proof: First consider 59.14.23. Letting \( A \in \mathcal{B}\left( E\right) \), the following computation\n\nholds from Fubini's theorem and Lemma 59.14.6\n\n\[ \mu * \nu \left( A\right) \equiv {\int }_{E}\nu \left( {A - x}\right) {d\mu }\left( x\right) = {\int }_{E}{\int }_{E}{\mathcal{X}}_{{S}_{A}}\left( {x, y}\right... | Yes |
Lemma 59.15.2 Let \( \\left( {\\Omega ,\\mathcal{F}, P}\\right) \) be a probability space and let \( \\mathbf{X} : \\Omega \\rightarrow E \) be a random variable, where \( E \) is a real separable Banach space. Also let \( \\mathcal{L}\\left( \\mathbf{X}\\right) = \\mu \), a probability measure defined on \( \\mathcal{... | Proof: First suppose \( A \) is a Borel set in \( E \) . Then\n\n\[ \n{\\int }_{E}{\\mathcal{X}}_{A}\\left( \\mathbf{x}\\right) {d\\mu } \\equiv \\mu \\left( A\\right) \\equiv P\\left( \\left\\lbrack {\\mathbf{X} \\in A}\\right\\rbrack \\right) \n\]\n\n\[ \n{\\int }_{\\Omega }\\left( {{\\mathcal{X}}_{A} \\circ \\mathbf... | Yes |
Lemma 59.15.4 Let \( \mathbf{X} \equiv \left( {{X}_{1},\cdots ,{X}_{n}}\right) \) and \( Y \) be random variables defined on a probability space, \( \left( {\Omega ,\mathcal{F}, P}\right) \) such that \( {X}_{i}, i = 1,2,\cdots, n \) and \( Y \) have values in \( E \) a separable Banach space. Thus \( \mathbf{X} \) has... | Proof: Denote by \( {\lambda }_{\mathbf{X}} \) and \( {\lambda }_{Y} \) the distribution measures for \( \mathbf{X} \) and \( Y \) respectively. Since the random variables are independent, the distribution for the random variable, \( \left( {\mathbf{X}, Y}\right) \) mapping into \( {E}^{n + 1} \) is \( {\lambda }_{\mat... | Yes |
Lemma 59.15.6 Let \( \\left\\{ {\\zeta }_{k}\\right\\} \) be a sequence of independent random variables having values in a separable real Banach space, \( E \) whose distributions are symmetric. Letting \( {S}_{k} \\equiv \\mathop{\\sum }\\limits_{{i = 1}}^{k}{\\zeta }_{i} \), suppose \( \\left\\{ {S}_{{n}_{k}}\\right\... | Proof: Let \( {n}_{k} \\leq l \\leq m \) . Then by Lemma 59.15.5\n\n\[ \nP\\left( \\left\\lbrack {\\mathop{\\sup }\\limits_{{{n}_{k} < l \\leq m}}\\begin{Vmatrix}{{S}_{l} - {S}_{{n}_{k}}}\\end{Vmatrix} > {2}^{-k}}\\right\\rbrack \\right) \\leq {2P}\\left( \\left\\lbrack {\\begin{Vmatrix}{{S}_{m} - {S}_{{n}_{k}}}\\end{V... | Yes |
Theorem 59.16.2 For \( \mathbf{X} \sim {N}_{p}\left( {\mathbf{m},\sum }\right) ,\mathbf{m} = E\left( \mathbf{X}\right) \) and\n\n\[ \sum = E\left( {\left( {\mathbf{X} - \mathbf{m}}\right) {\left( \mathbf{X} - \mathbf{m}\right) }^{ * }}\right) . \] | Proof: Let \( R \) be an orthogonal transformation such that\n\n\[ {R\sum }{R}^{ * } = D = \operatorname{diag}\left( {{\sigma }_{1}^{2},\cdots ,{\sigma }_{p}^{2}}\right) . \]\n\nChanging the variable by \( \mathbf{x} - \mathbf{m} = {R}^{ * }\mathbf{y} \) ,\n\n\[ E\left( \mathbf{X}\right) \equiv {\int }_{{\mathbb{R}}^{p... | Yes |
Corollary 59.16.5 Let \( \mathbf{X} = \left( {{X}_{1},\cdots ,{X}_{p}}\right) ,\mathbf{Y} = \left( {{Y}_{1},\cdots ,{Y}_{p}}\right) \) where each \( {X}_{i},{Y}_{i} \) is a real valued random variable. Suppose also that for every \( \mathbf{a} \in {\mathbb{R}}^{p},\mathbf{a} \cdot \mathbf{X} \) and \( \mathbf{a} \cdot ... | Proof: In the Proof of Theorem 59.16.4 the proof implies that the characteristic functions of \( \mathbf{a} \cdot \mathbf{X} \) and \( \mathbf{a} \cdot \mathbf{Y} \) are both of the form\n\n\[ \n{e}^{itm}{e}^{-\frac{1}{2}{\sigma }^{2}{t}^{2}}. \n\]\n\nThen as in the proof of that theorem, it must be the case that\n\n\[... | Yes |
Lemma 59.18.3 If \( {\phi }_{{\mathbf{X}}_{n}}\left( \mathbf{t}\right) \rightarrow {\phi }_{\mathbf{X}}\left( \mathbf{t}\right) \) for all \( \mathbf{t} \), then whenever \( \psi \in \mathfrak{S} \) , | Proof: Recall that if \( \mathbf{X} \) is any random vector, its characteristic function is given by\n\n\[ {\phi }_{\mathbf{X}}\left( \mathbf{y}\right) \equiv {\int }_{{\mathbb{R}}^{p}}{e}^{i\mathbf{y} \cdot \mathbf{x}}d{\lambda }_{\mathbf{X}}\left( x\right) \]\n\nAlso remember the inverse Fourier transform. Letting \(... | Yes |
Lemma 59.18.4 If \( {\phi }_{{\mathbf{X}}_{n}}\left( \mathbf{t}\right) \rightarrow {\phi }_{\mathbf{X}}\left( \mathbf{t}\right) \), then if \( \psi \) is any bounded uniformly continuous function, | Proof: Let \( \varepsilon > 0 \) be given, let \( \psi \) be a bounded function in \( {C}^{\infty }\left( {\mathbb{R}}^{p}\right) \) . Now let \( \eta \in {C}_{c}^{\infty }\left( {Q}_{r}\right) \) where \( {Q}_{r} \equiv {\left\lbrack -r, r\right\rbrack }^{p} \) satisfy the additional requirement that \( \eta = 1 \) on... | Yes |
Lemma 59.19.3 If \( {\phi }_{{\mu }_{n}}\left( \mathbf{t}\right) \rightarrow {\phi }_{\mu }\left( \mathbf{t}\right) \) for all \( \mathbf{t} \), then whenever \( \psi \in \mathfrak{S} \), the Schwartz class, \[ {\mu }_{n}\left( \psi \right) \equiv {\int }_{{\mathbb{R}}^{p}}\psi \left( \mathbf{y}\right) d{\mu }_{n}\left... | Proof: By definition, \[ {\phi }_{\mu }\left( \mathbf{y}\right) \equiv {\int }_{{\mathbb{R}}^{p}}{e}^{i\mathbf{y} \cdot \mathbf{x}}{d\mu }\left( x\right) \] Also remember the inverse Fourier transform. Letting \( \psi \in \mathfrak{S} \), the Schwartz class, \[ {F}^{-1}\left( \mu \right) \left( \psi \right) \equiv \mu ... | Yes |
Lemma 59.19.4 If \( {\phi }_{{\mu }_{n}}\left( \mathbf{t}\right) \rightarrow {\phi }_{\mu }\left( \mathbf{t}\right) \) where \( \left\{ {\mu }_{n}\right\} \) and \( \mu \) are probability measures defined on the Borel sets of \( {\mathbb{R}}^{p} \), then if \( \psi \) is any bounded uniformly continuous function, | Proof: Let \( \varepsilon > 0 \) be given, let \( \psi \) be a bounded function in \( {C}^{\infty }\left( {\mathbb{R}}^{p}\right) \) . Now let \( \eta \in {C}_{c}^{\infty }\left( {Q}_{r}\right) \) where \( {Q}_{r} \equiv {\left\lbrack -r, r\right\rbrack }^{p} \) satisfy the additional requirement that \( \eta = 1 \) on... | Yes |
Theorem 59.19.5 Let \( \Lambda = {\left\{ {\mu }_{n}\right\} }_{n = 1}^{\infty } \) be a sequence of probability measures defined on the Borel sets of \( {\mathbb{R}}^{p} \) . If \( \Lambda \) is tight then there exists a probability measure, \( \lambda \) and a subsequence of \( {\left\{ {\mu }_{n}\right\} }_{n = 1}^{... | Proof: By tightness, there exists an increasing sequence of compact sets, \( \left\{ {K}_{n}\right\} \) such that \[ \mu \left( {K}_{n}\right) > 1 - \frac{1}{n} \] for all \( \mu \in \Lambda \) . Now letting \( \mu \in \Lambda \) and \( \phi \in C\left( {K}_{n}\right) \) such that \( \parallel \phi {\parallel }_{\infty... | Yes |
Theorem 59.19.7 Suppose \( \left\{ {\mu }_{n}\right\} \) is a sequence of probability measures defined on the Borel sets of \( {\mathbb{R}}^{p} \) and let \( \left\{ {\phi }_{{\mu }_{n}}\right\} \) denote the corresponding sequence of characteristic functions. If there exists \( \psi \) which is continuous at \( \mathb... | Proof: By Lemma 59.19.2 \( \left\{ {\mu }_{n}\right\} \) is tight. Therefore, there exists a subsequence \( \left\{ {\mu }_{{n}_{k}}\right\} \) converging weakly to a probability measure, \( \lambda \) . In particular,\n\n\[{\phi }_{\lambda }\left( \mathbf{t}\right) = \int {e}^{i\mathbf{t} \cdot \mathbf{x}}{d\lambda }\... | Yes |
Lemma 59.21.1 Suppose \( M \) is an \( n \times n \) matrix. Suppose also that\n\n\[{\mathbf{\alpha }}^{ * }M\mathbf{\alpha } = 0\]\n\nfor all \( \mathbf{\alpha } \in {\mathbb{C}}^{n} \) . Then \( M = 0 \) . | Proof: Suppose \( \lambda \) is an eigenvalue for \( M \) and let \( \mathbf{\alpha } \) be an associated eigenvector.\n\n\[0 = {\mathbf{\alpha }}^{ * }M\mathbf{\alpha } = {\mathbf{\alpha }}^{ * }\lambda \mathbf{\alpha } = \lambda {\mathbf{\alpha }}^{ * }\mathbf{\alpha } = \lambda {\left| \mathbf{\alpha }\right| }^{2}\... | Yes |
Lemma 59.21.3 If \( f \) is positive definite then whenever \( {\left\{ {\mathbf{t}}_{k}\right\} }_{k = 1}^{p} \) are \( p \) points in \( {\mathbb{R}}^{n},\left| {f\left( {{\mathbf{t}}_{j} - {\mathbf{t}}_{k}}\right) }\right| \leq f\left( \mathbf{0}\right) \) . In particular, for all \( \mathbf{t},\left| {f\left( \math... | Proof: Let \( F \) be the \( p \times p \) matrix such that\n\n\[ \n{F}_{kj} = f\left( {{\mathbf{t}}_{j} - {\mathbf{t}}_{k}}\right) \n\]\n\nThen 59.21.52 is of the form\n\n\[ \n{\mathbf{\alpha }}^{ * }F\mathbf{\alpha } = \left( {F\mathbf{\alpha },\mathbf{\alpha }}\right) \geq 0 \n\]\n\n\( \left( {59.21.53}\right) \)\n\... | Yes |
Lemma 59.21.4 Let \( f \) be a positive definite function as defined above and let \( \mu \) be a finite Borel measure. Then \[ {\int }_{{\mathbb{R}}^{n}}{\int }_{{\mathbb{R}}^{n}}f\left( {\mathbf{x} - \mathbf{y}}\right) {d\mu }\left( x\right) {d\mu }\left( y\right) \geq 0. \] | Proof: By definition if \( {\left\{ {t}_{j}\right\} }_{j = 1}^{p} \subseteq {\mathbb{R}}^{n} \), and letting \( \mathbf{\alpha } = {\left( 1,\cdots ,1\right) }^{T} \in {\mathbb{R}}^{n} \) , \[ \mathop{\sum }\limits_{{j, k}}f\left( {{\mathbf{t}}_{j} - {\mathbf{t}}_{k}}\right) \geq 0 \] Therefore, integrating over each o... | Yes |
Lemma 59.21.5 Let \( {\mu }_{t} \) be the measure defined on \( \mathcal{B}\left( {\mathbb{R}}^{n}\right) \) by\n\n\[ \n{\mu }_{t}\left( F\right) \equiv {\int }_{F}\frac{1}{{\left( \sqrt{2\pi t}\right) }^{n}}{e}^{-\frac{1}{2t}{\left| \mathbf{x}\right| }^{2}}{dx} \n\]\n\nfor \( t > 0 \) . Then \( {\mu }_{t} * {\mu }_{t}... | Proof: By Theorem 59.14.7,\n\n\[ \n{\phi }_{{\mu }_{t} * {\mu }_{t}}\left( \mathbf{s}\right) = {\phi }_{{\mu }_{t}}\left( \mathbf{s}\right) {\phi }_{{\mu }_{t}}\left( \mathbf{s}\right) = {\left( {e}^{-\frac{1}{2}t{\left| \mathbf{s}\right| }^{2}}\right) }^{2} = {e}^{-\frac{1}{2}\left( {2t}\right) {\left| \mathbf{s}\righ... | Yes |
Theorem 59.21.7 Let \( \psi \) be positive definite, continuous at \( \mathbf{0} \), and \( \psi \left( \mathbf{0}\right) = 1 \) . Then there exists a unique Radon probability measure \( \mu \) such that \( \psi = {\phi }_{\mu } \) . | Proof: If \( \psi \in {L}^{1}\left( {{\mathbb{R}}^{n},{m}_{n}}\right) \), then the result follows from Lemma 59.21.6. By Lemma 59.21.3 \( \psi \) is bounded. Consider\n\n\[{\psi }_{t}\left( \mathbf{x}\right) \equiv \psi \left( \mathbf{x}\right) \frac{1}{{\left( 2\pi t\right) }^{n/2}}{e}^{-\frac{1}{2t}{\left| \mathbf{x}... | Yes |
Lemma 60.1.2 The above is well defined. Also, if \( \mathcal{S} \subseteq \mathcal{F} \) then\n\n\[ E\left( {X \mid \mathcal{S}}\right) = E\left( {E\left( {X \mid \mathcal{F}}\right) \mid \mathcal{S}}\right) . \]\n\n\( \left( {60.1.1}\right) \)\n\nIf \( Z \) is bounded and measurable in \( \mathcal{S} \) then\n\n\[ {ZE... | Proof: Let a finite measure on \( \mathcal{S},\mu \) be given by\n\n\[ \mu \left( E\right) \equiv {\int }_{E}{fdP} \]\n\nThen \( \mu \ll P \) and so by the Radon Nikodym theorem, there exists a unique \( \mathcal{S} \) measurable function, \( E\left( {f \mid \mathcal{S}}\right) \) such that\n\n\[ {\int }_{E}{fdP} \equi... | Yes |
Lemma 60.1.3 Let \( \\phi \) be a convex real valued function defined on an interval \( I \) . Then for each \( x \\in I \), there exists \( {a}_{x} \) such that for all \( t \\in I \) ,\n\n\[ \n\\phi \\left( t\\right) \\geq {a}_{x}\\left( {t - x}\\right) + \\phi \\left( x\\right) .\n\]\n\nAlso \( \\phi \) is continuou... | Proof: Let \( x \\in I \) and let \( t > x \) . Then by convexity of \( \\phi \) ,\n\n\[ \n\\frac{\\phi \\left( {x + \\lambda \\left( {t - x}\\right) }\\right) - \\phi \\left( x\\right) }{\\lambda \\left( {t - x}\\right) } \\leq \\frac{\\phi \\left( x\\right) \\left( {1 - \\lambda }\\right) + {\\lambda \\phi }\\left( t... | Yes |
Lemma 60.1.4 Let \( I \) be an open interval on \( \mathbb{R} \) and let \( \phi \) be a convex function defined on \( I \) . Then there exists a sequence \( \left\{ \left( {{a}_{n},{b}_{n}}\right) \right\} \) such that \[ \phi \left( t\right) = \sup \left\{ {{a}_{n}t + {b}_{n}, n = 1,\cdots }\right\} . \] | Proof: Let \( {a}_{x} \) be as defined in the above lemma. Let \[ \psi \left( x\right) \equiv \sup \left\{ {{a}_{r}\left( {x - r}\right) + \phi \left( r\right) : r \in \mathbb{Q} \cap I}\right\} . \] Thus if \( {r}_{1} \in \mathbb{Q} \) , \[ \psi \left( {r}_{1}\right) \equiv \sup \left\{ {{a}_{r}\left( {{r}_{1} - r}\ri... | Yes |
Lemma 60.1.5 If \( X \leq Y \), then \( E\left( {X \mid \mathcal{S}}\right) \leq E\left( {Y \mid \mathcal{S}}\right) \) a.e. Also\n\n\[ X \rightarrow E\left( {X \mid \mathcal{S}}\right) \] is linear. | Proof: Let \( A \in \mathcal{S} \). \n\n\[ {\int }_{A}E\left( {X \mid \mathcal{S}}\right) {dP} \equiv {\int }_{A}{XdP} \]\n\n\[ \leq {\int }_{A}{YdP} \equiv {\int }_{A}E\left( {Y \mid \mathcal{S}}\right) {dP} \]\n\nHence \( E\left( {X \mid \mathcal{S}}\right) \leq E\left( {Y \mid \mathcal{S}}\right) \) a.e. as claimed.... | Yes |
Theorem 60.1.6 (Jensen’s inequality) Let \( X\left( \omega \right) \in I \) and let \( \phi : I \rightarrow \mathbb{R} \) be convex. Suppose\n\n\[ E\left( \left| X\right| \right), E\left( \left| {\phi \left( X\right) }\right| \right) < \infty .\n\]\n\nThen\n\n\[ \phi \left( {E\left( {X \mid \mathcal{S}}\right) }\right)... | Proof: Let \( \phi \left( x\right) = \sup \left\{ {{a}_{n}x + {b}_{n}}\right\} \) . Letting \( A \in \mathcal{S \) ,\n\n\[ \frac{1}{P\left( A\right) }{\int }_{A}E\left( {X \mid \mathcal{S}}\right) {dP} = \frac{1}{P\left( A\right) }{\int }_{A}{XdP} \in I\text{ a.e. }\n\]\n\nwhenever \( P\left( A\right) \neq 0 \) . Hence... | Yes |
Proposition 60.2.3 Let \( {\left\{ {X}_{i}\right\} }_{i = 1}^{n} \) be a finite sequence of real random variables defined on \( \Omega \) where \( \left( {\Omega ,\mathcal{S}, P}\right) \) is a probability space. Let \( {U}_{\left\lbrack a, b\right\rbrack }\left( \omega \right) \) denote the number of upcrossings of \(... | Proof: Let \( {X}_{0}\left( \omega \right) \equiv a + 1 \), let \( {Y}_{0}\left( \omega \right) \equiv 0 \), and let \( {Y}_{k}\left( \omega \right) \) remain 0 for \( k = 0,\cdots, l \) until \( {X}_{l}\left( \omega \right) \leq a \) . When this happens (if ever), \( {Y}_{l + 1}\left( \omega \right) \equiv 1 \) . Then... | Yes |
Corollary 60.2.4 \( {U}_{\left\lbrack a, b\right\rbrack }\left( \omega \right) \leq \) the number of unbroken strings of ones in the sequence, \( \left\{ {{Y}_{k}\left( \omega \right) }\right\} \) there being at most one unbroken string of ones which produces no upcrossing. | \[ {Y}_{i}\left( \omega \right) = {\psi }_{i}\left( {\left\{ {X}_{j}\left( \omega \right) \right\} }_{j = 1}^{i - 1}\right) \] where \( {\psi }_{i} \) is some function of the past values of \( {X}_{j}\left( \omega \right) \). | No |
Lemma 60.2.5 Let \( \phi \) be a convex and increasing function and suppose\n\n\[ \left\{ \left( {{X}_{n},{\mathcal{S}}_{n}}\right) \right\} \]\n\nis a submartingale. Then if \( E\left( \left| {\phi \left( {X}_{n}\right) }\right| \right) < \infty \), it follows\n\n\[ \left\{ \left( {\phi \left( {X}_{n}\right) ,{\mathca... | Proof: It is given that \( E\left( {{X}_{n + 1},{\mathcal{S}}_{n}}\right) \geq {X}_{n} \) and so\n\n\[ \phi \left( {X}_{n}\right) \leq \phi \left( {E\left( {{X}_{n + 1} \mid {\mathcal{S}}_{n}}\right) }\right) \leq E\left( {\phi \left( {X}_{n + 1}\right) \mid {\mathcal{S}}_{n}}\right) \]\n\nby Jensen's inequality. | Yes |
Theorem 60.2.7 (submartingale convergence theorem) Let\n\n\\[ \n{\\left\\{ \\left( {X}_{i},{\\mathcal{S}}_{i}\\right) \\right\\} }_{i = 1}^{\\infty }\n\\]\n\nbe a submartingale with \\( K \\equiv \\sup E\\left( \\left| {X}_{n}\\right| \\right) < \\infty \\) . Then there exists a random variable, \\( X \\), such that \\... | Proof: Let \\( a, b \\in \\mathbb{Q} \\) and let \\( a < b \\) . Let \\( {U}_{\\left\\lbrack a, b\\right\\rbrack }^{n}\\left( \\omega \\right) \\) be the number of upcrossings of \\( {\\left\\{ {X}_{i}\\left( \\omega \\right) \\right\\} }_{i = 1}^{n} \\) . Then let\n\n\\[ \n{U}_{\\left\\lbrack a, b\\right\\rbrack }\\le... | Yes |
Theorem 60.2.8 Let \( {\left\{ \left( {X}_{i},{\mathcal{S}}_{i}\right) \right\} }_{i = 1}^{\infty } \) be a submartingale. Then for \( \lambda > 0 \) , \[ P\left( \left\lbrack {\mathop{\max }\limits_{{1 \leq k \leq n}}{X}_{k} \geq \lambda }\right\rbrack \right) \leq \frac{1}{\lambda }{\int }_{\Omega }{X}_{n}^{ + }{dP} ... | Proof: Let \[ {A}_{1} \equiv \left\lbrack {{X}_{1} \geq \lambda }\right\rbrack ,{A}_{2} \equiv \left\lbrack {{X}_{2} \geq \lambda }\right\rbrack \smallsetminus {A}_{1}, \] \[ \cdots ,{A}_{k} \equiv \left\lbrack {{X}_{k} \geq \lambda }\right\rbrack \smallsetminus \left( {{ \cup }_{i = 1}^{k - 1}{A}_{i}}\right) \cdots \]... | Yes |
Proposition 60.3.3 Let \( M \) be a martingale having values in some separable Banach space. Let \( \tau \) be a bounded stopping time and let \( \sigma \) be another stopping time. Then everything makes sense in the following formula and | \[ M\left( {\sigma \land \tau }\right) = E\left( {M\left( \tau \right) \mid {\mathcal{F}}_{\sigma }}\right) \text{ a.e. } \] | No |
Lemma 60.4.1 In the situation of Definition 60.3.1, if \( S \leq T \) for two stopping times, \( S \) and \( T \), then \( {\mathcal{F}}_{S} \subseteq {\mathcal{F}}_{T} \) . Also \( {\mathcal{F}}_{T} \) is a \( \sigma \) algebra. | Proof: Let \( A \in {\mathcal{F}}_{S} \) . Then this means\n\n\[ A \cap \left\lbrack {S \leq n}\right\rbrack \in {\mathcal{F}}_{n}\text{for all}n\text{.} \]\n\nThen I claim that\n\n\[ A \cap \left\lbrack {T \leq n}\right\rbrack = { \cup }_{i = 1}^{n}\left( {A \cap \left\lbrack {S \leq i}\right\rbrack }\right) \cap \lef... | Yes |
Lemma 60.4.2 Let \( T \) be a stopping time and let \( \left\{ {X}_{n}\right\} \) be a sequence of random variables such that \( {X}_{n} \) is \( {\mathcal{F}}_{n} \) measurable. Then \( {X}_{T}\left( \omega \right) \equiv {X}_{T\left( \omega \right) }\left( \omega \right) \) is also a random variable and it is measura... | Proof: I assume the \( {X}_{n} \) have values in some topological space and each is measurable because the inverse image of an open set is in \( {\mathcal{F}}_{n} \) . I need to show \( {X}_{T}^{-1}\left( U\right) \cap \left\lbrack {T \leq n}\right\rbrack \in {\mathcal{F}}_{n} \) for all \( n \) whenever \( U \) is ope... | Yes |
Theorem 60.4.4 Let \( \left\{ {T}_{n}\right\} \) be an increasing bounded sequence of stopping times and let \( \left\{ {X}_{n}\right\} \) be a submartingale (martingale) adapted to the increasing sequence of \( \sigma \) algebras, \( \left\{ {\mathcal{F}}_{n}\right\} \) . Then \( \left\{ {X}_{{T}_{n}}\right\} \) is a ... | Proof: This follows from Lemma 60.4.3 | No |
Example 60.4.5 Let \( \left\{ {X}_{n}\right\} \) be a sequence of real random variables such that \( {X}_{n} \) is \( {\mathcal{F}}_{n} \) measurable and let \( A \) be a Borel subset of \( \mathbb{R} \) . Let \( T\left( \omega \right) \) denote the first time \( {X}_{n}\left( \omega \right) \) is in \( A \) . Then \( ... | To see this is a stopping time,\n\n\[ \left\lbrack {T \leq l}\right\rbrack = { \cup }_{i = 1}^{l}{X}_{i}^{-1}\left( A\right) \in {\mathcal{F}}_{l} \] | Yes |
Proposition 60.5.2 For \( \tau \) a stopping time, \( {\mathcal{F}}_{\tau } \) is a \( \sigma \) algebra and if \( Y\left( k\right) \) is \( {\mathcal{F}}_{k} \) measurable for all \( k \), then\n\n\[ \omega \rightarrow Y\left( {\tau \left( \omega \right) }\right) \]\n\nis \( {\mathcal{F}}_{\tau } \) measurable. | Proof: Let \( {A}_{n} \in {\mathcal{F}}_{\tau } \) . I need to show \( { \cup }_{n}{A}_{n} \in {\mathcal{F}}_{\tau } \) . In other words, I need to show that\n\n\[ { \cup }_{n}{A}_{n} \cap \left\lbrack {\tau \leq k}\right\rbrack \in {\mathcal{F}}_{k} \]\n\nThe left side equals\n\n\[ { \cup }_{n}\left( {{A}_{n} \cap \le... | Yes |
Lemma 60.5.3 In the situation of Definition 60.5.1, let \( \sigma ,\tau \) be two stopping times. Then\n\n1. \( \tau \) is \( {\mathcal{F}}_{\tau } \) measurable\n\n2. \( {\mathcal{F}}_{\sigma } \cap \left\lbrack {\sigma \leq \tau }\right\rbrack \subseteq {\mathcal{F}}_{\sigma \land \tau } = {\mathcal{F}}_{\sigma } \ca... | Proof: Consider the first claim. I need to show that \( \left\lbrack {\tau \leq a}\right\rbrack \cap \left\lbrack {\tau \leq k}\right\rbrack \in {\mathcal{F}}_{k} \) for every \( k \) . However, this is easy if \( a \geq k \) because the left side is then \( \left\lbrack {\tau \leq k}\right\rbrack \) which is given to ... | Yes |
Theorem 60.5.4 Let \( \{ M\left( k\right) \} \) be a real valued martingale with respect to the increasing sequence of \( \sigma \) algebras, \( \left\{ {\mathcal{F}}_{k}\right\} \) and let \( \sigma ,\tau \) be two stopping times such that \( \tau \) is bounded. Then \( M\left( \tau \right) \) defined as\n\n\[ \omega ... | Proof: By Proposition 62.6.3 \( M\left( \tau \right) \) is \( {\mathcal{F}}_{\tau } \) measurable.\n\nNext note that since \( \tau \) is bounded by some \( l \) ,\n\n\[ {\int }_{\Omega }\parallel M\left( {\tau \left( \omega \right) }\right) \parallel {dP} \leq \mathop{\sum }\limits_{{i = 1}}^{l}{\int }_{\left\lbrack \t... | Yes |
Lemma 60.5.5 Let \( \{ X\left( k\right) {\} }_{k = 0}^{\infty } \) be a submartingale adapted to the increasing sequence of \( \sigma \) algebras, \( \left\{ {\mathcal{F}}_{k}\right\} \) . Then there exists a unique increasing process \( \{ A\left( k\right) {\} }_{k = 0}^{\infty } \) such that \( A\left( 0\right) = 0 \... | Proof: Define \( \mathop{\sum }\limits_{{k = 0}}^{{-1}} \neq 0 \) . First consider the uniqueness assertion. Suppose \( A \) is a process which does what is supposed to do.\n\n\[ \n\mathop{\sum }\limits_{{k = 0}}^{{n - 1}}E\left( {X\left( {k + 1}\right) - X\left( k\right) \mid {\mathcal{F}}_{k}}\right) = \mathop{\sum }... | Yes |
Theorem 60.5.6 Let \( \{ X\left( k\right) \} \) be a submartingale with respect to the increasing sequence of \( \sigma \) algebras, \( \left\{ {\mathcal{F}}_{k}\right\} \) and let \( \sigma ,\tau \) be two stopping times such that \( \tau \) is bounded. Then \( X\left( \tau \right) \) defined as\n\n\[ \omega \rightarr... | Proof: The claim about \( X\left( \tau \right) \) being integrable is the same as in Theorem 62.6.5. If \( \tau \leq l \) ,\n\n\[ E\left( \left| {X\left( {\tau \left( \omega \right) }\right) }\right| \right) = \mathop{\sum }\limits_{{i = 1}}^{l}{\int }_{\left\lbrack \tau = i\right\rbrack }\left| {X\left( i\right) }\rig... | Yes |
Lemma 60.6.1 Let \( \{ X\left( k\right) \} \) be a nonnegative submartingale. Let\n\n\[ \n{U}_{M}^{\left\lbrack a, b\right\rbrack } \equiv \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}\mathop{\sum }\limits_{{k = 0}}^{n}\frac{X\left( {\tau }_{{2k} + 1}\right) - X\left( {\tau }_{2k}\right) }{\varepsilon + X\left( {... | Suppose that there exists a constant \( C \geq E\left( {X\left( M\right) }\right) \) for all \( M \) . That is, \( \{ X\left( k\right) \} \) is bounded in \( {L}^{1}\left( \Omega \right) \) . Then letting\n\n\[ \n{U}^{\left\lbrack a, b\right\rbrack } \equiv \mathop{\lim }\limits_{{M \rightarrow \infty }}{U}_{M}^{\left\... | Yes |
Theorem 60.6.2 Let \( \{ X\left( k\right) \} \) be a submartingale which is bounded in \( {L}^{1}\left( \Omega \right) \), \[ \parallel X\left( k\right) {\parallel }_{{L}^{1}\left( \Omega \right) } \leq C \] Then there is a set of measure zero \( N \) such that for \( \omega \notin N,\mathop{\lim }\limits_{{k \rightarr... | Proof: Let \( a < b \) and consider the submartingale \( {\left( X\left( k\right) - a\right) }_{ + } \) . Let \( {U}^{\left\lbrack 0, b - a\right\rbrack } \) be the random variable of the above lemma which is associated with this submartingale. Thus \[ E\left( {U}^{\left\lbrack 0, b - a\right\rbrack }\right) \leq \frac... | Yes |
Lemma 60.6.3 Let \( \{ X\left( k\right) \} \) be real valued and adapted to the increasing sequence of \( \sigma \) algebras \( \left\{ {\mathcal{F}}_{k}\right\} \) . Let\n\n\[ T\left( \omega \right) \equiv \inf \{ k : X\left( k\right) \geq \lambda \}\]\n\nThen \( T \) is a stopping time. Similarly,\n\n\[ T\left( \omeg... | Proof: Is \( \left\lbrack {T \leq p}\right\rbrack \in {\mathcal{F}}_{p} \) for all \( p \) ?\n\n\[ \left\lbrack {T = p}\right\rbrack = \overset{ \in {\mathcal{F}}_{p - 1}}{\overbrace{{ \cap }_{i = 1}^{p - 1}\left\lbrack {X\left( i\right) < \lambda }\right\rbrack }} \cap \overset{ \in {\mathcal{F}}_{p}}{\overbrace{\left... | Yes |
Theorem 60.6.4 Let \( \\left\\{ {X}_{k}\\right\\} \) be a real valued submartingale with respect to the \( \\sigma \) algebras \( \\left\\{ {\\mathcal{F}}_{k}\\right\\} \). Then for \( \\lambda > 0 \)\n\n\[ \n{\\lambda P}\\left( \\left\\lbrack {\\mathop{\\max }\\limits_{{1 \\leq k \\leq n}}{X}_{k} \\geq \\lambda }\\rig... | Proof: Let \( T\\left( \\omega \\right) \) be the first time \( {X}_{k}\\left( \\omega \\right) \) is \( \\geq \\lambda \) or if this does not happen for \( k \\leq n \), then \( T\\left( \\omega \\right) \\equiv n \). Thus\n\n\[ \nT\\left( \\omega \\right) \\equiv \\min \\left( {\\min \\left\\{ {k : {X}_{k}\\left( \\o... | Yes |
Lemma 60.6.5 Let \( \left\{ {\mathcal{F}}_{k}\right\} \) be an increasing sequence of \( \sigma \) algebras and let \( \{ X\left( k\right) \} \) be adapted to this sequence. Suppose that \( X\left( k\right) \) has all values in \( \left\lbrack {a, b}\right\rbrack \) and suppose \( \sigma \) is a stopping time with the ... | Proof: Let \( I \) be an interval and consider \( X\left( {k \vee \sigma }\right) \) . Is \( k \rightarrow X\left( {k \vee \sigma }\right) \) adapted? Let \( I \) be an interval. Is\n\n\[ A \equiv X{\left( k \vee \sigma \right) }^{-1}\left( I\right) \in {\mathcal{F}}_{k}? \]\n\nWe know that this set is in \( {\mathcal{... | Yes |
Example 60.6.7 Let \( \\left\\{ {\\mathcal{F}}_{n}\\right\\} \) be an increasing sequence of \( \\sigma \) algebras contained in \( \\mathcal{F} \) where \( \\left( {\\Omega ,\\mathcal{F}, P}\\right) \) is a probability space and let \( \\left\\{ {X}_{n}\\right\\} \) be a sequence of real valued random variables such t... | It happens that the above gives an increasing sequence of stopping times. | No |
Lemma 60.6.8 The above example gives an increasing sequence of stopping times. | Proof: You could consider the modified random variables\n\n\[ Y\left( k\right) \equiv \left( {X\left( k\right) \vee a}\right) \land b \]\n\nThen these new random variables stay in \( \left\lbrack {a, b}\right\rbrack \) and if you replace \( X\left( n\right) \) in the above with \( Y\left( n\right) \), you get the same ... | Yes |
Theorem 60.6.9 Let \( \\left\\{ {X}_{n}\\right\\} \) be a real valued submartingale such that \( {X}_{n} \) is \( {\\mathcal{F}}_{n} \) measurable. Then letting \( {U}_{\\left\\lbrack a, b\\right\\rbrack }^{N} \) denote the upcrossings of \( \\left\\{ {X}_{n}\\right\\} \) from a to b for \( n \\leq N \)\n\n\[ E\\left( ... | Proof: The estimate 60.6.16 was based on the assumption that \( {X}_{0}\\left( \\omega \\right) \\leq a \) . If this is not so, modify \( {X}_{0} \) . Change it to \( \\min \\left( {{X}_{0}, a}\\right) \) . Then the inequality holds for the modified submartingale which has at least as many upcrossings. Therefore, the i... | No |
Theorem 60.7.1 Let \( \\left\\{ {X}_{n}\\right\\} \) be a real valued submartingale such that\n\n\[ E\\left( \\left| {X}_{n}\\right| \\right) < M \]\n\nfor all \( n \) . Then there exists \( X \\in {L}^{1}\\left( {\\Omega ,\\mathcal{F}}\\right) \) such that \( {X}_{n}\\left( \\omega \\right) \) converges to \( X\\left(... | Proof: Let \( a < b \) be two rational numbers. From Theorem 60.6.9 it follows that for all \( N \),\n\n\[ {\\int }_{\\Omega }{U}_{\\left\\lbrack a, b\\right\\rbrack }^{N}{dP} \\leq \\frac{1}{b - a}E\\left( {\\left( {X}_{N} - a\\right) }^{ + }\\right) \]\n\n\[ \\leq \\frac{1}{b - a}\\left( {E\\left( \\left| {X}_{N}\\ri... | Yes |
Theorem 60.7.2 Let \( \\left\\{ {X}_{k}\\right\\} \) be a sequence of independent real valued random variables such that \( E\\left( \\left| {X}_{k}\\right| \\right) < \\infty, E\\left( {X}_{k}\\right) = 0 \), and\n\n\[ \n\\mathop{\\sum }\\limits_{{k = 1}}^{\\infty }E\\left( {X}_{k}^{2}\\right) < \\infty \n\]\n\nThen \... | Proof: Let \( {\\mathcal{F}}_{n} \\equiv \\sigma \\left( {{X}_{1},\\cdots ,{X}_{n}}\\right) \) . Consider \( {S}_{n} \\equiv \\mathop{\\sum }\\limits_{{k = 1}}^{n}{X}_{k} \) .\n\n\[ \nE\\left( {{S}_{n + 1} \\mid {\\mathcal{F}}_{n}}\\right) = {S}_{n} + E\\left( {{X}_{n + 1} \\mid {\\mathcal{F}}_{n}}\\right) .\n\]\nLetti... | Yes |
Theorem 60.7.5 Suppose \( \left\{ {\mathbf{X}}_{k}\right\} \) are independent random variables and \( E\left( \left| {\mathbf{X}}_{k}\right| \right) < \) \( \infty \) for each \( k \) and \( E\left( {\mathbf{X}}_{k}\right) = {\mathbf{m}}_{k} \) . Suppose also\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{\infty }\frac{1}{{j}^... | Proof: Consider the sum\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{\infty }\frac{{\mathbf{X}}_{j} - {\mathbf{m}}_{j}}{j} \]\n\nThis sum converges a.e. because of 60.7.19 and Theorem 60.7.4 applied to the random vectors \( \left\{ \frac{{\mathbf{X}}_{j} - {\mathbf{m}}_{j}}{j}\right\} \) . Therefore, from Lemma 59.7.4 it fol... | Yes |
Theorem 60.8.3 Let \( {\left\{ {X}_{n},{\mathcal{F}}_{n}\right\} }_{n = 0}^{\infty } \) be a backwards submartingale as described above and suppose \( \mathop{\sup }\limits_{{n \geq 0}}E\left( \left| {X}_{n}\right| \right) < \infty \) . Then \( \left\{ {X}_{n}\right\} \) converges a.e. and in \( {L}^{1}\left( \Omega \r... | Proof: By the upcrossing lemma applied to the submartingale \( {\left\{ {X}_{k}\right\} }_{k = 0}^{N} \), the number of upcrossings (Downcrossings is probably a better term. They are upcross-ings as \( n \) gets smaller.) of the interval \( \left\lbrack {a, b}\right\rbrack \) satisfies the inequality\n\n\[ E\left( {U}_... | Yes |
Lemma 60.9.1 Let \( \\left\\{ {X}_{n}\\right\\} \) be a sequence of independent random variables such that \( E\\left( \\left| {X}_{k}\\right| \\right) < \\infty \) for all \( k \) and let \( {S}_{n} \\equiv \\mathop{\\sum }\\limits_{{k = 1}}^{n}{X}_{k} \) . Then for \( k \\leq n \) , | Proof: Note that \( {\\mathbb{R}}^{\\mathbb{N}} \) with the usual product topology has a countable basis. Here it is. Let \( {\\mathcal{B}}_{N} \) denote sets of the form \( \\mathop{\\prod }\\limits_{{i = 1}}^{\\infty }{D}_{i} \) where for \( i \\leq N,{D}_{i} \\in \\mathcal{B} \), a countable basis for \( \\mathbb{R}... | No |
Lemma 60.9.2 Let \( \\left\\{ {X}_{k}\\right\\} \) be a sequence of independent identically distributed random variables such that \( E\\left( \\left| {X}_{k}\\right| \\right) < \\infty \) . Then letting \( {S}_{n} = \\mathop{\\sum }\\limits_{{k = 1}}^{n}{X}_{k} \), it follows that for \( k \\leq n \)\n\n\[ E\\left( {{... | Proof: It was shown in Lemma 60.9.1 the first equality holds. It remains to show the second. Letting \( A = {S}_{n}^{-1}\\left( B\\right) \) where \( B \) is Borel, it follows there exists \( {B}^{\\prime } \\subseteq {\\mathbb{R}}^{n} \) a Borel set such that\n\n\[ {S}_{n}^{-1}\\left( B\\right) = {\\left( {X}_{1},\\cd... | Yes |
Theorem 60.9.3 Let \( \\left\\{ {X}_{k}\\right\\} \) be a sequence of independent identically distributed random variables such that \( E\\left( \\left| {X}_{k}\\right| \\right) < \\infty \) for all \( k \) . Letting \( m = E\\left( {X}_{k}\\right) \) ,\n\n\[ \n\\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}\\frac... | Proof: Consider the reverse submartingale \( \\left\\{ {E\\left( {{X}_{1} \\mid \\sigma \\left( {{S}_{n},{S}_{n + 1},\\cdots }\\right) }\\right) }\\right\\} \) . By Theorem 60.8.3, this converges a.e. and in \( {L}^{1}\\left( \\Omega \\right) \) to a random variable, \( {X}_{\\infty } \) . However, from Lemma 60.9.2, \... | Yes |
Theorem 61.1.1 Let \( E \) be a separable Banach space and let \( X \in {L}^{1}\left( {\Omega ;E,\mathcal{F}}\right) \) where \( X \) is measurable with respect to \( \mathcal{F} \) and let \( \mathcal{G} \) be a \( \sigma \) algebra which is contained in \( \mathcal{F} \) . Then there exists a unique \( Z \in {L}^{1}\... | Proof: First consider uniqueness. Suppose \( {Z}^{\prime } \) is another in \( {L}^{1}\left( {\Omega ;E,\mathcal{G}}\right) \) which works. Consider a dense subset of \( E{\left\{ {a}_{n}\right\} }_{n = 1}^{\infty } \) . Then the balls \( {\left\{ B\left( {a}_{n},\frac{\begin{Vmatrix}{a}_{n}\end{Vmatrix}}{4}\right) \ri... | Yes |
Theorem 61.1.2 An E valued function, \( X \), is Bochner integrable if and only if \( X \) is strongly measurable and\n\n\[{\int }_{\Omega }\parallel X\left( \omega \right) \parallel {dP} < \infty\] | In this case there exists a sequence of simple functions \( \left\{ {X}_{n}\right\} \) satisfying\n\n\[{\int }_{\Omega }\begin{Vmatrix}{{X}_{n}\left( \omega \right) - {X}_{m}\left( \omega \right) }\end{Vmatrix}{dP} \rightarrow 0\text{ as }m, n \rightarrow \infty .\]\n\n\( \left( {61.1.2}\right) \)\n\n\( {X}_{n}\left( \... | Yes |
Lemma 61.2.3 Let \( \mu \) be a finite measure on \( \mathcal{B}\left( E\right) \), the Borel sets of \( E \), a separable complete metric space. Then if \( C \) is a closed set, \[ \mu \left( C\right) = \sup \{ \mu \left( K\right) : K \subseteq C\text{ and }K\text{ is compact. }\} \] | Proof: Let \( \left\{ {a}_{k}\right\} \) be a countable dense subset of \( C \) . Thus \( { \cup }_{k = 1}^{\infty }B\left( {{a}_{k},\frac{1}{n}}\right) \supseteq C \) . Therefore, there exists \( {m}_{n} \) such that \[ \mu \left( {C \smallsetminus { \cup }_{k = 1}^{{m}_{n}}\overline{B\left( {{a}_{k},\frac{1}{n}}\righ... | Yes |
Lemma 61.3.2 Let \( E \) be a separable complete metric space and let \( \Lambda \) be a set of Borel probability measures. Then \( \Lambda \) is tight if and only if for every \( \varepsilon > 0 \) and \( r > 0 \) there exists a finite collection of balls, \( {\left\{ B\left( {a}_{i}, r\right) \right\} }_{i = 1}^{m} \... | Proof: If \( \Lambda \) is tight, then there exists a compact set, \( {K}_{\varepsilon } \) such that\n\n\[ \mu \left( {K}_{\varepsilon }\right) > 1 - \varepsilon \]\n\nfor all \( \mu \in \Lambda \) . Then consider the open cover, \( \left\{ {B\left( {x, r}\right) : x \in {K}_{\varepsilon }}\right\} \) . Finitely many ... | Yes |
Theorem 61.3.3 Let \( H \) be a compact metric space. Then there exists a compact subset of \( \left\lbrack {0,1}\right\rbrack, K \) and a continuous function, \( \theta \) which maps \( K \) onto \( H \) . | Proof: Without loss of generality, it can be assumed \( H \) is an infinite set since otherwise the conclusion is trivial. You could pick finitely many points of \( \left\lbrack {0,1}\right\rbrack \) for \( K \) . Since \( H \) is compact, it is totally bounded. Therefore, there exists a 1 net for \( H{\left\{ {h}_{i}\... | Yes |
Corollary 61.3.4 Let \( H \) be a compact metric space and let \( C\left( H\right) \) denote the continuous functions defined on \( H \) with the usual norm,\n\n\[ \n\parallel f{\parallel }_{\infty } \equiv \max \{ \left| {f\left( x\right) }\right| : x \in H\}\n\]\n\nThen \( C\left( H\right) \) is separable. | Proof: The proof is by contradiction. Suppose \( C\left( H\right) \) is not separable. Let \( {\mathcal{H}}_{k} \) denote a maximal collection of functions of \( C\left( H\right) \) with the property that if \( f, g \in {\mathcal{H}}_{k} \), then \( \parallel f - g{\parallel }_{\infty } \geq 1/k \) . The existence of s... | Yes |
Lemma 61.4.1 Let \( {\mu }_{n} \) converge weakly to \( \mu \) and let \( U \) be an open set with \( \mu \left( {\partial U}\right) = 0 \). Then \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{\mu }_{n}\left( U\right) = \mu \left( U\right) \] | Proof: Let \( \left\{ {\psi }_{k}\right\} \) be a sequence of bounded continuous functions which decrease to \( {\mathcal{X}}_{\bar{U}} \). Also let \( \left\{ {\phi }_{k}\right\} \) be a sequence of bounded continuous functions which increase to \( {\mathcal{X}}_{U} \). For example, you could let \[ {\psi }_{k}\left( ... | Yes |
Lemma 61.5.1 Let \( \mathcal{A} \) denote sets of the form\n\n\[ \left\{ {{x}^{ * } : \theta \left( {x}^{ * }\right) \in U}\right\} \]\n\nwhere \( U \in \mathcal{E} \) . Then \( \mathcal{A} \) is an algebra and \( \sigma \left( \mathcal{A}\right) = \sigma \left( X\right) \) . Also\n\n\[ \left\{ {{\theta }^{-1}\left( U\... | Proof: Since \( \mathcal{E} \) is an algebra it is clear \( \mathcal{A} \) is also an algebra. Also, \( \mathcal{A} \subseteq \sigma \left( X\right) \) because you could let \( U \) have only one \( {A}_{y} \) not equal to \( \mathbb{R} \) and all the others equal to \( \mathbb{R} \) and then\n\n\[ \left\{ {{x}^{ * } :... | Yes |
Theorem 61.5.4 Let \( X \) be a real vector space and let \( {X}^{ * } \) be the space of linear functionals defined on \( X \) . Also let \( \psi : X \rightarrow \mathbb{C} \) . Then \( \psi \) is a characteristic function if and only if \( \psi \left( 0\right) = 1 \) and \( \psi \) is pseudo continuous at 0 . | Proof: Suppose first \( \psi \) is a characteristic function as just described. I need to show it is positive definite and pseudo continuous. It is obvious \( \psi \left( 0\right) = 1 \) in this case. Also\n\n\[ \psi \left( {\mathop{\sum }\limits_{k}{a}_{k}{x}_{k}}\right) = {\int }_{{X}^{ * }}\exp \left( {i{x}^{ * }\le... | Yes |
Lemma 61.6.7 Let \( \mu \) be a probability measure on \( \mathcal{B}\left( E\right) \), the Borel subsets of a separable real Banach space. Then there exists a probability space \( \left( {\Omega ,\mathcal{F}, P}\right) \) and two independent random variables, \( X, Y \) mapping \( \Omega \) to \( E \) such that \( \m... | Proof: First note that if \( A, B \) are Borel sets of \( E \) then \( A \times B \) is a Borel set in \( E \times E \) where the norm on \( E \times E \) is given by\n\n\[ \parallel \left( {x, y}\right) \parallel \equiv \max \left( {\parallel x\parallel ,\parallel y\parallel }\right) .\n\]\n\nThis can be proved by let... | Yes |
Theorem 61.6.9 Let \( \mathbf{X} \) and \( \mathbf{Y} \) be random vectors having values in \( {\mathbb{R}}^{p} \) and \( {\mathbb{R}}^{q} \) respectively. Suppose also that \( \left( {\mathbf{X},\mathbf{Y}}\right) \) is multivariate normally distributed and\n\n\[ E\left( {\left( {\mathbf{X} - E\left( \mathbf{X}\right)... | Proof: Let \( \mathbf{Z} = \left( {\mathbf{X},\mathbf{Y}}\right), m = p + q \) . Then by hypothesis, the characteristic function of \( \mathbf{Z} \) is of the form\n\n\[ E\left( {e}^{i\mathbf{t} \cdot \mathbf{Z}}\right) = {e}^{i\mathbf{t} \cdot \mathbf{m}}{e}^{-\frac{1}{2}i{\mathbf{t}}^{ * }\sum \mathbf{t}} \]\n\nwhere... | Yes |
Lemma 61.7.3 Let \( \mu = \mathcal{L}\left( X\right) \) where \( X \) is a random variable defined on a probability space, \( \left( {\Omega ,\mathcal{F}, P}\right) \) which has values in \( E \), a Banach space. Suppose also that for all \( \phi \in {E}^{\prime },\phi \circ X \) is normally distributed. Then \( \mu \)... | Proof: First suppose \( \mu \) is a Gaussian measure and \( X \) is a random variable such that \( \mathcal{L}\left( X\right) = \mu \) . Then if \( F \) is a Borel set in \( \mathbb{R} \), and \( h \in {E}^{\prime } \n\n\[ \nP\left( {{\left( h \circ X\right) }^{-1}\left( F\right) }\right) = P\left( {{X}^{-1}\left( {{h}... | Yes |
Lemma 61.8.2 There exists a sequence, \( {\left\{ {\xi }_{k}\right\} }_{k = 1}^{\infty } \) of random variables such that\n\n\[ \mathcal{L}\left( {\xi }_{k}\right) = N\left( {0,1}\right) \]\n\nand \( {\left\{ {\xi }_{k}\right\} }_{k = 1}^{\infty } \) is independent. | Proof: Let \( {i}_{1} < {i}_{2}\cdots < {i}_{n} \) be positive integers and define\n\n\[ {\mu }_{{i}_{1}\cdots {i}_{n}}\left( {{F}_{1} \times \cdots \times {F}_{n}}\right) \equiv \frac{1}{{\left( \sqrt{2\pi }\right) }^{n}}{\int }_{{F}_{1} \times \cdots \times {F}_{n}}{e}^{-{\left| \mathbf{x}\right| }^{2}/2}{dx}. \]\n\n... | Yes |
Theorem 61.8.5 A measure \( \mu \) on \( \mathcal{B}\left( U\right) \) is Gaussian if and only if there exists \( m \in U \) and \( Q \in \mathcal{L}\left( U\right) \) such that \( Q \) is nonnegative symmetric with finite trace, \[ \mathop{\sum }\limits_{k}\left( {Q{e}_{k},{e}_{k}}\right) < \infty \] for a complete or... | Proof: First of all suppose 61.8.32 holds. Why is \( \mu \) Gaussian? Consider the random variable \( {u}^{\prime } \) defined by \( {u}^{\prime }\left( v\right) \equiv \left( {v, u}\right) \) . Why is \( {\lambda }_{{u}^{\prime }} \) a Gaussian measure on \( \mathbb{R} \) ? By the definition in 61.8.31, \[ {\int }_{U}... | Yes |
Lemma 61.8.6 Let \( U \) be a real separable Hilbert space and let \( \mu \) be a probability measure defined on \( \mathcal{B}\left( U\right) \). Suppose for some positive integer, \( k \)\n\n\[{\int }_{U}{\left| \left( x, z\right) \right| }^{k}{d\mu }\left( x\right) < \infty\n\]\n\nfor all \( z \in U \). Then the tra... | Proof: I need to show that for each \( \mathbf{h} \in {U}^{k} \), the integral in 61.8.36 exists. From this it is obvious it is \( k \) - linear, meaning linear in each argument. Then it is shown it is continuous.\n\nFirst note\n\n\[\n\left| {\left( {{h}_{1}, x}\right) \cdots \left( {{h}_{k}, x}\right) }\right| \leq {\... | Yes |
Proposition 61.9.2 The cylinder sets form an algebra of sets. | Proof: First note the complement of a cylinder set is a cylinder set.\n\n\[ \n{\left\{ x \in H : \left( \left( x,{e}_{1}\right) ,\cdots ,\left( x,{e}_{n}\right) \right) \in F\right\} }^{C} \n\]\n\n\[ \n= \left\{ {x \in H : \left( {\left( {x,{e}_{1}}\right) ,\cdots ,\left( {x,{e}_{n}}\right) }\right) \in {F}^{C}}\right\... | Yes |
Lemma 61.9.4 \( \sigma \left( \mathcal{C}\right) \), the smallest \( \sigma \) algebra containing \( \mathcal{C} \), contains the Borel sets of \( H,\mathcal{B}\left( H\right) \) . | Proof: It follows from the definition of these cylinder sets that if \( {f}_{i}\left( x\right) \equiv \left( {x,{e}_{i}}\right) \) , so that \( {f}_{i} \in {H}^{\prime } \), then with respect to \( \sigma \left( \mathcal{C}\right) \), each \( {f}_{i} \) is measurable. It follows that every linear combination of the \( ... | Yes |
Proposition 61.9.7 For \( Q = I,\nu \) cannot be extended to a measure defined on \( \sigma \left( \mathcal{C}\right) \) whenever \( H \) is infinite dimensional. | Proof: Let \( \left\{ {e}_{n}\right\} \) be a complete orthonormal set of vectors in \( H \) . Then first note that \( H \) is a cylinder set.\n\n\[ H = \left\{ {x \in H : \left( {x,{e}_{1}}\right) \in \mathbb{R}}\right\} \]\n\nand so\n\n\[ \nu \left( H\right) = \frac{1}{\sqrt{2\pi }}{\int }_{\mathbb{R}}{e}^{-\frac{1}{... | Yes |
Proposition 61.9.8 \( \mu \) is finitely additive on \( \mathcal{C} \) the algebra of cylinder sets. | Proof: Let\n\n\[ A \equiv \left\{ {x \in H : \left( {\left( {x,{e}_{1}}\right) ,\cdots ,\left( {x,{e}_{n}}\right) }\right) \in E}\right\} ,\n\]\n\n\[ B \equiv \left\{ {x \in H : \left( {\left( {x,{f}_{1}}\right) ,\cdots ,\left( {x,{f}_{m}}\right) }\right) \in F}\right\} \]\n\nbe two disjoint cylinder sets. Then writing... | Yes |
Lemma 61.9.13 There exists a countably additive Gaussian measure, \( \lambda \) defined on \( \mathcal{B}\left( E\right) \) . This measure is the law of the random variable,\n\n\[ X\left( \omega \right) \equiv \mathop{\sum }\limits_{{k = 1}}^{\infty }{\xi }_{k}\left( \omega \right) {e}_{k} \]\n\nwhere \( \left\{ {\xi }... | Proof: Observe that \( \mathop{\sum }\limits_{{k = 1}}^{\infty }\frac{1}{{k}^{2}}\left( {k{e}_{k}}\right) \otimes \left( {k{e}_{k}}\right) \) is a nuclear operator on the Hilbert space, \( E \) . Letting \( \left\{ {\xi }_{k}\right\} \) be a sequence of independent random variables each normally distributed with mean 0... | Yes |
Corollary 61.9.15 Let \( \\left( {i, H, B}\\right) \) be an abstract Wiener space. Then there exists a Gaussian measure on the Borel sets of \( B \) . This Gaussian measure equals \( \\mathcal{L}\\left( S\\right) \) where \( S\\left( \\omega \\right) \) is the a.e. limit of a subsequence of the sequence of partial sums... | \[ {S}_{{p}_{n}}\\left( \\omega \\right) \\equiv \\mathop{\\sum }\\limits_{{k = 1}}^{{p}_{n}}{\\xi }_{k}\\left( \\omega \\right) {e}_{k} \] for \( \\left\\{ {\\xi }_{k}\\right\\} \) a sequence of independent random variables which are normal with mean 0 and variance 1 which are defined on a probability space, \( \\left... | Yes |
Theorem 61.10.1 Let \( \\left( {i, H, B}\\right) \) be an abstract Wiener space and \( \\left\\{ {e}_{k}\\right\\} \) is a complete orthonormal sequence in \( H \) . Then there exists a Gaussian measure on the Borel sets of \( B \) . This Gaussian measure equals \( \\mathcal{L}\\left( S\\right) \) where \( S\\left( \\o... | Proof: By Corollary 61.9.15 there is a subsequence, \( \\left\\{ {S}_{{p}_{n}}\\right\\} \) of these partial sums which converge pointwise a.e. to \( S\\left( \\omega \\right) \) . However, this corollary also states that \[ P\\left( \\left\\{ {\\omega \\in \\Omega : \\begin{Vmatrix}{{S}_{k}\\left( \\omega \\right) - {... | Yes |
Lemma 61.11.2 Let \( E \) be a separable Banach space. Then there exists a sequence \( \left\{ {e}_{n}\right\} \) of points of \( E \) such that whenever \( \left| \mathbf{\beta }\right| \leq 1 \) for \( \mathbf{\beta } \in {\mathbb{F}}^{n} \) , \[ \mathop{\sum }\limits_{{k = 1}}^{n}{\beta }_{k}{e}_{k} \in B\left( {0,1... | Proof: By Lemma 61.11.1, let \( \left\{ {{z}_{1},\cdots ,{z}_{n}}\right\} \) be a basis for \( {F}_{n} \) where \( { \cup }_{n = 1}^{\infty }{F}_{n} \) is dense in \( E \) . Then let \( {\alpha }_{1} \) be such that \( {e}_{1} \equiv {\alpha }_{1}{z}_{1} \in B\left( {0,1}\right) \) . Thus \( {\beta }_{1}{e}_{1} \in B\l... | Yes |
Corollary 61.11.4 Let \( E \) be any real separable Banach space. Then there exists a sequence, \( \left\{ {e}_{k}\right\} \subseteq E \) such that for any \( \left\{ {\xi }_{k}\right\} \) a sequence of independent random variables such that \( \mathcal{L}\left( {\xi }_{k}\right) = N\left( {0,1}\right) \), it follows\n... | Proof: From the proof of Theorem 61.11.3 a basis for \( H \) is \( \left\{ {{\lambda }_{k}{e}_{k}}\right\} \) . Therefore, by Theorem 61.10.1, if \( \left\{ {\xi }_{k}\right\} \) is a sequence of independent \( N\left( {0,1}\right) \) random variables, then \( \mathop{\sum }\limits_{{k = 1}}^{\infty }{\xi }_{k}\left( \... | Yes |
Lemma 62.1.1 A stochastically continuous process on \( \left\lbrack {a, b}\right\rbrack \equiv I \) is uniformly stochastically continuous on \( \left\lbrack {a, b}\right\rbrack \equiv I \) . | Proof: If this is not so, there exists \( \varepsilon ,\delta > 0 \) and points of \( I,{s}_{n},{t}_{n} \) such that even though\n\n\[ \left| {{t}_{n} - {s}_{n}}\right| < \frac{1}{n} \]\n\n\[ P\left( \left\lbrack {\begin{Vmatrix}{X\left( {s}_{n}\right) - X\left( {t}_{n}\right) }\end{Vmatrix} \geq \varepsilon }\right\rb... | Yes |
Proposition 62.1.2 Let \( X \) be a stochastically continuous process defined on a closed interval, \( I \equiv \left\lbrack {a, b}\right\rbrack \) . Then there exists a measurable version of \( X \) . | Proof: By Lemma 62.1.1 \( X \) is uniformly stochastically continuous and so there exists a sequence of positive numbers, \( \left\{ {\rho }_{n}\right\} \) such that if \( \left| {s - t}\right| < {\rho }_{n} \), then\n\n\[ P\left( \left\lbrack {\parallel X\left( t\right) - X\left( s\right) \parallel \geq \frac{1}{{2}^{... | Yes |
Lemma 62.1.3 Let \( D \) be a dense subset of an interval, \( I = \left\lbrack {0, T}\right\rbrack \) and suppose \( X : D \rightarrow E \) satisfies\n\n\[ \begin{Vmatrix}{X\left( d\right) - X\left( {d}^{\prime }\right) }\end{Vmatrix} \leq C{\left| d - {d}^{\prime }\right| }^{\gamma } \]\n\nfor all \( {d}^{\prime }, d ... | Proof: Let \( t \in I \) and let \( {d}_{k} \rightarrow t \) where \( {d}_{k} \in D \) . Then \( \left\{ {X\left( {d}_{k}\right) }\right\} \) is a Cauchy sequence because \( \begin{Vmatrix}{X\left( {d}_{k}\right) - X\left( {d}_{m}\right) }\end{Vmatrix} \leq C{\left| {d}_{k} - {d}_{m}\right| }^{\gamma } \) . Therefore, ... | Yes |
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