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Theorem 6.5.4. For each \( \alpha \) in the range \( (0,2\rbrack \) , \[ {f}_{\alpha }\left( t\right) = {e}^{-{\left| t\right| }^{\alpha }} \] is a ch.f. | PROOF. For \( 0 < \alpha \leq 1 \), this is a quick consequence of Pólya’s theorem above. Other conditions there being obviously satisfied, we need only check that \( {f}_{\alpha } \) is convex in \( \lbrack 0,\infty ) \) . This is true because its second derivative is equal to \[ {e}^{-{t}^{\alpha }}\left\{ {{\alpha }... | Yes |
Theorem 6.5.5. There exist two distinct ch.f.'s that coincide in an interval containing the origin. | That the interval of coincidence can be \ | No |
Theorem 6.5.6. If \( f \) is a ch.f., then so is \( {e}^{\lambda \left( {f - 1}\right) } \) for each \( \lambda \geq 0 \) . | PROOF. For each \( \lambda \geq 0 \), as soon as the integer \( n \geq \lambda \), the function\n\n\[ 1 - \frac{\lambda }{n} + \frac{\lambda }{n}f = 1 + \frac{\lambda \left( {f - 1}\right) }{n} \]\n\nis a ch.f., hence so is its \( n \) th power; see propositions (iv) and (v) of Sec. 6.1.\n\nAs \( n \rightarrow \infty \... | Yes |
Theorem 6.6.1. Two r.v.’s \( X \) and \( Y \) are independent if and only if\n\n\[ \forall s,\forall t : \;{f}_{\left( X, Y\right) }\left( {s, t}\right) = {f}_{X}\left( s\right) {f}_{Y}\left( t\right) ,\] \n\nwhere \( {f}_{X} \) and \( {f}_{Y} \) are the ch.f.’s of \( X \) and \( Y \), respectively. | PROOF OF THEOREM 6.6.1. If \( X \) and \( Y \) are independent, then so are \( {e}^{isX} \) and \( {e}^{itY} \) for every \( s \) and \( t \), hence\n\n\[ \mathcal{E}\left( {e}^{i\left( {{sX} + {tY}}\right) }\right) = \mathcal{E}\left( {{e}^{isX} \cdot {e}^{itY}}\right) = \mathcal{E}\left( {e}^{isX}\right) \mathcal{E}\... | Yes |
Theorem 6.6.2. Let \( {\widehat{F}}_{j} \) be the Laplace transform of the d.f. \( {F}_{j} \) with support in \( {\mathcal{R}}_{ + }, j = 1,2 \) . If \( {\widehat{F}}_{1} = {\widehat{F}}_{2} \), then \( {F}_{1} = {F}_{2} \) . | PROOF. We shall apply the Stone-Weierstrass theorem to the algebra generated by the family of functions \( \left\{ {{e}^{-{\lambda x}},\lambda \geq 0}\right\} \), defined on the closed positive real line: \( {\overline{\mathcal{R}}}_{ + } = \left\lbrack {0,\infty }\right\rbrack \), namely the one-point compactification... | Yes |
Theorem 6.6.3. Let \( \left\{ {{F}_{n},1 \leq n < \infty }\right\} \) be a sequence of s.d.f.’s with supports in \( {\mathcal{R}}_{ + } \) and \( \left\{ {\widehat{F}}_{n}\right\} \) the corresponding Laplace transforms. Then \( {F}_{n}\overset{v}{ \rightarrow }{F}_{\infty } \), where \( {F}_{\infty } \) is a d.f., if ... | PROOF. The \ | No |
Theorem 6.6.4. A function \( f \) on \( \left( {0,\infty }\right) \) is the Laplace transform of a d.f. \( F \) :\n\n\[ f\left( \lambda \right) = {\int }_{\mathcal{R} + }{e}^{-{\lambda x}}{dF}\left( x\right) \]\n\nif and only if it is completely monotonic in \( \left( {0,\infty }\right) \) with \( f\left( {0 + }\right)... | PROOF. The \ | No |
Theorem 6.6.5. The function \( h \) of the complex variable \( z \) given by\n\n\[ h\left( z\right) = {\int }_{{\mathcal{R}}_{ + }}{e}^{zx}{dF}\left( x\right) \]\n\nis analytic in \( \mathbf{R}z < 0 \) and continuous in \( \mathbf{R}z \leq 0 \) . Suppose that \( g \) is another function of \( z \) that is analytic in \... | PROOF. For each integer \( m \geq 1 \), the function \( {h}_{m} \) defined by\n\n\[ {h}_{m}\left( z\right) = {\int }_{\left\lbrack 0, m\right\rbrack }{e}^{zx}{dF}\left( x\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{z}^{n}{\int }_{\left\lbrack 0, m\right\rbrack }\frac{{x}^{n}}{n!}{dF}\left( x\right) \]\n\nis clea... | Yes |
Theorem 7.1.1. A necessary and sufficient condition for (2) to be holospoudic is:\n\n\[ \forall t \in {\mathcal{R}}^{1} : \mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\max }\limits_{{1 \leq j \leq {k}_{n}}}\left| {{f}_{nj}\left( t\right) - 1}\right| = 0. \] | PROOF. Assuming that (b) is true, we have\n\n\[ \left| {{f}_{nj}\left( t\right) - 1}\right| \leq \int \left| {{e}^{itx} - 1}\right| d{F}_{nj}\left( x\right) = {\int }_{\left| x\right| > \epsilon } + {\int }_{\left| x\right| \leq \epsilon } \]\n\n\[ \leq {\int }_{\left| x\right| > \epsilon }{2d}{F}_{nj}\left( x\right) +... | Yes |
Theorem 7.1.2. Assume that (4) and (5) hold for the double array (2) and that \( {\gamma }_{nj} \) is finite for every \( n \) and \( j \) . If (10) \( {\Gamma }_{n} \rightarrow 0 \) as \( n \rightarrow \infty \), then \( {S}_{n} \) converges in dist. to \( \Phi \) . | PROOF. For each \( n \), the range of \( j \) below will be from 1 to \( {k}_{n} \) . It follows from the assumption (10) and Liapounov's inequality that (11) \( \mathop{\max }\limits_{j}{\sigma }_{nj}^{3} \leq \mathop{\max }\limits_{j}{\gamma }_{nj} \leq {\Gamma }_{n} \rightarrow 0. \) By \( \left( {3}^{\prime }\right... | Yes |
Lemma 1. Let \( u\left( {m, n}\right) \) be a function of positive integers \( m \) and \( n \) such that\n\n\[ \forall m : \mathop{\lim }\limits_{{n \rightarrow \infty }}u\left( {m, n}\right) = 0.\]\n\nThen there exists a sequence \( \left\{ {m}_{n}\right\} \) increasing to \( \infty \) such that\n\n\[ \mathop{\lim }\... | PROOF. It is the statement of the lemma and its necessity in our application that requires a certain amount of sophistication; the proof is easy. For each \( m \), there is an \( {n}_{m} \) such that \( n \geq {n}_{m} \Rightarrow u\left( {m, n}\right) \leq 1/m \) . We may choose \( \left\{ {{n}_{m}, m \geq 1}\right\} \... | Yes |
Lemma 2. For each \( n \), the r.v.’s \( \left\{ {{X}_{nj},1 \leq j \leq n}\right\} \) are independent with the following distributions:\n\n\[ \mathcal{P}\left\{ {{X}_{nj} = m}\right\} = \frac{1}{j}\;\text{ for }0 \leq m \leq j - 1. \] | The lemma is a striking example that stochastic independence need not be an obvious phenomenon even in the simplest problems and may require verification as well as discovery. It is often disposed of perfunctorily but a formal proof is lengthier than one might think. Observe first that the values of \( {X}_{n1},\ldots ... | Yes |
Lemma 1. Let \( F \) be a d.f., \( G \) a real-valued function satisfying the conditions below:\n\n(i) \( \mathop{\lim }\limits_{{x \rightarrow - \infty }}G\left( x\right) = 0,\mathop{\lim }\limits_{{x \rightarrow + \infty }}G\left( x\right) = 1 \) ;\n\n(ii) \( G \) has a derivative that is bounded everywhere: \( \math... | PROOF. Clearly the \( \Delta \) in (2) is finite, since \( G \) is everywhere bounded by (i) and (ii). We may suppose that the left member of (3) is strictly positive, for otherwise there is nothing to prove; hence \( \Delta > 0 \) . Since \( F - G \) vanishes at \( \pm \infty \) by (i), there exists a sequence of numb... | Yes |
Lemma 2. In addition to the assumptions of Lemma 1, we assume that\n\n(iii) \( G \) is of bounded variation in \( \\left( {-\\infty ,\\infty }\\right) \) ;\n\n(iv) \( {\\int }_{-\\infty }^{\\infty }\\left| {F\\left( x\\right) - G\\left( x\\right) }\\right| {dx} < \\infty \) .\n\nLet\n\[ f\\left( t\\right) = {\\int }_{-... | PROOF. That the integral on the right side of (4) is finite will soon be apparent, and as a Lebesgue integral the value of the integrand at \( t = 0 \) may be overlooked. We have, by a partial integration, which is permissible on account of condition (iii):\n\n(5)\n\[ f\\left( t\\right) - g\\left( t\\right) = - {it}{\\... | Yes |
Lemma 3. For \( \left| t\right| < 1/\left( {2{\Gamma }_{n}^{1/3}}\right) \), we have\n\n\[ \left| {{f}_{n}\left( t\right) - {e}^{-{t}^{2}/2}}\right| \leq {\Gamma }_{n}{\left| t\right| }^{3}{e}^{-{t}^{2}/2}. \] | PROOF. We shall denote by \( \theta \) below a \ | No |
Lemma 4. For \( \\left| t\\right| < 1/\\left( {4{\\Gamma }_{n}}\\right) \\), we have\n\n(10)\n\n\\[ \n\\left| {{f}_{n}\\left( t\\right) }\\right| \\leq {e}^{-{t}^{2}/3} \n\\] | PROOF. We symmetrize (see end of Sec. 6.2) to facilitate the estimation.\n\nWe have\n\n\\[ \n{\\left| {f}_{nj}\\left( t\\right) \\right| }^{2} = {\\int }_{-\\infty }^{\\infty }{\\int }_{-\\infty }^{\\infty }\\cos t\\left( {x - y}\\right) d{F}_{nj}\\left( x\\right) d{F}_{nj}\\left( y\\right) , \n\\]\n\nsince \( {\\left|... | Yes |
Lemma 5. For \( \left| t\right| < 1/\left( {4{\Gamma }_{n}}\right) \), we have\n\n(11)\n\n\[ \left| {{f}_{n}\left( t\right) - {e}^{-{t}^{2}/2}}\right| \leq {16}{\Gamma }_{n}{\left| t\right| }^{3}{e}^{-{t}^{2}/3}. \] | PROOF. If \( \left| t\right| < 1/\left( {2{\Gamma }_{n}{}^{1/3}}\right) \), this is implied by (8). If \( 1/\left( {2{\Gamma }_{n}{}^{1/3}}\right) \leq \left| t\right| < \) \( 1/\left( {4{\Gamma }_{n}}\right) \), then \( 1 \leq 8{\Gamma }_{n}{\left| t\right| }^{3} \), and so by (10):\n\n\[ \left| {{f}_{n}\left( t\right... | Yes |
Lemma 1. Suppose that for some \( \epsilon ,0 < \epsilon < 1 \), we have\n\n(3)\n\[ \frac{{\Gamma }_{n}}{{s}_{n}^{3}} \leq \frac{A}{{\left( \log {s}_{n}\right) }^{1 + \epsilon }} \]\n\nThen for each \( \delta ,0 < \delta < \epsilon \), we have\n\n(4)\n\[ \mathcal{P}\left\{ {{S}_{n} > \varphi \left( {1 + \delta ,{s}_{n}... | PROOF. By Theorem 7.4.1, we have for each \( x \) :\n\n(6)\n\[ \mathcal{P}\left\{ {{S}_{n} > x{s}_{n}}\right\} = \frac{1}{\sqrt{2\pi }}{\int }_{x}^{\infty }{e}^{-{y}^{2}/2}{dy} + \Lambda \frac{{\Gamma }_{n}}{{s}_{n}^{3}}. \]\n\nWe have as \( x \rightarrow \infty \) :\n\n(7)\n\[ {\int }_{x}^{\infty }{e}^{-{y}^{2}/2}{dy}... | No |
Lemma 2. Let \( \\left\\{ {E}_{j}\\right\\} \) and \( \\left\\{ {F}_{j}\\right\\} ,1 \\leq j \\leq n < \\infty \\), be two sequences of events. Suppose that for each \( j \), the event \( {F}_{j} \) is independent of \( {E}_{1}^{c}\\cdots {E}_{j - 1}^{c}{E}_{j} \), and\n\nthat there exists a constant \( A > 0 \) such t... | PROOF. The left member in (12) is equal to\n\n\[ \n\\mathcal{P}\\left( {\\mathop{\\bigcup }\\limits_{{j = 1}}^{n}\\left\\lbrack {{\\left( {E}_{1}{F}_{1}\\right) }^{c}\\cdots {\\left( {E}_{j - 1}{F}_{j - 1}\\right) }^{c}\\left( {{E}_{j}{F}_{j}}\\right) }\\right\\rbrack }\\right) \n\]\n\n\[ \n\\geq \\mathcal{P}\\left( {\... | Yes |
Theorem 7.5.1. Under the condition (3), the lim sup and lim inf, as \( n \rightarrow \infty \) , of \( {S}_{n}/\sqrt{2{s}_{n}^{2}\log \log {s}_{r}} \) are respectively +1 and -1, with probability one. | The assertion about liminf follows, of course, from (2) if we apply it to \( \left\{ {-{X}_{j}, j \geq 1}\right\} \) . Recall that (3) is more than sufficient to ensure the validity of the central limit theorem, namely that \( {S}_{n}/{s}_{n} \) converges in dist. to \( \Phi \) . Thus the law of the iterated logarithm ... | No |
Theorem 7.6.1. An infinitely divisible ch.f. never vanishes (for real \( t \) ). | PROOF. We shall see presently that a complex-valued ch.f. is trouble-some when its \ | No |
Theorem 7.6.2. Let a complex-valued function \( f \) of the real variable \( t \) be given. Suppose that \( f\left( 0\right) = 1 \) and that for some \( T > 0, f \) is continuous in \( \left\lbrack {-T, T}\right\rbrack \) and does not vanish in the interval. Then there exists a unique (single-valued) function \( \lambd... | PROOF. Consider the range of \( f\left( t\right), t \in \left\lbrack {-T, T}\right\rbrack \) ; this is a closed set of points in the complex plane. Since it does not contain the origin, we have\n\n\[ \mathop{\inf }\limits_{{-T \leq t \leq T}}\left| {f\left( t\right) - 0}\right| = {\rho }_{T} > 0 \]\n\nNext, since \( f ... | Yes |
Theorem 7.6.3. For a fixed \( T \), let each \( {}_{k}f, k \geq 1 \), as well as \( f \) satisfy the conditions for \( f \) in Theorem 7.6.2, and denote the corresponding \( \lambda \) by \( {}_{k}\lambda \) . Suppose that \( {}_{k}f \) converges uniformly to \( f \) in \( \left\lbrack {-T, T}\right\rbrack \), then \( ... | PROOF. Let \( L \) be as in (7), then there exists a \( \delta ,0 < \delta < \frac{1}{2} \), such that\n\n\[ \left| {L\left( z\right) }\right| \leq 1,\;\text{ if }\left| {z - 1}\right| \leq \delta . \]\n\nBy the hypothesis of uniformity, there exists \( {k}_{1}\left( T\right) \) such that if \( k \geq {k}_{1}\left( T\r... | Yes |
Theorem 7.6.4. For each \( n \), the \( {f}_{n} \) in (1) is just the distinguished \( n \) th root of \( f \) . | PROOF. It follows from Theorem 7.6.1 and (1) that the ch.f. \( {f}_{n} \) never vanishes in \( \left( {-\infty ,\infty }\right) \), hence its distinguished logarithm \( {\lambda }_{n} \) is defined. Taking multiple-valued logarithms in (1), we obtain as in (10):\n\n\[\n\forall t : \lambda \left( t\right) - n{\lambda }_... | Yes |
Theorem 7.6.5. Let \( \left\{ {{}_{k}f, k \geq 1}\right\} \) be a sequence of infinitely divisible ch.f.’s converging everywhere to the ch.f. \( f \) . Then \( f \) is infinitely divisible. | PROOF. The difficulty is to prove first that \( f \) never vanishes. Consider, as in the proof of Theorem 7.6.1: \( g = {\left| f\right| }^{2},{}_{k}g = {\left| {}_{k}f\right| }^{2} \) . For each \( n > 1 \), let \( {x}^{1/n} \) denote the real positive \( n \) th root of a real positive \( x \) . Then we have, by the ... | Yes |
Theorem 7.6.6. For each infinitely divisible ch.f. \( f \), there exists a double array of pairs of real constants \( \left( {{a}_{nj},{u}_{nj}}\right) ,1 \leq j \leq {k}_{n},1 \leq n \), where \( {a}_{j} > 0 \) , such that\n\n\[ f\left( t\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\prod }\limits_{... | PROOF. Let \( f \) and \( {f}_{n} \) be as in (1) and let \( \lambda \) be the distinguished logarithm of \( f,{F}_{n} \) the d.f. corresponding to \( {f}_{n} \) . We have for each \( t \), as \( n \rightarrow \infty \) :\n\n\[ n\left\lbrack {{f}_{n}\left( t\right) - 1}\right\rbrack = n\left\lbrack {{e}^{\lambda \left(... | Yes |
Theorem 7.6.7. Every infinitely divisible ch.f. \( f \) has the following canonical representation: | \[ f\left( t\right) = \exp \left\lbrack {{ait} + {\int }_{-\infty }^{\infty }\left( {{e}^{itu} - 1 - \frac{itu}{1 + {u}^{2}}}\right) \frac{1 + {u}^{2}}{{u}^{2}}{dG}\left( u\right) }\right\rbrack \] where \( a \) is a real constant, \( G \) is a bounded increasing function in \( \left( {-\infty ,\infty }\right) \) , and... | Yes |
Theorem 8.1.1. Given \( \epsilon > 0 \) and \( \Lambda \in {\mathcal{F}}_{\infty } \), there exists \( {\Lambda }_{\epsilon } \in \mathop{\bigcup }\limits_{{n = 1}}^{\infty }{\mathcal{F}}_{n} \) such that\n\n(2)\n\[ \mathcal{P}\left( {{\Lambda }_{\Delta }{\Lambda }_{\epsilon }}\right) \leq \epsilon \] | PROOF. Let \( \mathcal{G} \) be the collection of sets \( \Lambda \) for which the assertion of the theorem is true. Suppose \( {\Lambda }_{k} \in \mathcal{G} \) for each \( k \in N \) and \( {\Lambda }_{k} \uparrow \Lambda \) or \( {\Lambda }_{k} \downarrow \Lambda \) . Then \( \Lambda \) also belongs to \( \mathcal{G... | Yes |
Theorem 8.1.2. For an independent process, each remote event has probability zero or one. | PROOF. Let \( \Lambda \in \mathop{\bigcap }\limits_{{n = 1}}^{\infty }{\mathcal{F}}_{n}^{\prime } \) and suppose that \( \mathcal{P}\left( \Lambda \right) > 0 \) ; we are going to prove that \( \mathcal{P}\left( \Lambda \right) = 1 \) . Since \( {\mathcal{F}}_{n} \) and \( {\mathcal{F}}_{n}^{\prime } \) are independent... | Yes |
Theorem 8.1.3. For a stationary independent process, if \( \Lambda \in \mathcal{F} \) and \( \sigma \) is any finite permutation, we have\n\n(7)\n\[ \mathcal{P}\left( {{\tau }^{-1}\Lambda }\right) = \mathcal{P}\left( \Lambda \right) \]\n\n(8)\n\[ \mathcal{P}\left( {\sigma \Lambda }\right) = \mathcal{P}\left( \Lambda \r... | PROOF. Define a set function \( \mathcal{P} \) on \( \mathcal{F} \) as follows:\n\n\[ \widetilde{\mathcal{P}}\left( \Lambda \right) = \mathcal{P}\left( {{\tau }^{-1}\Lambda }\right) \]\n\nSince \( {\tau }^{-1} \) maps disjoint sets into disjoint sets, it is clear that \( \mathcal{P} \) is a p.m. For a finite-product se... | Yes |
Theorem 8.1.4. For a stationary independent process, each permutable event has probability zero or one. | PROOF. Let \( \Lambda \) be a permutable event. Given \( \epsilon > 0 \), we may choose \( {\epsilon }_{k} > 0 \) so that\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{\infty }{\epsilon }_{k} \leq \epsilon \]\n\nBy Theorem 8.1.1, there exists \( {\Lambda }_{k} \in {\mathcal{F}}_{{n}_{k}} \) such that \( \mathcal{P}\left( {\La... | Yes |
Theorem 8.2.1. Let \( {B}_{n} \in {\mathcal{B}}^{1} \) for each \( n \in N \) . Then\n\n\[ \mathcal{P}\left\{ {{S}_{n} \in {B}_{n}\text{ i.o. }}\right\} \]\n\nis equal to zero or one. | PROOF. If \( \sigma \) is a permutation on \( {N}_{m} \), then \( {S}_{n}\left( {\sigma \omega }\right) = {S}_{n}\left( \omega \right) \) for \( n \geq m \) ,\n\nhence the set\n\n\[ {\Lambda }_{m} = \mathop{\bigcup }\limits_{{n = m}}^{\infty }\left\{ {{S}_{n} \in {B}_{n}}\right\} \]\n\nis unchanged under \( {\sigma }^{... | No |
For a stationary independent process and an almost everywhere finite optional r.v. \( \alpha \) relative to it, the pre- \( \alpha \) and post- \( \alpha \) fields are independent. Furthermore the post- \( \alpha \) process is a stationary independent process with the same common distribution as the original one. | Both assertions are summarized in the formula below. For any \( \Lambda \in {\mathcal{F}}_{\alpha }, k \in N,{B}_{j} \in {\mathcal{B}}^{1},1 \leq j \leq k \), we have\n\n\[ \mathcal{P}\left\{ {\Lambda ;{X}_{\alpha + j} \in {B}_{j},1 \leq j \leq k}\right\} = \mathcal{P}\{ \Lambda \} \mathop{\prod }\limits_{{j = 1}}^{k}\... | Yes |
Theorem 8.2.3. Let \( \alpha \) be an a.e. finite optional r.v. relative to a stationary independent process. Then the random vectors \( \left\{ {{V}_{k}, k \in N}\right\} \), where\n\n\[ \n{V}_{k}\left( \omega \right) = \left( {{\alpha }^{k}\left( \omega \right) ,{X}_{{\beta }_{k - 1} + 1}\left( \omega \right) ,\ldots... | PROOF. The independence follows from Theorem 8.2.2 by our showing that \( {V}_{1},\ldots ,{V}_{k - 1} \) belong to the pre- \( {\beta }_{k - 1} \) field, while \( {V}_{k} \) belongs to the post- \( {\beta }_{k - 1} \) field. The details are left to the reader; cf. Exercise 6 below.\n\nTo prove that \( {V}_{k} \) and \(... | No |
Theorem 8.2.4. The statements (a), (b), and (c) below are equivalent; the statements \( \left( {a}^{\prime }\right) ,\left( {b}^{\prime }\right) \), and \( \left( {c}^{\prime }\right) \) are equivalent.\n\n(a) \( \mathcal{P}\{ \alpha < + \infty \} = 1;\;\left( {\mathrm{a}}^{\prime }\right) \mathcal{P}\{ \alpha < + \inf... | PROOF. If (a) is true, we may suppose \( \alpha < \infty \) everywhere. Consider the r.v. \( {S}_{\alpha } \) : it is strictly positive by definition and so \( 0 < \mathcal{E}\left( {S}_{\alpha }\right) \leq + \infty \) . By the Corollary to Theorem 8.2.3, \( \left\{ {{S}_{{\beta }_{k + 1}} - {S}_{{\beta }_{k}}, k \geq... | Yes |
Theorem 8.2.5. For the general random walk, there are four mutually exclusive possibilities, each taking place a.e.:\n\n(i) \( \forall n \in N : {S}_{n} = 0 \) ;\n\n(ii) \( {S}_{n} \rightarrow - \infty \) ;\n\n(iii) \( {S}_{n} \rightarrow + \infty \) ;\n\n(iv) \( - \infty = \mathop{\lim }\limits_{{n \rightarrow \infty ... | PROOF. If \( X = 0 \) a.e., then (i) happens. Excluding this, let \( {\varphi }_{1} = {\overline{\lim }}_{n}{S}_{n} \) .\n\nThen \( {\varphi }_{1} \) is a permutable r.v., hence a constant \( c \), possibly \( \pm \infty \), a.e. by\n\nTheorem 8.1.4. Since | No |
Theorem 8.1.4. Since\n\n\[\n\\mathop{\\lim }\\limits_{n}{S}_{n} = {X}_{1} + \\mathop{\\lim }\\limits_{n}\\left( {{S}_{n} - {X}_{1}}\\right)\n\]\n\nwe have \( {\\varphi }_{1} = {X}_{1} + {\\varphi }_{2} \), where \( {\\varphi }_{2}\\left( \\omega \\right) = {\\varphi }_{1}\\left( {\\tau \\omega }\\right) = c \) a.e. Sin... | This last possibility will be elaborated upon in the next section. | No |
Theorem 8.3.1. The set \( \Re \) is either empty or a closed additive group of real numbers. In the latter case it reduces to the singleton \( \{ 0\} \) if and only if \( X \equiv 0 \) a.e.; otherwise \( \mathfrak{R} \) is either the whole \( {\mathcal{R}}^{1} \) or the infinite cyclic group generated by a nonzero numb... | PROOF. Suppose \( \Re \neq \phi \) throughout the proof. To prove that \( \Re \) is a group, let us show that if \( x \) is a possible value and \( y \in \Re \), then \( y - x \in \Re \) . Suppose not; then there is a strictly positive probability that from a certain value of \( n \) on, \( {S}_{n} \) will not be in a ... | No |
Theorem 8.3.2. If for some \( \epsilon > 0 \) we have\n\n\[ \mathop{\sum }\limits_{n}\mathcal{P}\left\{ {\left| {S}_{n}\right| < \epsilon }\right\} < \infty \]\n\nthen\n\n\[ \mathcal{P}\left\{ {\left| {S}_{n}\right| < \epsilon \text{ i.o. }}\right\} = 0 \]\n\n(for the same \( \epsilon \) ) so that \( 0 \notin \Re \) . ... | PROOF. The first assertion follows at once from the convergence part of the Borel-Cantelli lemma (Theorem 4.2.1). To prove the second part consider\n\n\[ F = \mathop{\liminf }\limits_{n}\left\{ {\left| {S}_{n}\right| \geq \epsilon }\right\} \]\n\nnamely \( F \) is the event that \( \left| {S}_{n}\right| < \epsilon \) f... | Yes |
Theorem 8.3.3. If the weak law of large numbers holds for the random walk \( \left\{ {{S}_{n}, n \in N}\right\} \) in the form that \( {S}_{n}/n \rightarrow 0 \) in pr., then \( \Re \neq \phi \) . | PROOF. We need two lemmas, the first of which is also useful elsewhere. | No |
Lemma 1. For any \( \epsilon > 0 \) and \( m \in N \) we have\n\n\[ \mathop{\sum }\limits_{{n = 0}}^{\infty }\mathcal{P}\left\{ {\left| {S}_{n}\right| < {m\epsilon }}\right\} \leq {2m}\mathop{\sum }\limits_{{n = 0}}^{\infty }\mathcal{P}\left\{ {\left| {S}_{n}\right| < \epsilon }\right\} \] | PROOF OF LEMMA 1. It is sufficient to prove that if the right member of (8) is finite, then so is the left member and (8) is true. Put\n\n\[ I = \left( {-\epsilon ,\epsilon }\right) ,\;J = \lbrack {j\epsilon },\left( {j + 1}\right) \epsilon ), \]\n\nfor a fixed \( j \in N \) ; and denote by \( {\varphi }_{I},{\varphi }... | Yes |
Lemma 2. Let the positive numbers \( \left\{ {{u}_{n}\left( m\right) }\right\} \), where \( n \in N \) and \( m \) is a real number \( \geq 1 \), satisfy the following conditions:\n\n(i) \( {\forall }_{n} : {u}_{n}\left( m\right) \) is increasing in \( m \) and tends to 1 as \( m \rightarrow \infty \) ;\n\n(ii) \( \exi... | PROOF OF LEMMA 2. Suppose not; then for every \( A > 0 \) :\n\n\[ \infty > \mathop{\sum }\limits_{{n = 0}}^{\infty }{u}_{n}\left( 1\right) \geq \frac{1}{cm}\mathop{\sum }\limits_{{n = 0}}^{\infty }{u}_{n}\left( m\right) \geq \frac{1}{cm}\mathop{\sum }\limits_{{n = 0}}^{\left\lbrack Am\right\rbrack }{u}_{n}\left( m\righ... | Yes |
Theorem 8.3.4. Suppose that at least one of \( \mathcal{E}\left( {X}^{ + }\right) \) and \( \mathcal{E}\left( {X}^{ - }\right) \) is finite. The \( \Re \neq \phi \) if and only if \( \mathcal{E}\left( X\right) = 0 \) ; otherwise case (ii) or (iii) of Theorem 8.2.5 happens according as \( \mathcal{E}\left( X\right) < 0 ... | PROOF. If \( - \infty \leq \mathcal{E}\left( X\right) < 0 \) or \( 0 < \mathcal{E}\left( X\right) \leq + \infty \), then by the strong law of large numbers (as amended by Exercise 1 of Sec. 5.4), we have\n\n\[ \frac{{S}_{n}}{n} \rightarrow \mathcal{E}\left( X\right) \text{ a.e.,} \]\n\nso that either (ii) or (iii) happ... | No |
Theorem 8.4.1. Let\n\n\[ \nP\left( {r, t}\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{r}^{n}{p}_{n}\left( t\right) ,\;Q\left( {r, t}\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{r}^{n}{q}_{n}\left( t\right) , \]\n\n\[ \n{P}^{ * }\left( {r, t}\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{r}^{n}{p}_{n}... | PROOF. It follows from (9) and the identity theorem for power series that for every \( n \geq 0 \) :\n\n(10) \n\[ \n\mathop{\sum }\limits_{{k = 0}}^{n}{p}_{k}\left( t\right) {q}_{n - k}^{ * }\left( t\right) = \mathop{\sum }\limits_{{k = 0}}^{n}{p}_{k}^{ * }\left( t\right) {q}_{n - k}\left( t\right) . \]\n\nThen for \( ... | Yes |
Theorem 8.4.3. The generating function of \( \alpha \) in Theorem 8.4.2 is given by\n\n(13)\n\n\[{\mathcal{E}}^{\mathcal{E}}\left\{ {r}^{\alpha }\right\} = 1 - \exp \left\{ {-\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{r}^{n}}{n}\mathcal{P}\left\lbrack {{S}_{n} \in A}\right\rbrack }\right\} = 1 - \left( {1 - r}\rig... | PROOF. Setting \( t = 0 \) in (11), we obtain the first equation in (13), from which the second follows at once through\n\n\[ \frac{1}{1 - r} = \exp \left\{ {\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{r}^{n}}{n}}\right\} = \exp \left\{ {\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{r}^{n}}{n}\mathcal{P}\left\lbr... | Yes |
Theorem 8.4.4. Suppose that \( X ≢ 0 \) and at least one of \( \mathcal{E}\left( {X}^{ + }\right) \) and \( {\mathcal{E}}^{c}\left( {X}^{ - }\right) \) is finite; then\n\n(18)\n\n\[ \mathcal{E}\left( X\right) > 0 \Rightarrow \mathcal{E}\left\{ {\alpha }_{\left( 0,\infty \right) }\right\} < \infty \]\n\n(19)\n\n\[ \math... | PROOF. If \( \mathcal{E}\left( X\right) > 0 \), then \( \mathcal{P}\left\{ {{S}_{n} \rightarrow + \infty }\right\} = 1 \) by the strong law of large numbers. Hence \( \mathcal{P}\left\{ {\mathop{\lim }\limits_{{n \rightarrow \infty }}{S}_{n} = - \infty }\right\} = 0 \), and this implies by the dual of Theorem 8.2.4 tha... | Yes |
Theorem 8.4.5. \( {S}_{n}/n \rightarrow m \) a.e. for a finite constant \( m \) if and only if for every \( \epsilon > 0 \) we have\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{1}{n}\mathcal{P}\left\{ {\left| {\frac{{S}_{n}}{n} - m}\right| > \epsilon }\right\} < \infty . \]\n | PROOF. Without loss of generality we may suppose \( m = 0 \) . We know from Theorem 5.4.2 that \( {S}_{n}/n \rightarrow 0 \) a.e. if and only if \( \mathcal{E}\left( \left| X\right| \right) < \infty \) and \( \mathcal{E}\left( X\right) = 0 \) . If this is so, consider the stationary independent process \( \left\{ {{X}_... | No |
Theorem 8.4.7. Suppose that \( X ≢ 0 \) and at least one of \( \mathcal{E}\left( {X}^{ + }\right) \) and \( \mathcal{E}\left( {X}^{ - }\right) \) is finite; and let \( \alpha = {\alpha }_{\left( 0,\infty \right) },\beta = {\alpha }_{( - \infty ,0\rbrack } \) . (i) If \( \mathcal{E}\left( X\right) > 0 \) but may be \( +... | PROOF. The assertion (i) is a consequence of (18) and Wald's equation (Theorem 5.5.3 and Exercise 8 of Sec. 5.5). The \ | No |
Theorem 8.5.1. We have for \( 0 < r < 1 \) :\n\n\[ \mathop{\sum }\limits_{{n = 0}}^{\infty }{r}^{n}\mathcal{E}\left\{ {e}^{{it}{M}_{n}}\right\} = \exp \left\{ {\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{r}^{n}}{n}\mathcal{E}\left( {e}^{{it}{S}_{n}^{ + }}\right) }\right\} . \] | PROOF. Observe the basic equation that follows at once from the meaning\n\n\[ \left\{ {{L}_{n} = k}\right\} = \left\{ {{L}_{k} = k}\right\} \cap \left\{ {{L}_{n - k} \circ {\tau }^{k} = 0}\right\} ,\]\n\nwhere \( {\tau }^{k} \) is the \( k \) th iterate of the shift. Since the two events on the right side\n\nof (10) ar... | Yes |
Theorem 8.5.3. We have for \( k \in {N}_{n}^{0} \) :\n\n(24)\n\n\[ \mathcal{P}\left\{ {{v}_{n} = k}\right\} = \mathcal{P}\left\{ {{v}_{k} = k}\right\} \mathcal{P}\left\{ {{v}_{n - k} = 0}\right\} .\n\]\n\nIf the common distribution of each \( {X}_{n} \) is symmetric with no atom at zero,\n\nthen\n\n(25)\n\n\[ \forall k... | PROOF. Let us denote the number on right side of (25), which is equal to\n\n\[ \frac{1}{{2}^{2n}}\left( \begin{matrix} {2k} \\ k \end{matrix}\right) \left( \begin{matrix} {2n} - {2k} \\ n - k \end{matrix}\right) ,\n\]\n\nby \( {a}_{n}\left( k\right) \) . Then for each \( n \in N,\left\{ {{a}_{n}\left( k\right), k \in {... | Yes |
Theorem 8.5.4. If the common distribution of a stationary independent process is symmetric, then we have\n\n\\[ \n\\forall x \\in \\left\\lbrack {0,1}\\right\\rbrack : \\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}\\mathcal{P}\\left\\{ {\\frac{{v}_{n}}{n} \\leq x}\\right\\} = \\frac{2}{\\pi }\\arcsin \\sqrt{x} = ... | This can be proved by the same method (invariance principle) as Theorem 7.3.3. | No |
Theorem 9.1.1. If \( \ell \left( \left| Y\right| \right) < \infty \) and \( \mathcal{G} \) is a Borel subfield of \( \mathcal{F} \), then there exists a unique equivalence class of integrable r.v.’s \( \mathcal{E}\left( {Y \mid \mathcal{G}}\right) \) belonging to \( \mathcal{G} \) such that (6) holds. | PROOF. Consider the set function \( v \) on \( \mathcal{G} \) :\n\n\[ \forall \Lambda \in \mathcal{G} : v\left( \Lambda \right) = {\int }_{\Lambda }{Yd}\mathcal{P} \]\n\nIt is finite-valued and countably additive, hence a \ | Yes |
Theorem 9.1.2. One version of the conditional expectation \( \mathcal{E}\left( {Y \mid X}\right) \) is given by \( \varphi \left( X\right) \), where \( \varphi \) is a Borel measurable function on \( {\mathcal{R}}^{1} \). Furthermore, if we define the signed measure \( \lambda \) on \( {\mathcal{B}}^{1} \) by\n\n\[ \fo... | PROOF. The first assertion of the theorem is a particular case of the following lemma.\n\nLemma. If \( Z \in \mathcal{F}\{ X\} \), then \( Z = \varphi \left( X\right) \) for some extended-valued Borel measurable function \( \varphi \) .\n\nPROOF OF THE LEMMA. It is sufficient to prove this for a bounded positive \( Z \... | Yes |
Theorem 9.1.3. Let \( Y \) and \( {YZ} \) be integrable r.v.’s and \( Z \in \mathcal{G} \) ; then we have\n\n\[ \mathcal{E}\left( {{YZ} \mid \mathcal{G}}\right) = Z\mathcal{E}\left( {Y \mid \mathcal{G}}\right) \text{ a.e. } \] | PROOF. As usual we may suppose \( Y \geq 0, Z \geq 0 \) (see property (ii) below). The proof consists in observing that the right member of (8) belongs to \( \mathcal{G} \) and satisfies the defining relation for the left member, namely:\n\n\[ \forall \Lambda \in \mathcal{G} : {\int }_{\Lambda }Z\mathcal{E}\left( {Y \m... | Yes |
Theorem 9.1.4. If \( \varphi \) is a convex function on \( {\mathcal{R}}^{1} \) and \( X \) and \( \varphi \left( X\right) \) are integrable r.v.’s, then for each \( \mathcal{G} \) :\n\n(11)\n\[ \varphi \left( {\mathcal{E}\left( {X \mid \mathcal{G}}\right) }\right) \leq \mathcal{E}\left( {\varphi \left( X\right) \mid \... | PROOF. If \( X \) is a simple r.v. taking the values \( \left\{ {y}_{j}\right\} \) on the sets \( \left\{ {\Lambda }_{j}\right\} ,1 \leq \nj \leq n \), which forms a partition of \( \Omega \), we have\n\n\[ \mathcal{E}\left( {X \mid \mathcal{G}}\right) = \mathop{\sum }\limits_{{j = 1}}^{n}{y}_{j}\mathcal{P}\left( {{\La... | Yes |
Theorem 9.1.5. If \( Y \) is integrable and \( {\mathcal{T}}_{1} \subset {\mathcal{T}}_{2} \), then\n\n(13) \( \;{c}_{{\mathcal{F}}_{1}}\left( Y\right) = {\mathcal{E}}_{{\mathcal{F}}_{2}}\left( Y\right) \; \) if and only if \( {\mathcal{E}}_{{\mathcal{F}}_{2}}\left( Y\right) \in {\mathcal{F}}_{1} \) ;\n\nand\n\n(14)\n\... | PROOF. Since \( Y \) satisfies trivially the defining relation for \( \mathcal{E}\left( {Y \mid {\mathcal{T}}_{1}}\right) \), it will be equal to the latter if and only if \( Y \in {\mathcal{T}}_{1} \) . Now if we replace our basic \( \mathcal{F} \) by \( {\mathcal{T}}_{2} \) and \( Y \) by \( {\mathcal{C}}_{{\mathcal{... | Yes |
Theorem 9.2.1. For each \( \alpha \in A \) let \( {\mathcal{F}}^{\left( \alpha \right) } \) denote the smallest B.F. containing all \( {\mathcal{F}}_{\beta },\beta \in A - \{ \alpha \} \) . Then the \( {\mathcal{F}}_{\alpha } \) ’s are conditionally independent relative to \( \mathcal{G} \) if and only if for each \( \... | PROOF. It is sufficient to prove this for two B.F.’s \( {\mathcal{F}}_{1} \) and \( {\mathcal{F}}_{2} \), since the general result follows by induction (how?). Suppose then that for each \( \Lambda \in {\mathcal{F}}_{1} \) we have\n\n(1)\n\[ \mathcal{P}\left( {\Lambda \mid {\mathcal{F}}_{2} \vee \mathcal{G}}\right) = \... | Yes |
Theorem 9.2.2. Let \( {X}_{1} \) and \( {X}_{2} \) be independent r.v.’s with p.m.’s \( {\mu }_{1} \) and \( {\mu }_{2} \) ; then for each \( B \in {\mathcal{B}}^{1} \) : (4) \( \mathcal{P}\left\{ {{X}_{1} + {X}_{2} \in B \mid {X}_{1}}\right\} = {\mu }_{2}\left( {B - {X}_{1}}\right) \;\text{ a.e. } \) | PROOF. To prove (4), since its right member belongs to \( \mathcal{F}\left\{ {X}_{1}\right\} \), it is sufficient to verify that it satisfies the defining relation for its left member. Let \( \Lambda \in \mathcal{F}\left\{ {X}_{1}\right\} \), then \( \Lambda = {X}_{1}^{-1}\left( A\right) \) for some \( A \in {\mathcal{... | Yes |
Theorem 9.2.3. Let \( \left\{ {{X}_{n}, n \geq 1}\right\} \) be an independent (but not necessarily stationary) process such that for \( A > 0 \) there exists \( \delta > 0 \) satisfying\n\n\[ \inf \mathcal{P}\left\{ {{X}_{n} \geq A}\right\} > \delta \text{.}\]\n\nThen we have\n\n\[ \forall n \geq 1 : \mathcal{P}\left\... | PROOF. We write \( {\Lambda }_{n} \) for the event that \( {S}_{j} \in (0, A\rbrack \) for \( 1 \leq j \leq n \) ; then\n\n\[ \mathcal{P}\left\{ {\Lambda }_{n}\right\} = \mathcal{P}\left\{ {{\Lambda }_{n - 1};0 < {S}_{n} \leq A}\right\} \]\n\nBy the definition of conditional probability and (5), the last-written probab... | No |
Theorem 9.3.1. Let \( \left\{ {{X}_{n},{\mathcal{F}}_{n}}\right\} \) be a submartingale and let \( \varphi \) be an increasing convex function defined on \( {\mathcal{R}}^{1} \) . If \( \varphi \left( {X}_{n}\right) \) is integrable for every \( n \), then \( \left\{ {\varphi \left( {X}_{n}\right) ,{\mathcal{T}}_{n}}\r... | PROOF. Since \( \varphi \) is increasing, and\n\n\[ \n{X}_{n} \leq \mathcal{E}\left\{ {{X}_{n + 1} \mid {\mathcal{F}}_{n}}\right\} \n\] \n\nwe have\n\n(5) \n\[ \n\varphi \left( {X}_{n}\right) \leq \varphi \left( {\mathcal{E}\left\{ {{X}_{n + 1} \mid {\mathcal{F}}_{n}}\right\} }\right) \n\] \n\nBy Jensen's inequality (S... | No |
If \( \left\{ {{X}_{n},{\mathcal{T}}_{n}}\right\} \) is a martingale, then \( \left\{ {\left| {X}_{n}\right| ,{\mathcal{T}}_{n}}\right\} \) is a submartingale; and \( \left\{ {{\left| {X}_{n}\right| }^{p},{\mathcal{F}}_{n}}\right\} ,1 < p < \infty \), is a submartingale provided that every \( {X}_{n} \in {L}^{p} \) ; .... | For a martingale we have equality in (5) for any convex \( \varphi \), hence we may take \( \varphi \left( x\right) = \left| x\right| ,{\left| x\right| }^{p} \) or \( \left| x\right| {\log }^{ + }\left| x\right| \) in the proof above. | Yes |
Corollary 3. If \( \left\{ {{X}_{n},{\mathcal{T}}_{n}}\right\} \) is a supermartingale, then so is \( \left\{ {{X}_{n} \land A,{\mathcal{T}}_{n}}\right\} \) where \( A \) is any constant. | PROOF. We leave it to the reader to deduce this from the theorem, but here is a quick direct proof:\n\n\[ \n{X}_{n} \land A \geq \mathcal{E}\left( {{X}_{n + 1} \mid {\mathcal{F}}_{n}}\right) \land \mathcal{E}\left( {A \mid {\mathcal{F}}_{n}}\right) \geq \mathcal{E}\left( {{X}_{n + 1} \land A \mid {\mathcal{F}}_{n}}\rig... | No |
Any submartingale \( \left\{ {{X}_{n},{\mathcal{T}}_{n}}\right\} \) can be written as\n\n\[ \n{X}_{n} = {Y}_{n} + {Z}_{n} \]\n\nwhere \( \left\{ {{Y}_{n},{\mathcal{T}}_{n}}\right\} \) is a martingale, and \( \left\{ {Z}_{n}\right\} \) is an increasing process. | From \( \left\{ {X}_{n}\right\} \) we define its difference sequence as follows:\n\n\[ \n{x}_{1} = {X}_{1},\;{x}_{n} = {X}_{n} - {X}_{n - 1},\;n \geq 2, \]\n\nso that \( {X}_{n} = \mathop{\sum }\limits_{{j = 1}}^{n}{x}_{j}, n \geq 1 \) (cf. the notation in the first paragraph of this section). The defining relation for... | Yes |
Theorem 9.3.3. For any optional \( \alpha \), we have\n\n(15)\n\n\[ \n{X}_{\alpha } = \mathcal{E}\left( {Y \mid {\mathcal{F}}_{\alpha }}\right) \]\n\nIf \( \alpha \leq \beta \) where \( \beta \) is also optional, then \( \left\{ {{X}_{\alpha },{\mathcal{F}}_{\alpha };{X}_{\beta },{\mathcal{F}}_{\beta }}\right\} \) form... | PROOF. Let us first show that \( {X}_{\alpha } \) is integrable. It follows from (13) and Jensen's inequality that\n\n\[ \n\left| {X}_{n}\right| \leq \mathcal{E}\left( {\left| Y\right| \mid {\mathcal{F}}_{n}}\right) \]\n\nSince \( \{ \alpha = n\} \in {\mathcal{T}}_{n} \), we may apply this to get\n\n\[ \n{\int }_{\Omeg... | Yes |
Theorem 9.3.4. Let \( \alpha \) and \( \beta \) be two bounded optional r.v.’s such that \( \alpha \leq \beta \) . Then for any [super]martingale \( \left\{ {X}_{n}\right\} ,\left\{ {{X}_{\alpha },{\mathcal{T}}_{\alpha };{X}_{\beta },{\mathcal{T}}_{\beta }}\right\} \) forms a [super]martingale. | PROOF. Let \( \Lambda \in {\mathcal{T}}_{\alpha } \) ; using (10) again we have for each \( k \geq j \) :\n\n\[{\Lambda }_{j} \cap \{ \beta > k\} \in {\mathcal{T}}_{k}\]\n\nbecause \( {\Lambda }_{j} \in {\mathcal{F}}_{j} \subset {\mathcal{F}}_{k} \), whereas \( \{ \beta > k\} = \{ \beta \leq k{\} }^{c} \in {\mathcal{F}... | Yes |
Theorem 9.3.5. Let \( \alpha \) and \( \beta \) be two arbitrary optional r.v.’s such that \( \alpha \leq \beta \) . Then the conclusion of Theorem 9.3.4 holds true for any supermartingale \( \left\{ {{X}_{n},{\mathcal{F}}_{n};n \in {N}_{ \propto }}\right\} \) | PROOF. (a) Suppose first that the supermartingale is positive with \( {X}_{\infty } = 0 \) a.e. The inequality (17) is true for every \( m \in N \), but now the second integral there is positive so that we have\n\n\[{\int }_{{\Lambda }_{j}}{X}_{\alpha }d\mathcal{P} \geq {\int }_{{\Lambda }_{j}\cap \{ \beta \leq m\} }{X... | Yes |
Theorem 9.4.1. If \( \left\{ {{X}_{j},{\mathcal{F}}_{j}, j \in {N}_{n}}\right\} \) is a submartingale, then for each real \( \lambda \) we have\n\n(1)\n\n\[ \lambda \mathcal{P}\left\{ {\mathop{\max }\limits_{{1 \leq j \leq n}}{X}_{j} \geq \lambda }\right\} \leq {\int }_{\left\{ \mathop{\max }\limits_{{1 \leq j \leq n}}... | PROOF. Let \( \alpha \) be the first \( j \) such that \( {X}_{j} \geq \lambda \) if there is such a \( j \) in \( {N}_{n} \) , otherwise let \( \alpha = n \) (optional stopping at \( n \) ). It is clear that \( \alpha \) is optional;\n\nsince it takes only a finite number of values, Theorem 9.3.4 shows that the pair \... | Yes |
If \( \left\{ {X}_{n}\right\} \) is a martingale, then for each \( \lambda > 0 \) : | \[ \mathcal{P}\left\{ {\mathop{\max }\limits_{{1 \leq j \leq n}}\left| {X}_{j}\right| \geq \lambda }\right\} \leq \frac{1}{\lambda }{\int }_{\left\{ \mathop{\max }\limits_{{1 \leq j \leq n}}\left| {X}_{j}\right| \geq \lambda \right\} }\left| {X}_{n}\right| d\mathcal{P} \leq \frac{1}{\lambda }\mathcal{E}\left( \left| {X... | Yes |
Corollary 2. Let \( 1 \leq m \leq n,{\Lambda }_{m} \in {\mathcal{T}}_{m} \) and \( \mathrm{M} = \left\{ {\mathop{\max }\limits_{{m \leq j \leq n}}{X}_{j} \geq \lambda }\right\} \), then \[ \lambda \mathcal{P}\left\{ {{\Lambda }_{m} \cap \mathrm{M}}\right\} \leq {\int }_{{\Lambda }_{m} \cap \mathrm{M}}{X}_{n}d\mathcal{P... | This is proved just as (1) and will be needed later. | No |
Theorem 9.4.3. Let \( \left\{ {{X}_{j},{\mathcal{F}}_{j};j \in {N}_{n}}\right\} \) be a supermartingale and let \( - \infty < \) \( {ab} < \infty \) . Let \( {\widetilde{v}}_{\left\lbrack a, b\right\rbrack }^{\left( n\right) } \) be the number of downcrossings of \( \left\lbrack {a, b}\right\rbrack \) by the sample seq... | PROOF. \( \left\{ {-{X}_{j}, j \in {N}_{n}}\right\} \) is a submartingale and \( {\widetilde{\nu }}_{\left\lbrack a, b\right\rbrack }^{\left( n\right) } \) is \( {\nu }_{\left\lbrack -b, - a\right\rbrack }^{\left( n\right) } \) for this submartingale. Hence the first part of (5) becomes\n\n\[ {\widetilde{c}}^{c}\left\{... | Yes |
Theorem 9.4.5. The three propositions below are equivalent for a sub-martingale \( \left\{ {{X}_{n},{\mathcal{F}}_{n};n \in N}\right\} \) :\n\n(a) it is a uniformly integrable sequence;\n\n(b) it converges in \( {L}^{1} \) ;\n\n(c) it converges a.e. to an integrable \( {X}_{\infty } \) such that \( \left\{ {{X}_{n},{\m... | PROOF. (a) \( \Rightarrow \) (b): under (a) the condition in Theorem 9.4.4 is satisfied so that \( {X}_{n} \rightarrow {X}_{\infty } \) a.e. This together with uniform integrability implies \( {X}_{n} \rightarrow {X}_{\infty } \) in \( {L}^{1} \) by Theorem 4.5.4 with \( r = 1 \) .\n\n(b) \( \Rightarrow \) (c): under (... | Yes |
Theorem 9.4.6. In the case of a martingale, propositions (a) and (b) above are equivalent to \( \left( {\mathrm{c}}^{\prime }\right) \) or \( \left( \mathrm{d}\right) \) below:\n\n\( \left( {\mathrm{c}}^{\prime }\right) \) it converges a.e. to an integrable \( {X}_{\infty } \) such that \( \left\{ {{X}_{n},{\mathcal{F}... | PROOF. (b) \( \Rightarrow \left( {\mathrm{c}}^{\prime }\right) \) as before; \( \left( {\mathrm{c}}^{\prime }\right) \Rightarrow \) (a) as before if we observe that \( \ell \left( {X}_{n}\right) = \ell \left( {X}_{\infty }\right) \) for every \( n \) in the present case, or more rapidly by considering \( \left| {X}_{n}... | Yes |
Theorem 9.4.7. Let \( \left\{ {{X}_{n}, n \in - N}\right\} \) be a submartingale. Then\n\n(11)\n\[ \mathop{\lim }\limits_{{n \rightarrow - \infty }}{X}_{n} = {X}_{-\infty },\;\text{ where }\; - \infty \leq {X}_{-\infty } < \infty \;\text{ a.e. } \] | PROOF. Let \( {v}_{\left\lbrack a, b\right\rbrack }^{\left( n\right) } \) be the number of upcrossings of \( \left\lbrack {a, b}\right\rbrack \) by the sequence\n\n\( \left\{ {X}_{-n,\ldots ,{X}_{-1}}\right\} \) . We have from Theorem 9.4.2:\n\n\[ \mathcal{E}\left\{ {v}_{\left\lbrack a, b\right\rbrack }^{\left( n\right... | Yes |
Theorem 9.4.8. Suppose that the \( {Y}_{n} \) ’s are dominated by an integrable r.v. \( Z \) :\n\n(13)\n\n\[ \mathop{\sup }\limits_{n}\left| {Y}_{n}\right| \leq Z \]\n\nand \( \mathop{\lim }\limits_{n}{Y}_{n} = {Y}_{\infty } \) or \( {Y}_{-\infty } \) as \( n \rightarrow \infty \) or \( - \infty \) . Then we have\n\n(1... | PROOF. We prove (15) first. Let \( {X}_{n} = \mathcal{E}\left\{ {Y \mid {\mathcal{T}}_{n}}\right\} \) . For \( n \in N,\left\{ {{X}_{n},{\mathcal{T}}_{n}}\right\} \n\nis a martingale already introduced in (13) of Sec. 9.3; the same is true for \( n \in - N \) . To prove (15a), we apply Theorem 9.4.6 to deduce \( \left(... | Yes |
Theorem 9.5.1. We have\n\n\[ \text{(1)}\;\mathop{\lim }\limits_{{n \rightarrow \infty }}\mathcal{P}\left\{ {{\Lambda }_{n + 1} \mid {\mathcal{T}}_{\left\lbrack 0, n\right\rbrack }}\right\} = {1}_{\mathrm{M}}\;\text{a.e.,} \]\n\nwhere \( {\mathcal{F}}_{\left\lbrack 0, n\right\rbrack } \) may be replaced by \( {\mathcal{... | PROOF. By Theorem 9.4.8, (14a), the limit is\n\n\[ \mathcal{P}\left\{ {M \mid {\mathcal{F}}_{\lbrack 0,\infty )}}\right\} = {1}_{M} \] | No |
Theorem 9.5.3. Suppose that the remote field of \( \\left\\{ {{X}_{n}, n \\in {N}^{0}}\\right\\} \) is trivial. Then each bounded harmonic function is a constant a.e. with respect to each \( {\\mu }_{n} \), where \( {\\mu }_{n} \) is the p.m. of \( {X}_{n} \) . | PROOF. By Theorem 9.4.5, \( f\\left( {X}_{n}\\right) \) converges a.e. to \( Z \) such that\n\n\[ \n\\left\\{ {f\\left( {X}_{n}\\right) ,{\\mathcal{F}}_{\\left\\lbrack 0, n\\right\\rbrack };Z,{\\mathcal{F}}_{\\[0,\\infty )}}\\right\\}\n\]\n\nis a martingale. Clearly \( Z \) belongs to the remote field and so is a const... | Yes |
Theorem 9.5.4. Let \( 1 < p < \infty \) and \( 1/p + 1/q = 1 \) . Suppose that \( \left\{ {{X}_{n}, n \in }\right. \) \( N\} \) is a positive submartingale satisfying the condition\n\n(5)\n\[ \mathop{\sup }\limits_{n}\mathcal{E}\left\{ {X}_{n}^{p}\right\} < \infty \]\n\nThen \( \mathop{\sup }\limits_{{n \in N}}{X}_{n} ... | PROOF. The condition (5) implies that \( \left\{ {X}_{n}\right\} \) is uniformly integrable (Exercise 8 of Sec. 4.5), hence by Theorem 9.4.5, \( {X}_{n} \rightarrow {X}_{\infty } \) a.e. and \( \left\{ {{X}_{n}, n \in }\right. \) \( \left. {N}_{\infty }\right\} \) is a submartingale. Writing \( Y \) for sup \( {X}_{n} ... | No |
Theorem 9.5.6. Let \( \\left\\{ {{S}_{n}, n \\in N}\\right\\} \) be a random walk (in the sense of Chapter 8) with \( \\mathcal{E}\\left\\{ \\left| {S}_{1}\\right| \\right\\} < \\infty \) . Then we have\n\n\[ \n\\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}\\frac{{S}_{n}}{n} = \\mathcal{E}\\left\\{ {S}_{1}\\right... | PROOF. Recall that \( {S}_{n} = \\mathop{\\sum }\\limits_{{j = 1}}^{n}{X}_{j} \) and consider for \( 1 \\leq k \\leq n \) :\n\n(9)\n\n\[ \n\\mathcal{E}\\left\\{ {{X}_{k} \\mid {S}_{n},{S}_{n + 1},\\ldots }\\right\\} = \\mathcal{E}\\left\\{ {{X}_{k} \\mid {\\mathcal{G}}_{n}}\\right\\} \n\]\n\nwhere \( {\\mathcal{G}}_{n}... | No |
Theorem 1. We have \( {\mu }^{ * } = \mu \) on \( {\mathcal{T}}_{0};{\mu }^{ * } \) on \( \mathcal{S} \) is an outer measure. | PROOF. Let \( A \in {\mathcal{T}}_{0} \), then the single set \( A \) serves as a covering of \( A \) ; hence\n\n\( {\mu }^{ * }\left( A\right) \leq \mu \left( A\right) \) . For any covering \( \left\{ {B}_{j}\right\} \) of \( A \), we have \( A{B}_{j} \in {\mathcal{T}}_{0} \) and\n\n\[ \mathop{\bigcup }\limits_{j}A{B}... | Yes |
Theorem 2. \( {\mathcal{F}}^{ * } \) is a Borel field and contains \( {\mathcal{T}}_{0} \) . On \( {\mathcal{F}}^{ * },{\mu }^{ * } \) is a measure. | PROOF. Let \( A \in {\mathcal{T}}_{0} \) . For any \( Z \subset \Omega \) and any \( \epsilon > 0 \), there exists a covering\n\n\( \left\{ {B}_{j}\right\} \) of \( Z \) such that\n\n(4)\n\[ \mathop{\sum }\limits_{j}\mu \left( {B}_{j}\right) \leq {\mu }^{ * }\left( Z\right) + \epsilon \]\n\nSince \( A{B}_{j} \in {\math... | Yes |
Theorem 3. Let \( {\mathcal{T}}_{0} \) be a field and \( \mathcal{F} \) the Borel field generated by \( {\mathcal{T}}_{0} \). Let \( {\mu }_{1} \) and \( {\mu }_{2} \) be two measures on \( \mathcal{F} \) that agree on \( {\mathcal{T}}_{0} \). If one of them, hence both are \( \sigma \) -finite on \( {\mathcal{T}}_{0} ... | PROOF. Let \( \left\{ {\Omega }_{n}\right\} \) be as in Definition 5. Define a class &of subsets of \( \Omega \) as follows:\n\n\[ \mathcal{C} = \left\{ {A \subset \Omega : {\mu }_{1}\left( {{\Omega }_{n}A}\right) = {\mu }_{2}\left( {{\Omega }_{n}A}\right) \text{ for all }n \in N}\right\} .\n\]\n\nSince \( {\Omega }_{n... | Yes |
Theorem 4. Let \( A \in {\mathcal{F}}^{ * } \) . There exists \( B \in {\mathcal{T}}_{0\sigma \delta } \) such that\n\n\[ A \subset B;\;{\mu }^{ * }\left( A\right) = {\mu }^{ * }\left( B\right) . \] | PROOF. For each \( m \), there exists \( \left\{ {B}_{mn}\right\} \) in \( \mathcal{F} \) such that\n\n\[ A \subset \mathop{\bigcup }\limits_{n}{B}_{mn};\;\mathop{\sum }\limits_{n}{\mu }^{ * }\left( {B}_{mn}\right) \leq {\mu }^{ * }\left( A\right) + \frac{1}{m}. \]\n\nPut\n\n\[ {B}_{m} = \mathop{\bigcup }\limits_{n}{B}... | Yes |
Theorem 6. The measure space \( \left( {\Omega ,{\mathcal{F}}^{ * },{\mu }^{ * }}\right) \) is complete. Let \( \left( {\Omega ,\mathcal{G}, v}\right) \) be a complete measure space; \( \mathcal{G} \supset {\mathcal{T}}_{0} \) and \( v = \mu \) on \( {\mathcal{T}}_{0} \) . If \( \mu \) is \( \sigma \) -finite on \( {\m... | PROOF. Let \( A \in {\mathcal{F}}^{ * } \), then by Theorem 4 there exists \( B \in \mathcal{F} \) and \( C \in \mathcal{F} \) such that \[ C \subset A \subset B;\;\mu \left( C\right) = {\mu }^{ * }\left( A\right) = \mu \left( B\right) . \] Since \( v = \mu \) on \( {\mathcal{T}}_{0} \), we have by Theorem \( 3, v = \m... | Yes |
Theorem 7. \( \overline{\mathcal{F}} \) is a Borel field and \( \bar{\mu } \) is a measure on \( \overline{\mathcal{F}} \) . | PROOF. Let \( {A}_{n} \in \overline{\mathcal{F}}, n \in N \) ; so that \( {A}_{n} = {B}_{n}{C}_{n}^{c} \) as in (9). We have then\n\n\[ \mathop{\bigcap }\limits_{{n = 1}}^{\infty }{A}_{n} = \left( {\mathop{\bigcap }\limits_{{n = 1}}^{\infty }{B}_{n}}\right) \cap {\left( \mathop{\bigcup }\limits_{{n = 1}}^{\infty }{C}_{... | Yes |
Theorem 8. Let \( \left\{ {f}_{n}\right\} \) and \( \left\{ {g}_{n}\right\} \) be two increasing sequences of basic functions such that\n\n(36)\n\n\[ \mathop{\lim }\limits_{n} \uparrow {f}_{n} = \mathop{\lim }\limits_{n} \uparrow {g}_{n} \]\n\n(everywhere in \( \Omega \) ). Then we have\n\n(37)\n\n\[ \mathop{\lim }\lim... | PROOF. Denote the common limit function in (36) by \( f \) and put\n\n\[ A = \{ \omega \in \Omega : f\left( \omega \right) > 0\} \]\n\nthen \( A \in \mathcal{F} \) . Since \( 0 \leq {g}_{n} \leq f \), we have \( {1}_{{A}^{c}}{g}_{n} = 0 \) identically; hence by prop-\n\nerty (iii):\n\n(38)\n\n\[ E\left( {g}_{n}\right) ... | Yes |
Theorem 9. Let \( \\left\\{ {f}_{n}\\right\\} \) be an increasing sequence of functions in \( {\\mathcal{F}}_{ + } \) with limit \( f : {f}_{n} \\uparrow f \) . Then we have\n\n\[ \n\\mathop{\\lim }\\limits_{n} \\uparrow E\\left( {f}_{n}\\right) = E\\left( f\\right) \\leq + \\infty .\n\] | PROOF. We have \( f \\in {\\mathcal{F}}_{ + } \) ; hence by Definition 8(b),(41) holds. For each\n\n\( {f}_{n} \), we have, using analogous notation:\n\n(42)\n\n\[ \n\\mathop{\\lim }\\limits_{m} \\uparrow E\\left( {f}_{n}^{\\left( m\\right) }\\right) = E\\left( {f}_{n}\\right)\n\]\n\nSince \( \\left. {\\left. {{{f}_{n}... | Yes |
Theorem 10. Let \( {f}_{n} \in {\mathcal{F}}_{ + }, n \in N \) . Suppose\n\n(a) \( \mathop{\lim }\limits_{n}{f}_{n} = 0 \) ;\n\n(b) \( E\left( {\mathop{\sup }\limits_{n}{f}_{n}}\right) < \infty \) .\n\nThen we have\n\n(44)\n\n\[ \mathop{\lim }\limits_{n}E\left( {f}_{n}\right) = 0 \] | PROOF. Put for \( n \in N \) :\n\n(45) \[ {g}_{n} = \mathop{\sup }\limits_{{k \geq n}}{f}_{k} \]\n\nThen \( {g}_{n} \in {\mathcal{F}}_{ + } \), and as \( n \uparrow \infty ,{g}_{n} \downarrow \lim \mathop{\sup }\limits_{n}{f}_{n} = 0 \) by (a); and \( {g}_{1} = \mathop{\sup }\limits_{n}{f}_{n} \n\nso that \( E\left( {g... | Yes |
Theorem 11. Let \( \left\{ {f}_{n}\right\} \) be an arbitrary sequence of functions in \( {f}_{ + } \) . Then we have\n\n\[ E\left( {\mathop{\liminf }\limits_{n}{f}_{n}}\right) \leq \mathop{\liminf }\limits_{n}E\left( {f}_{n}\right) \] | PROOF. Put for \( n \in N \) :\n\n\[ {g}_{n} = \mathop{\inf }\limits_{{k \geq n}}{f}_{k} \]\n\nthen\n\n\[ \mathop{\liminf }\limits_{n}{f}_{n} = \mathop{\lim }\limits_{n} \uparrow {g}_{n} \]\n\nHence by Theorem 9,\n\n\[ E\left( {\mathop{\liminf }\limits_{n}{f}_{n}}\right) = \mathop{\lim }\limits_{n} \uparrow E\left( {g}... | Yes |
Theorem 12. (i) The function \( f \) in \( \mathcal{F} \) is integrable if and only if \( \left| f\right| \) is integrable; we have\n\n\[ \left| {E\left( f\right) }\right| \leq E\left( \left| f\right| \right) \] | PROOF. (i) is trivial from (48) and (49); | No |
Question: does the numerical series above converge? and if so is the sum of integrals equal to the integral of the sum:\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{\infty }{\int }_{I}{u}_{k}\left( x\right) {dx} = {\int }_{I}\mathop{\sum }\limits_{{k = 1}}^{\infty }{u}_{k}\left( x\right) {dx} = {\int }_{I}s\left( x\right) {d... | A very special but important case is when the interval \( I = \left\lbrack {a, b}\right\rbrack \) is compact and the functions \( {u}_{k} \) are all continuous in \( I \) . If we assume that the series \( \mathop{\sum }\limits_{{k = 1}}^{\infty }{u}_{k}\left( x\right) \) converges uniformly in \( I \), then it follows ... | Yes |
Let \( {u}_{k} \geq 0,{u}_{k} \in {L}^{1} \), then\n\n\[ \n{\int }_{a}^{b}\left( {\mathop{\sum }\limits_{{k = 1}}^{\infty }{u}_{k}}\right) {d\mu } = \mathop{\sum }\limits_{{k = 1}}^{\infty }{\int }_{a}^{b}{u}_{k}{d\mu }.\n\] | Let \( {f}_{n} = \mathop{\sum }\limits_{{k = 1}}^{n}{u}_{k} \), then \( {f}_{n} \in {L}^{1},{f}_{n} \uparrow f = \mathop{\sum }\limits_{{k = 1}}^{\infty }{u}_{k} \) . Hence by monotone convergence\n\n\[ \nE\left( f\right) = \mathop{\lim }\limits_{n}E\left( {f}_{n}\right)\n\]\n\nthat is (54). | Yes |
Let \( \left( {I,{\mathcal{B}}^{ * }, m}\right) \) be as in the preceding example, but let \( I = \left\lbrack {a, b}\right\rbrack \) be compact. Let \( f \) be a continuous function on \( I \) . Denote by \( P \) a partition of \( I \) as follows:\n\n\[ a = {x}_{0} < {x}_{1} < {x}_{2} < \cdots < {x}_{n} = b; \]\n\nand... | Now let \( \{ P\left( n\right), n \in N\} \) be a sequence of partitions such that \( \delta \left( {P\left( n\right) }\right) \rightarrow 0 \) as \( n \rightarrow \infty \) . Since \( f \) is continuous on a compact set, it is bounded. It follows that there is a constant \( C \) such that\n\n\[ \mathop{\sup }\limits_{... | Yes |
The square of the function \( f \) in (56):\n\n\[ f{\left( x\right) }^{2} = {\left( \frac{\sin x}{x}\right) }^{2},\;x \in \mathbf{R} \] | is integrable in the Lebesgue sense, and is also improperly integrable in the Riemann sense.\n\nWe have\n\n\[ f{\left( x\right) }^{2} \leq {1}_{\left( -1, + 1\right) } + {1}_{\left( {-\infty , - 1}\right) \cup \left( {+1, + \infty }\right) }\frac{1}{{x}^{2}} \]\n\nand the function on the right side is integrable, hence... | Yes |
For \( z \in \mathbb{C} \), the correlation length satisfies the following equality | \[ \mathop{\lim }\limits_{{\beta \nearrow {\beta }_{c}}}\frac{{\tau }_{\beta }\left( z\right) }{\left( {\beta }_{c} - \beta \right) } = 4\left| z\right| \] | Yes |
Proposition 2.5. Let \( G \) be a finite graph and \( a, b \) be two sites of \( G \) . At inverse-temperature \( \beta > 0 \) , \[ {Z}_{\beta, G}^{f} = {2}^{\# \text{ vertices }G}\cosh {\left( \beta \right) }^{\# \text{ edges in }G}\mathop{\sum }\limits_{{\omega \in {\mathcal{E}}_{G}}}\tanh {\left( \beta \right) }^{\l... | Proof Let us start with the partition function (2.8). Let \( E \) be the set of edges of \( G \) . We know \[ {Z}_{\beta, G}^{f} = \mathop{\sum }\limits_{\sigma }\mathop{\prod }\limits_{{\left\lbrack {xy}\right\rbrack \in E}}{\mathrm{e}}^{\beta {\sigma }_{x}{\sigma }_{y}} \] \[ = \cosh {\left( \beta \right) }^{\# \text... | Yes |
Proposition 2.6 (Kramers-Wannier duality). Let \( \beta > 0 \) and define \( {\beta }^{ \star } \in \left( {0,\infty }\right) \) such that \( \tanh \left( {\beta }^{ \star }\right) = {\mathrm{e}}^{-{2\beta }} \), then for every graph \( G \) , \[ {2}^{\# \text{ vertices }{G}^{ \star }}\cosh \left( {\beta }^{ \star }\ri... | Proof When writing the contour of connected components for the Ising model with + boundary conditions, the only edges of \( {\mathbb{L}}^{ \star } \) used are those of \( {G}^{ \star } \) . Indeed, edges between boundary sites cannot be present since boundary spins are + . Thus, the right and left-hand side terms of (2... | Yes |
Theorem 2.10. Let \( \left( {\Omega, a, b}\right) \) be a simply connected domain with two marked points on the boundary. Let \( {\gamma }_{\delta } \) be the interface of the critical Ising model with Dobrushin boundary conditions on the spin Dobrushin domain \( \left( {{\Omega }_{\delta }^{ \circ },{a}_{\delta },{b}_... | The proof of Theorem 2.10 follows the program below, see Section 6\n\n- Prove that the family of interfaces \( {\left( {\gamma }_{\delta }\right) }_{\delta > 0} \) is tight.\n\n- Prove that \( {M}_{t}^{{z}_{\delta }} = {F}_{{\Omega }_{\delta }^{\diamond } \smallsetminus {\gamma }_{\delta }\left\lbrack {0, t}\right\rbra... | No |
Theorem 3.2. For any \( q \geq 1 \), there exists \( {p}_{c}\left( q\right) \in \left( {0,1}\right) \) such that for any infinite volume measure \( {\phi }_{p, q} \) , - if \( p < {p}_{c}\left( q\right) \), there is almost surely no infinite cluster under \( {\phi }_{p, q} \) , - if \( p > {p}_{c}\left( q\right) \), th... | Note that \( q = 1 \) is simply bond percolation. In this case, the existence of a phase transition is a well-known fact. The existence of a critical point in the general case \( q \geq 1 \) is not much harder to prove: a coupling between two measures \( {\phi }_{{p}_{1}, q, G} \) and \( {\phi }_{{p}_{2}, q, G} \) can ... | No |
The dual model of the FK percolation with parameters \( \left( {p, q}\right) \) with wired boundary conditions is the FK percolation with parameters \( \left( {{p}^{ \star }, q}\right) \) and free boundary conditions on \( {G}^{ \star } \), where\n\n\[ \n{p}^{ \star } = {p}^{ \star }\left( {p, q}\right) \mathrel{\text{... | Proof Note that the state of edges between two sites of \( \partial G \) is not relevant when boundary conditions are wired. Indeed, sites on the boundary are connected via boundary conditions anyway, so that the state of each boundary edge does not alter the connectivity properties of the subgraph, and is independent ... | Yes |
Proposition 3.8. Let \( p \in \left( {0,1}\right) \) and \( G \) a finite graph. If the configuration \( \omega \) is distributed according to a \( {FK} \) measure with parameters \( \left( {p,2}\right) \) and free boundary conditions, then the spin configuration \( \sigma \) is distributed according to an Ising measur... | Proof Consider a finite graph \( G \), let \( p \in \left( {0,1}\right) \) . Consider a measure \( P \) on pairs \( \left( {\omega ,\sigma }\right) \) , where \( \omega \) is a FK configuration with free boundary conditions and \( \sigma \) is the corresponding random spin configuration, constructed as explained above.... | Yes |
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