Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Proposition 21.1.2. Assume that\n\n(1) \( \operatorname{char}\left( F\right) \) does not divide \( n \), and\n\n(2) \( F \) contains all \( n \) th roots of unity.\n\nThen \( K = F\left( \sqrt[n]{a}\right) \) is a cyclic extension of degree dividing \( n \) . | Proof. We may factor\n\n\[ {x}^{n} - a = \left( {x - \sqrt[n]{a}}\right) \left( {x - {\zeta }_{n}\sqrt[n]{a}}\right) \cdots \left( {x - {\zeta }_{n}^{n - 1}\sqrt[n]{a}}\right) \]\n\nSo \( K \) is the splitting field of \( {x}^{n} - a \) over \( F \) . As \( \left( {{x}^{n} - a, D\left( {{x}^{n} - a}\right) }\right) = \... | Yes |
Proposition 21.1.3 (Kummer). If \( F \) is a field such that \( \operatorname{char}\left( F\right) \nmid n \) and \( F \) contains all \( n \) th roots of unity. Then any cyclic field extension \( K \) of \( F \) is of the form \( K = F\left( \sqrt[n]{a}\right) \) for some \( a \in {F}^{ \times } \) . | Proof. Write \( \operatorname{Gal}\left( {K/F}\right) \cong {\mathbf{Z}}_{n} = \langle \sigma \rangle \) . For each \( \alpha \in K \), we define\n\n(21.1.3.1)\n\n\[ b \mathrel{\text{:=}} \alpha + {\zeta }_{n}\sigma \left( \alpha \right) + \cdots + {\zeta }_{n}^{n - 1}{\sigma }^{n - 1}\left( \alpha \right) . \]\n\nBy l... | Yes |
Proposition 21.2.3. An element \( \alpha \) can be expressed by radicals over a field \( F \) if \( \alpha \) is contained in a Galois extension \( K \) of \( F \) which admits a tower of subfields of the form (21.2.1.1). | Proof. By definition of expressing elements by radicals, there exists a finite extension \( {K}^{\prime } \) of \( F \) which admits a tower of subfields of the form (21.2.1.1). Let \( K \) be the Galois closure of \( {K}^{\prime } \) over \( F \) . This implies that for each \( \sigma \in {\operatorname{Hom}}_{F}\left... | No |
An (irreducible) polynomial \( f\left( x\right) \) can be solved by radicals if and only if its Galois group (meaning the Galois group of its splitting field) is a solvable group. | Proof. \ | No |
The Galois group for \( {x}^{7} - 5 \) over \( \mathbb{Q} \) (irreducible by Eisenstein criterion). | The splitting field is \( \mathbb{Q}\left( {\sqrt[7]{5},{\zeta }_{7}}\right) \). The associated Galois group is \( {\mathbf{Z}}_{7} \rtimes {\mathbf{Z}}_{7}^{ \times } \). | No |
Lemma 21.3.4. The group \( G \) acts transitively on the set \( \left\{ {{\alpha }_{1},\ldots ,{\alpha }_{n}}\right\} \) . | Proof. Suppose now and suppose that \( \left\{ {{\alpha }_{1},\ldots ,{\alpha }_{r}}\right\} \) (with \( r < n \) ) is an orbit under \( G \), then \( \left( {x - {\alpha }_{1}}\right) \cdots \left( {x - {\alpha }_{r}}\right) \in F\left\lbrack x\right\rbrack \) is a factor of \( f\left( x\right) \) . Yet \( f\left( x\r... | No |
Proposition 21.3.6. The field \( M = F\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) is a Galois extension over \( L = F\left( {{s}_{1},\ldots ,{s}_{n}}\right) \) with Galois group \( {S}_{n} \) . | Proof. Consider the polynomial\n\n\[ f\left( x\right) = \left( {x - {x}_{1}}\right) \cdots \left( {x - {x}_{n}}\right) = {x}^{n} - {s}_{1}{x}^{n - 1} + \cdots + {\left( -1\right) }^{n}{s}_{n} \in L\left\lbrack x\right\rbrack .\n\]\n\nThis \( M \) is the splitting field of \( f\left( x\right) \) over \( L \) . In partic... | Yes |
Lemma 21.3.9. We have \( G = \operatorname{Gal}\left( {K/F}\right) \subseteq {A}_{n} \) if and only if \( D \) is a square in \( F \) . More precisely, we have the following diagram of Galois extensions: | Proof. The second statement implies the first one because its shows that \( F\left( \sqrt{D}\right) = F \) if and only if \( G \subseteq {A}_{n} \) . Indeed, \( \delta \mathrel{\text{:=}} \mathop{\prod }\limits_{{1 \leq i < j \leq n}}\left( {{\alpha }_{i} - {\alpha }_{j}}\right) \in K \) is a square root of \( D \), an... | Yes |
Take one example: for \( p = 7 \), we show that 2 is invertible in \( {\mathbb{Z}}_{7} \) as follows: | \[ 2{x}_{1} \equiv 1{\;\operatorname{mod}\;7} \Rightarrow {x}_{1} \equiv 1{\;\operatorname{mod}\;7} \] \[ 2{x}_{2} \equiv 1{\;\operatorname{mod}\;{49}} \Rightarrow {x}_{2} \equiv {25}{\;\operatorname{mod}\;{49}}\left( { \equiv 4{\;\operatorname{mod}\;7}}\right) \] \[ \ldots \] We can always solve \( 2{x}_{i} \equiv 1{\... | No |
Why did we call this a limit? We can see this as\n\n\[ \mathbb{C}\llbracket x\rrbracket \mathrel{\text{:=}} \mathop{\lim }\limits_{n}\mathbb{C}\left\lbrack x\right\rbrack /\left( {x}^{n}\right) \] | Given a complex function \( f \) on \( \mathbb{C} \), holomorphic at 0, then the Taylor expansion at 0 :\n\n\[ f\left( 0\right) + x{f}^{\prime }\left( 0\right) + \frac{{f}^{\prime \prime }\left( 0\right) }{2}{x}^{2} + \cdots + \frac{{f}^{\left( n\right) }\left( 0\right) }{n!}{x}^{n} + \cdots \]\n\ndefines an element in... | No |
We define \( \widehat{\mathbb{Z}} \mathrel{\text{:=}} \mathop{\lim }\limits_{{ \leftarrow n}}\mathbb{Z}/n\mathbb{Z} \) by divisibility (i.e. for \( m \mid n,\mathbb{Z}/n\mathbb{Z} \rightarrow \mathbb{Z}/m\mathbb{Z} \) ).\n\nIt is fact that \( \widehat{\mathbb{Z}} \cong \mathop{\prod }\limits_{{p\text{ prime }}}{\mathbb... | We give the following proof: for each prime \( p \), we have\n\n\[ \n{\varphi }_{p} : \widehat{\mathbb{Z}} \rightarrow {\mathbb{Z}}_{p} \n\] \n\n\[ \n{\left( {a}_{n}\right) }_{n} \mapsto {\left( {a}_{{p}^{r}}\right) }_{r} \n\] \n\nThis together give a map \n\n\[ \n\varphi = \mathop{\prod }\limits_{p}{\varphi }_{p} : \w... | Yes |
Lemma 22.3.2. The two above definitions of topology on \( \mathop{\lim }\limits_{{i \rightarrow i \in I}}{A}_{i} \) are equivalent. | Proof. Clearly, the open subset in (1) is clearly open in the sense of (2). Conversely, we note that an open subset as defined in (2) takes the form of \( {\pi }_{{i}_{1}}^{-1}\left( {a}_{{i}_{1}}\right) \cap \cdots {\pi }_{{i}_{n}}^{-1}\left( {a}_{{i}_{n}}\right) \), where \( {i}_{1},\ldots ,{i}_{n} \in I \) and \( {a... | Yes |
Theorem 22.3.3. If each \( {A}_{i} \) is finite, then \( \mathop{\lim }\limits_{{i \in I}}{A}_{i} \) is compact and Hausdorff. In this case, we say that \( \mathop{\lim }\limits_{{i \in I}}{A}_{i} \) profinite. | Proof. We consider the subspace \n\nThe latter space is compact and Hausdorff (as the product of compact spaces is compact). The subspace is determined by taking the conditions \( {\varphi }_{ji}\left( {a}_{j}\right)... | Yes |
Lemma 22.3.6. If \( H \leq G \) is an open subset of a topological group \( G \), then \( H \) is also closed! | Proof. Note that we have a disjoint union\n\n\[ G = \mathop{\coprod }\limits_{{{gH} \in G/H}}{gH} \]\n\nof subsets. But each \( {gH} \) is open. It implies that\n\n\[ H = G \smallsetminus \left( {\mathop{\coprod }\limits_{{{gH} \neq H}}{gH}}\right) \]\n\nis closed. | Yes |
Lemma 22.3.7. If \( G \) is a compact topological group, then a subgroup \( H \leq G \) is open if and only if it is closed and of finite index in \( G \) . | Proof. \ | No |
Lemma 22.3.9. For a profinite group \( G \), we have\n\n\[ G \mathrel{\text{:=}} \mathop{\lim }\limits_{\substack{ \leftarrow \\ {H \leq G} }}\mathop{\lim }\limits_{\text{open normal }}G/H. \] | Proof. There is an obvious map \( G \rightarrow \mathop{\lim }\limits_{{H \leq G\text{ open normal }}}G/H = : {G}^{\prime } \) .\n\nBy definition, \( G = \mathop{\lim }\limits_{{i \leftarrow i \in I}}{G}_{i} \) . We want to construct the reserve arrow:\n\n\[ {G}^{\prime } = \mathop{\lim }\limits_{\substack{ \leftarrow ... | Yes |
Example 22.4.2. (1) Write \( \mathbb{Q}\left( {\mu }_{{p}^{\infty }}\right) \mathrel{\text{:=}} \mathbb{Q}\left( {{\zeta }_{{p}^{n}};n \in \mathbb{N}}\right) \) . We have | \[ \operatorname{Gal}\left( {\mathbb{Q}\left( {\mu }_{{p}^{\infty }}\right) /\mathbb{Q}}\right) = \mathop{\lim }\limits_{n}\operatorname{Gal}\left( {\mathbb{Q}\left( {\zeta }_{{p}^{n}}\right) /\mathbb{Q}}\right) \cong \underset{n}{\underbrace{\lim }}{\left( \mathbb{Z}/{p}^{n}\mathbb{Z}\right) }^{ \times } = {\mathbb{Z}... | Yes |
Proposition 22.5.1. If \( G \) is a profinite group, then any continuous representation \( \rho : G \rightarrow \) \( {\mathrm{{GL}}}_{n}\left( \mathbb{C}\right) \) has finite image. | Proof. Take a very small open neighborhood \( U \) of \( {I}_{n} \in {\mathrm{{GL}}}_{n}\left( \mathbb{C}\right) \) . The preimage \( {\rho }^{-1}\left( U\right) \) is an open subset of \( G \) containing \( {I}_{n} \) . This implies that \( {\rho }^{-1}\left( U\right) \) contains an open subgroup \( H \) of \( G \) .\... | Yes |
Theorem 6.1. Assume an irreducible Markov process with matrix \( Q \) has a stationary distribution \( \pi .\pi \) is detailed balanced if and only if for every sequence of distinct states \( {i}_{0},{i}_{1},\cdots ,{i}_{n - 1},{i}_{n} \in \mathcal{S} \) :\n\n\[ \n{q}_{{i}_{0}{i}_{1}}{q}_{{i}_{1}{i}_{2}}\cdots {q}_{{i}... | Proof. Necessity: From detailed balance, we have \n\n\[ \n1 = \left( {\mathop{\prod }\limits_{{k = 0}}^{{n - 1}}\frac{{\pi }_{{i}_{k}}{q}_{{i}_{k}{i}_{k + 1}}}{{\pi }_{{i}_{k + 1}}{q}_{{i}_{k + 1}{i}_{k}}}}\right) \frac{{\pi }_{{i}_{n}}{q}_{{i}_{n}{i}_{0}}}{{\pi }_{{i}_{0}}{q}_{{i}_{0}{i}_{n}}} = \left( {\mathop{\prod ... | Yes |
Theorem 6.2. The following six statements regarding an irreducible stationary Markov process with matrix \( Q \) and stationary distribution \( \pi \) are equivalent [137].\n\n(i) Its stationary distribution satisfies detailed balance: \( {\pi }_{i}{q}_{ij} = {\pi }_{j}{q}_{ji},\forall i, j \in \mathcal{S} \).\n\n(ii) ... | Proof. \( \left( i\right) \Rightarrow \left( {ii}\right) \) :\n\nUsing \( \left( i\right) \), we have\n\n\[ \ln \left( \frac{{q}_{{i}_{0}{i}_{1}}}{{q}_{{i}_{1}{i}_{0}}}\right) + \ln \left( \frac{{q}_{{i}_{1}{i}_{2}}}{{q}_{{i}_{2}{i}_{1}}}\right) + \cdots + \ln \left( \frac{{q}_{{i}_{n - 1}{i}_{n}}}{{q}_{{i}_{n}{i}_{n -... | Yes |
Lemma 2 For each \( \varepsilon \), assume\n\n\[ \n{k}^{\varepsilon }\left( {V,\mathbf{x}}\right) = {\pi }_{V}\left( {{B}_{\varepsilon }\left( \mathbf{x}\right) }\right) \exp \left\{ {V\mathop{\inf }\limits_{{\mathbf{y} \in {B}_{\varepsilon }\left( \mathbf{x}\right) }}{\varphi }^{ss}\left( \mathbf{y}\right) }\right\} \... | Proof. According to these assumptions, we know that for each \( \mathbf{x} \), given any \( \delta > 0 \) , there exists a constant \( {V}_{0} \) and \( {\varepsilon }_{0} \), such that when \( V > {V}_{0} \) and \( \varepsilon < {\varepsilon }_{0} \), \n\n\[ \n\left| {\frac{{k}^{\varepsilon }\left( {V,\mathbf{x}}\righ... | Yes |
Under the assumptions in Theorem 7.2 and Lemma 2 and further assuming that the limit\n\n\[ \mathop{\lim }\limits_{{V \rightarrow \infty }}\frac{{r}_{\pm \ell }\left( {V\mathbf{x};V}\right) }{V} = {R}_{\pm \ell }\left( \mathbf{x}\right) \]\n\nis locally uniform for any sufficiently small neighborhood of each \( \mathbf{... | Given \( V \), denote \( \mathbf{n}\left( {\mathbf{x}, V}\right) \) as the nearest integer vector to point \( \mathbf{x}V \) . Then, the CME (7.5) at steady state can be rewritten as\n\n\[ \mathop{\sum }\limits_{{\ell = 1}}^{M}\frac{{p}_{V}^{ss}\left( {\mathbf{n}\left( {\mathbf{x}, V}\right) - {v}_{\ell }}\right) }{{p}... | Yes |
Theorem 7.5. Assume Eqns. (7.38), (7.43), (7.44), and (7.47) as well as the assumptions of Lemma 2 hold, and assume that the limit\n\n\[ \mathop{\lim }\limits_{{V \rightarrow \infty }}\frac{{r}_{\pm \ell }\left( {V\mathbf{x};V}\right) }{V} = {R}_{\pm \ell }\left( \mathbf{x}\right) \]\n\nis locally uniform for any suffi... | Proof. According to the strong law of large numbers (Eqn. (7.38)), we know that given any small \( \varepsilon > 0 \), for sufficiently large \( V \), the probability is concentrated on the integers satisfying \( \frac{\mathbf{n}}{V} \in {B}_{\varepsilon }\left( {\mathbf{x}\left( t\right) }\right) \).\n\nGiven any smal... | Yes |
Proposition 3 The three terms \( {\sigma }^{\text{tot }}\left\lbrack \mathbf{x}\right\rbrack ,{f}_{d}^{\text{macro }}\left\lbrack \mathbf{x}\right\rbrack \) and \( {E}_{\text{in }}^{\text{macro }}\left\lbrack \mathbf{x}\right\rbrack \) are all nonnegative. Furthermore,\n\n(a) \( {\sigma }^{\text{tot }}\left\lbrack \mat... | Proof. (a) is straightforward.\n\n(b) Note that \( \forall x \geq 0,\ln x \geq 1 - \frac{1}{x} \) and \( \ln x \leq x - 1 \) . Thus, we have\n\n\[ \n\ln \left( {\frac{{R}_{-\ell }\left( \mathbf{x}\right) }{{R}_{+\ell }\left( \mathbf{x}\right) }{\mathrm{e}}^{-{\mathbf{V}}_{\ell } \cdot {\nabla }_{\mathbf{x}}{\varphi }^{... | Yes |
Theorem 7.6. \( \mathbf{q} \) is any stable fixed point of Eqn. (7.2). Assume \( {\varphi }^{ss}\left( \mathbf{x}\right) \) is at least twice differentiable. Define\n\n\[ \n{\Xi }_{ij} = \frac{{\partial }^{2}{\varphi }^{ss}\left( \mathbf{q}\right) }{\partial {\mathbf{x}}_{i}\partial {\mathbf{x}}_{j}},{A}_{ij} = \mathop... | Proof. Taking the second derivative of the left-hand side of Eqn. (7.47) and noticing that \( {v}_{\ell } \cdot {\nabla }_{\mathbf{x}}{\varphi }^{ss}\left( \mathbf{q}\right) = {B}_{i}\left( \mathbf{q}\right) = 0 \) for each \( \ell \) and \( i \), we can obtain\n\n\[ \n\mathop{\sum }\limits_{{\ell = 1}}^{M}\left\{ {\le... | Yes |
Proposition 3 (Abel’s formula). The fundmental matrix (2.10) satisfies\n\n(2.12)\n\n\[ \frac{d}{dt}\left| {\Psi \left( t\right) }\right| = \operatorname{trace}\left( {A\left( t\right) }\right) \left| {\Psi \left( t\right) }\right| \]\n\n\[ \left( {Abel}\right) \]\n\nwhere \( \left| {\Psi \left( t\right) }\right| = \det... | Proof.\n\n\[ \frac{d}{dt}\left| {\Psi \left( t\right) }\right| = \frac{d}{dt}\left| \begin{array}{ll} {\psi }_{11}\left( t\right) & {\psi }_{21}\left( t\right) \\ {\psi }_{12}\left( t\right) & {\psi }_{22}\left( t\right) \end{array}\right| = \left| \begin{array}{ll} {\dot{\psi }}_{11}\left( t\right) & {\dot{\psi }}_{21... | Yes |
Proposition 3 (Abel's formula). The fundmental matrix (2.10) satisfies \[ \frac{d}{dt}\left| {\Psi \left( t\right) }\right| = \operatorname{trace}\left( {A\left( t\right) }\right) \left| {\Psi \left( t\right) }\right| \] where \( \left| {\Psi \left( t\right) }\right| = \det \Psi \left( t\right) \). | Proof. \[ \frac{d}{dt}\left| {\Psi \left( t\right) }\right| = \frac{d}{dt}\left| \begin{array}{ll} {\psi }_{11}\left( t\right) & {\psi }_{21}\left( t\right) \\ {\psi }_{12}\left( t\right) & {\psi }_{22}\left( t\right) \end{array}\right| = \left| \begin{array}{ll} {\dot{\psi }}_{11}\left( t\right) & {\dot{\psi }}_{21}\l... | Yes |
Proposition 7 (Gronwall's Inequality). Assume that for some constant \( C \geq 0 \) and non-negative integrable functions \( f \) and \( g \), we have\n\n(2.16)\n\n\[ f\left( t\right) \leq C + {\int }_{0}^{t}f\left( s\right) g\left( s\right) {ds} \]\n\nthen\n\n(2.17)\n\n\[ f\left( t\right) \leq C{e}^{{\int }_{0}^{t}g\l... | Proof. Let \( F\left( t\right) = C + {\int }_{0}^{t}f\left( s\right) g\left( s\right) {ds} \), then \( F\left( t\right) \) is differentiable\n\nand\n\n\[ {F}^{\prime }\left( t\right) = f\left( t\right) g\left( t\right) \]\n\nSince \( g\left( t\right) \geq 0 \) and \( f\left( t\right) \leq F\left( t\right) \), we have\n... | Yes |
Proposition 9. If \( t \in \left\lbrack {{t}_{0} - h,{t}_{0} + h}\right\rbrack \), then\n\n\[ \left| {{r}_{n}\left( t\right) }\right| \leq M{L}^{n}{t}^{n + 1}/\left( {n + 1}\right) ! \] | Proof. We prove this by induction. When \( n = 0 \) ,\n\n\[ \left| {{r}_{0}\left( t\right) }\right| = \left| {{y}_{1}\left( t\right) - {y}^{0}}\right| = \left| {{\int }_{0}^{t}f\left( {s,{y}^{0}}\right) {ds}}\right| \leq {\int }_{0}^{t}\left| {f\left( {s,{y}^{0}}\right) }\right| {ds} \leq {Mt}. \]\n\nSuppose that \( \l... | Yes |
Proposition 11. Assume that \( f\left( x\right) \) is a continuous function If over any interval \( \left\lbrack {a, b}\right\rbrack \left( {a < b}\right) \), the integration \( {\int }_{a}^{b}f\left( x\right) \mathrm{d}x = 0 \) then, \( f\left( x\right) \equiv 0. \) | Proof. If at some point \( {x}_{0}, f\left( {x}_{0}\right) \neq 0 \), assume \( f\left( {x}_{0}\right) > 0 \) without loss of generality, then there is a \( \delta > 0 \) such that\n\n(3.4)\n\n\[ \left| {f\left( x\right) - f\left( {x}_{0}\right) }\right| \leq \frac{1}{2}f\left( {x}_{0}\right) \]\n\nfor \( x \in \left\l... | Yes |
Proposition 16. \( \; \bullet {G}^{\mu } \) is symmetric and decreasing. | \[ {G}^{\mu }\left( x\right) = \left\{ \begin{array}{ll} \frac{1}{2\mu }{e}^{-\mu \left| x\right| } & d = 1, \\ \frac{1}{{4\pi }\left| x\right| }{e}^{-\mu \left| x\right| } & d = 3. \end{array}\right. \] | Yes |
The probability space for the outcome of one trial can be defined as follows. The sample space \( \Omega = \{ H, T\} \) where \( H \) and \( T \) represent head and tail, respectively. The \( \sigma \) -algebra | \[ \mathcal{F} = \text{ all subsets of }\Omega = \{ \varnothing ,\{ H\} ,\{ T\} ,\Omega \} \] and \[ \mathbb{P}\left( \varnothing \right) = 0,\;\mathbb{P}\left( {\{ H\} }\right) = \mathbb{P}\left( {\{ T\} }\right) = \frac{1}{2},\;\mathbb{P}\left( \Omega \right) = 1. \] | Yes |
Example 1.5 (Uniform orientation distribution on \( {\mathbb{S}}^{2} \) ). In this case, the sample space \( \Omega = {\mathbb{S}}^{2} \) . Let \( \mathcal{B} \) be the set of all open sets of \( {\mathbb{S}}^{2} \), defined as the intersection of any open set \( B \subset {\mathbb{R}}^{3} \) and \( {\mathbb{S}}^{2} \)... | Within this framework, the standard rules of set theory are used to answer probability questions. For instance, if both \( A, B \in \mathcal{F} \), the probalility that both \( A \) and \( B \) occurs is given by \( \mathbb{P}\left( {A \cap B}\right) \), the probability that either \( A \) or \( B \) occurs is given by... | Yes |
Example 1.7 (Bernoulli distribution). The Bernoulli distribution has the form\n\n\[ \n\\mathbb{P}\\left( {X = j}\\right) = \\left\\{ \\begin{array}{ll} p, & j = 1 \\\\ q, & j = 0 \\end{array}\\right.\n\]\n\n\( p + q = 1 \) and \( p, q \\geq 0 \) . When \( p = q = 1/2 \), it corresponds to the toss of a fair coin. The m... | \n\[ \n\\mathbb{E}X = p,\\;\\operatorname{Var}\\left( X\\right) = {pq}.\n\] | Yes |
Example 1.8 (Binomial distribution \( B\left( {n, p}\right) \) ). The binomial distribution \( B\left( {n, p}\right) \) has the form\n\n(1.9)\n\n\[ \mathbb{P}\left( {X = k}\right) = \left( \begin{array}{l} n \\ k \end{array}\right) {p}^{k}{q}^{n - k},\;k = 0,1,\ldots, n. \] | It is straightforward to obtain\n\n\[ \mathbb{E}X = {np},\;\operatorname{Var}\left( X\right) = {npq}. \] | No |
Lemma 1.15 (Chebyshev’s inequality). Let \( \mathbf{X} \) be a random variable such that \( \mathbb{E}{\left| \mathbf{X}\right| }^{p} < \infty \) for some \( p > 0 \) . Then\n\n\[ \mathbb{P}\{ \left| \mathbf{X}\right| \geq \lambda \} \leq \frac{1}{{\lambda }^{p}}\mathbb{E}{\left| \mathbf{X}\right| }^{p} \]\n\nfor any p... | Proof. For any \( \lambda > 0 \) ,\n\n\[ \mathbb{E}{\left| \mathbf{X}\right| }^{p} = {\int }_{{\mathbb{R}}^{d}}{\left| \mathbf{x}\right| }^{p}\mu \left( {d\mathbf{x}}\right) \geq {\int }_{\left| \mathbf{x}\right| \geq \lambda }{\left| \mathbf{x}\right| }^{p}\mu \left( {d\mathbf{x}}\right) \geq {\lambda }^{p}{\int }_{\l... | Yes |
Lemma 1.16 (Jensen’s inequality). Let \( \mathbf{X} \) be a random variable such that \( \mathbb{E}\left| \mathbf{X}\right| < \infty \) and \( \phi : \mathbb{R} \rightarrow \mathbb{R} \) is a convex function such that \( \mathbb{E}\left| {\phi \left( \mathbf{X}\right) }\right| < \infty \) . Then\n\n(1.21)\n\n\[ \mathbb... | This follows directly from the definition of convex functions. Readers can also refer to \( \mathbf{{Chu01}} \) for the details. | No |
Example 1.17 (Uniform distribution). The uniform distribution on a domain \( B \) (in \( {\mathbb{R}}^{d} \) ) is defined by the probability density function:\n\n\[ \rho \left( x\right) = \left\{ \begin{array}{ll} \frac{1}{\operatorname{vol}\left( B\right) }, & \text{ if }\mathbf{x} \in B, \\ 0, & \text{ otherwise. } \... | In one dimension if \( B = \left\lbrack {0,1}\right\rbrack \) (denoted as \( \mathcal{U}\left\lbrack {0,1}\right\rbrack \) later), this reduces to\n\n\[ \rho \left( x\right) = \left\{ \begin{array}{ll} 1, & \text{ if }x \in \left\lbrack {0,1}\right\rbrack \\ 0, & \text{ otherwise. } \end{array}\right. \]\n\nFor the uni... | Yes |
Example 1.18 (Exponential distribution). The exponential distribution \( \mathcal{E}\left( \lambda \right) \) is defined by the probability density function:\n\n\[ \rho \left( x\right) = \left\{ \begin{array}{ll} 0, & \text{ if }x < 0 \\ \lambda {e}^{-{\lambda x}}, & \text{ if }x \geq 0 \end{array}\right. \] | The mean and variance of \( E\left( \lambda \right) \) are\n\n(1.22)\n\n\[ \mathbb{E}X = \frac{1}{\lambda },\;\operatorname{Var}\left( X\right) = \frac{1}{{\lambda }^{2}}. \] | Yes |
The one-dimensional normal distribution (also called Gaussian distribution) \( N\left( {\mu ,{\sigma }^{2}}\right) \) is defined by the probability density function:\n\n\[ \rho \left( x\right) = \frac{1}{\sqrt{{2\pi }{\sigma }^{2}}}\exp \left( {-\frac{1}{2{\sigma }^{2}}{\left( x - \mu \right) }^{2}}\right) \] | with mean \( \mu \) and variance \( {\sigma }^{2} \). | No |
Example 1.20 (Gibbs distribution). In equilibrium statistical mechanics, we are concerned with a probability distribution \( \pi \) over a state space \( S \) . In the case of an \( n \) -particle system with continuous states, we have \( \mathbf{x} = \) \( \left( {{\mathbf{x}}_{1},\ldots ,{\mathbf{x}}_{n},{\mathbf{p}}... | \[ \pi \left( \mathbf{x}\right) = \frac{1}{Z}{e}^{-{\beta H}\left( \mathbf{x}\right) },\;\mathbf{x} \in {\mathbb{R}}^{6n},\beta = {\left( {k}_{B}T\right) }^{-1}, \] where \( H \) is the energy of the considered system, \( T \) is the absolute temperature, \( {k}_{B} \) is the Boltzmann constant, and \[ Z = {\int }_{{\m... | Yes |
Proposition 1.26. Let \( g \) be a measurable function. Then\n\n\[ \mathbb{E}{\left( X - \mathbb{E}\left( X \mid Y\right) \right) }^{2} \leq \mathbb{E}{\left( X - g\left( Y\right) \right) }^{2}. \] | Proof. We have\n\n\[ \mathbb{E}{\left( X - g\left( Y\right) \right) }^{2} = \mathbb{E}{\left( X - E\left( X \mid Y\right) \right) }^{2} + \mathbb{E}{\left( E\left( X \mid Y\right) - g\left( Y\right) \right) }^{2} \]\n\n\[ + 2\mathbb{E}\left\lbrack {\left( {X - E\left( {X \mid Y}\right) }\right) \left( {E\left( {X \mid ... | Yes |
(i) Almost sure convergence implies convergence in probability. | Note that\n\n\[ \mathbb{P}\left( {\left| {{X}_{n}\left( \omega \right) - X\left( \omega \right) }\right| > \epsilon }\right) = {\int }_{\Omega }{\chi }_{\left\{ \left| {X}_{n} - X\right| > \epsilon \right\} }\left( \omega \right) \mathbb{P}\left( {d\omega }\right) \rightarrow 0 \]\n\nby the almost sure convergence and ... | Yes |
Proposition 1.33. The characteristic function has the following properties:\n\n(1) \( \forall \xi \in \mathbb{R},\left| {f\left( \xi \right) }\right| \leq 1, f\left( \xi \right) = \overline{f\left( {-\xi }\right) }, f\left( 0\right) = 1 \) ;\n\n(2) \( f \) is uniformly continuous on \( \mathbb{R} \) . | Proof. The proof of the first statements is straightforward. For the second statement, we have\n\n\[ \left| {f\left( {\xi }_{1}\right) - f\left( {\xi }_{2}\right) }\right| = \left| {\mathbb{E}\left( {{e}^{i{\xi }_{1}X} - {e}^{i{\xi }_{2}X}}\right) }\right| = \left| {\mathbb{E}\left( {{e}^{i{\xi }_{1}X}\left( {1 - {e}^{... | Yes |
Theorem 1.35 (Lévy’s continuity theorem). Let \( {\left\{ {\mu }_{n}\right\} }_{n \in \mathbb{N}} \) be a sequence of probability measures, and let \( {\left\{ {f}_{n}\right\} }_{n \in \mathbb{N}} \) be their corresponding characteristic functions. Assume that:\n\n(1) \( {f}_{n} \) converges everywhere on \( \mathbb{R}... | For a proof, see [Chu01]. | No |
Theorem 1.37 (Bochner’s theorem). A function \( f \) is the characteristic function of a probability measure if and only if it is positive semidefinite and continuous at 0 with \( f\left( 0\right) = 1 \) . | Proof. We only prove the necessity part. The other part is less trivial and readers may consult Chu01. Assume that \( f \) is a characteristic function. Then\n\n(1.50)\n\n\[ \mathop{\sum }\limits_{{i, j = 1}}^{n}f\left( {{\xi }_{i} - {\xi }_{j}}\right) {v}_{i}{\bar{v}}_{j} = {\int }_{\mathbb{R}}{\left| \mathop{\sum }\l... | No |
Theorem 1.40. Denote \( {M}_{X}\left( t\right) ,{M}_{Y}\left( t\right) \), and \( {M}_{X + Y}\left( t\right) \) the moment generating functions of the random variables \( X, Y \), and \( X + Y \), respectively. If \( X, Y \) are independent, then\n\n(1.56)\n\n\[ \n{M}_{X + Y}\left( t\right) = {M}_{X}\left( t\right) {M}... | Proof. The proof is straightforward by noticing\n\n\[ \n{M}_{X + Y}\left( t\right) = \mathbb{E}{e}^{t\left( {X + Y}\right) } = \mathbb{E}{e}^{tX}\mathbb{E}{e}^{tY} = {M}_{X}\left( t\right) {M}_{Y}\left( t\right) .\n\] | Yes |
(1) If \( \mathop{\sum }\limits_{{n = 1}}^{\infty }\mathbb{P}\left( {A}_{n}\right) < \infty \), then \( \mathbb{P}\left( \left\{ {{A}_{n}\text{i.o.}}\right\} \right) = 0 \) . | Proof. (1) We have\n\n\[ \mathbb{P}\left( \left\{ {\mathop{\bigcap }\limits_{{n = 1}}^{\infty }\mathop{\bigcup }\limits_{{k = n}}^{\infty }{A}_{k}}\right\} \right) \leq \mathbb{P}\left( \left\{ {\mathop{\bigcup }\limits_{{k = n}}^{\infty }{A}_{k}}\right\} \right) \leq \mathop{\sum }\limits_{{k = n}}^{\infty }\mathbb{P}... | Yes |
Lemma 1.42. Let \( {\left\{ {X}_{n}\right\} }_{n \in \mathbb{N}} \) be a sequence of identically distributed (not necessarily independent) random variables, such that \( \mathbb{E}\left| {X}_{n}\right| < \infty \) . Then | Proof. For any \( \epsilon > 0 \), define\n\n\[ \n{A}_{n}^{\epsilon } = \left\{ {\omega \in \Omega : \left| {{X}_{n}\left( \omega \right) /n}\right| > \epsilon }\right\} \n\]\n\nThen\n\n\[ \n\mathop{\sum }\limits_{{n = 1}}^{\infty }\mathbb{P}\left( {A}_{n}^{\epsilon }\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\... | Yes |
Theorem 2.1 (Weak law of large numbers (WLLN)). Let \( {\left\{ {X}_{j}\right\} }_{j = 1}^{\infty } \) be a sequence of i.i.d. random variables such that \( \mathbb{E}\left| {X}_{j}\right| < \infty \) . Then\n\n\[ \frac{{S}_{n}}{n} \rightarrow \eta \;\text{ in probability. } \] | Proving this result under the stated assumption is quite involved. We will give a proof of the WLLN under the stronger assumption that \( \mathbb{E}{\left| {X}_{j}\right| }^{2} < \) \( \infty \) .\n\nProof. Without loss of generality, we can assume \( \eta = 0 \) . From the Cheby-shev inequality, we have\n\n\[ \mathbb{... | Yes |
Theorem 2.2 (Strong law of large numbers (SLLN)). Let \( {\left\{ {X}_{j}\right\} }_{j = 1}^{\infty } \) be a sequence of i.i.d. random variables such that \( \mathbb{E}\left| {X}_{j}\right| < \infty \) . Then\n\n\[ \frac{{S}_{n}}{n} \rightarrow \eta \text{ a.s. } \] | Proof. We will only give a proof of the SLLN under the stronger assumption that \( \mathbb{E}{\left| {X}_{j}\right| }^{4} < \infty \) . The proof under the stated assumption can be found in Chu01\n\nWithout loss of generality, we can assume \( \eta = 0 \) . Using the Chebyshev inequality, we have\n\n\[ \mathbb{P}\left(... | No |
For the distribution with PDF\n\n\[ \n p\left( x\right) = \frac{1}{\pi \left( {1 + {x}^{2}}\right) },\;x \in \mathbb{R}, \]\n\nwe have \( \mathbb{E}\left| {X}_{j}\right| = \infty \) . For a sequence of i.i.d. random variables \( {\left\{ {X}_{j}\right\} }_{j = 1}^{\infty } \) with Cauchy-Lorentz distribution, one can e... | In fact, the converse statement for SLLN is also true: If \( {S}_{n}/n \rightarrow a < \) \( \infty \) a.s. holds, then \( \mathbb{E}\left| {X}_{j}\right| < \infty \) . The proof can be found in Chu01. | No |
Theorem 2.4 (Lindeberg-Lévy central limit theorem (CLT)). Let \( {\left\{ {X}_{j}\right\} }_{j = 1}^{\infty } \) be a sequence of i.i.d. random variables. Assume that \( \mathbb{E}{X}_{j}^{2} < \infty \) and let \( {\sigma }^{2} = \operatorname{Var}\left( {X}_{j}\right) \) . Then\n\n\[ \frac{{S}_{n} - {n\eta }}{\sqrt{n... | Outline of proof. Assume without loss of generality that \( \eta = 0 \) and \( \sigma = 1 \) ; otherwise we can shift and rescale \( {X}_{j} \) . Let \( f \) be the characteristic function of \( {X}_{1} \) and let \( {g}_{n} \) be the characteristic function of \( {S}_{n}/\sqrt{n} \) . Then\n\n\[ {g}_{n}\left( \xi \rig... | Yes |
Lemma 2.9. The rate function \( I\left( x\right) \) has the following properties:\n\n(i) \( I\left( x\right) \) is convex and lower semicontinuous.\n\n(ii) \( I\left( x\right) \) is nonnegative and \( I\left( \eta \right) = 0 \).\n\n(iii) \( I\left( x\right) \) is nondecreasing in \( \left\lbrack {\eta ,\infty )\text{a... | Proof. (i) The convexity of \( \Lambda \left( \lambda \right) \) follows from Hölder’s inequality. For any \( 0 \leq \theta \leq 1, \)\n\n\[ \Lambda \left( {\theta {\lambda }_{1} + \left( {1 - \theta }\right) {\lambda }_{2}}\right) = \log \mathbb{E}\left( {\exp \left( {\theta {\lambda }_{1}{X}_{j}}\right) \exp \left( {... | Yes |
Example 2.10 (Cramér's theorem applied to the Bernoulli distribution with parameter \( p\left( {0 < p < 1}\right) \) ). We have \( \Lambda \left( \lambda \right) = \log \left( {p{e}^{\lambda } + q}\right) \) where \( q = 1 - p \) . The rate function | \[ I\left( x\right) = \left\{ \begin{array}{ll} x\log \frac{x}{p} + \left( {1 - x}\right) \log \frac{1 - x}{q}, & x \in \left\lbrack {0,1}\right\rbrack , \\ \infty , & \text{ otherwise. } \end{array}\right. \] It is obvious that \( I\left( x\right) \geq 0 \), and \( I\left( x\right) \) achieves its global minimum 0 at ... | Yes |
Assume that \( {X}_{j} \) is exponentially distributed; i.e., if we denote by \( \rho \left( x\right) \) the probability density function of \( {X}_{j} \), then\n\n\[ \rho \left( x\right) = \left\{ \begin{array}{ll} {e}^{-x}, & \text{ if }x > 0 \\ 0, & \text{ if }x \leq 0 \end{array}\right. \]\n\nThen \( \mathbb{P}\lef... | \[ = \mathop{\prod }\limits_{{j = 1}}^{n}\mathbb{P}\left( {{X}_{j} \leq x}\right) = {\left( 1 - {e}^{-x}\right) }^{n}. \]\n\nThis remains true even if \( x \) depends on \( n \) . We will choose \( x = {x}_{n} \) such that \( {\left( 1 - {e}^{-{x}_{n}}\right) }^{n} \) has a nontrivial limit. For this purpose, we let\n\... | Yes |
Assume that \( {X}_{j} \) is uniformly distributed on \( \left\lbrack {0,1}\right\rbrack \) ; i.e., \n\n\[ \n\rho \left( x\right) = \left\{ \begin{array}{ll} 1, & \text{ if }x \in \left\lbrack {0,1}\right\rbrack \\ 0, & \text{ otherwise. } \end{array}\right. \n\] \n\nWe expect that \( {M}_{n} \rightarrow 1 \) as \( n \... | Notice that if we let \( {x}_{n} = 1 - x/n \), then \n\n\[ \n\mathbb{P}\left( {{M}_{n} \leq {x}_{n}}\right) = {\left( 1 - \frac{x}{n}\right) }^{n} \rightarrow {e}^{-x} \n\] \nas \( n \rightarrow \infty \), or equivalently \n\n\[ \n\mathbb{P}\left( {n\left( {{M}_{n} - 1}\right) \leq x}\right) \rightarrow {e}^{-\left| x\... | Yes |
Example 3.1 (Symmetric random walk). Consider the sequence of random variables \( {\left\{ {\xi }_{j}\right\} }_{j = 1}^{\infty } \), where the \( {\left\{ {\xi }_{j}\right\} }^{\prime } \) s are i.i.d and \( {\xi }_{j} = \pm 1 \) with probability \( 1/2 \) . Let\n\n\[ \n{X}_{n} = \mathop{\sum }\limits_{{j = 1}}^{n}{\x... | \[ \n\mathbb{P}\left( {{X}_{n + 1} = i \pm 1 \mid {X}_{n} = i}\right) = \mathbb{P}\left( {{\xi }_{n + 1} = \pm 1}\right) = \frac{1}{2} \n\]\n\nand \( \mathbb{P}\left( {{X}_{n + 1} = }\right. \) anything else \( \left. {\mid {X}_{n} = i}\right) = 0 \) . We see that, knowing \( {X}_{n} \), the distribution of \( {X}_{n +... | Yes |
Example 3.2 (Ehrenfest’s diffusion model). Consider a container separated by a permeable membrane in the middle and filled with a total of \( K \) particles. At each time \( n = 1,2\ldots \), pick one particle in random among the \( K \) particles and place it into the other part of the container. Let \( {X}_{n} \) be ... | \[ \mathbb{P}\left( {{X}_{n + 1} = {i}_{n + 1} \mid {\left\{ {X}_{m} = {i}_{m}\right\} }_{m = 0}^{n}}\right) = \mathbb{P}\left( {{X}_{n + 1} = {i}_{n + 1} \mid {X}_{n} = {i}_{n}}\right) ,\] for \( {i}_{k} \in \{ 0,1,\ldots, K\} \), and this is also a Markov process. | Yes |
Example 3.3 (Autoregressive model). The autoregressive process \( {\left\{ {Y}_{n}\right\} }_{n \in \mathbb{N}} \) of order \( k \) is defined as follows. For each \( n \geq 1 \) ,\n\n\[ \n{Y}_{n} = {\alpha }_{1}{Y}_{n - 1} + {\alpha }_{2}{Y}_{n - 2} + \cdots + {\alpha }_{k}{Y}_{n - k} + {R}_{n}, \n\]\n\nwhere \( {\alp... | In this example, \( {Y}_{n} \) itself is not Markovian since \( {Y}_{n} \) depends not only on \( {Y}_{n - 1} \), but also on \( {Y}_{n - 2},\ldots ,{Y}_{n - k} \) . But if we introduce the new variable\n\n\[ \n{\mathbf{X}}_{n} = {\left( {Y}_{n},\ldots ,{Y}_{n - k + 1}\right) }^{T},\;n = 0,1,\ldots , \n\]\n\nthen we ob... | Yes |
Proposition 3.4 (Chapman-Kolmogorov equation).\n\n\[ \mathbb{P}\left( {{X}_{n} = j \mid {X}_{0} = i}\right) \]\n\n\[ = \mathop{\sum }\limits_{{k \in S}}\mathbb{P}\left( {{X}_{n} = j \mid {X}_{m} = k}\right) \mathbb{P}\left( {{X}_{m} = k \mid {X}_{0} = i}\right) ,\;1 \leq m \leq n - 1. \] | The proof is a straightforward application of Bayes's rule and the Markov property. | No |
Lemma 3.5. The spectral radius of \( \mathbf{P} \) is equal to 1 :\n\n\[ \rho \left( \mathbf{P}\right) = \mathop{\max }\limits_{\lambda }\left| \lambda \right| = 1 \]\n\nwhere the maximum is taken over the eigenvalues of \( \mathbf{P} \) . | Proof. We already know that 1 is an eigenvalue of \( \mathbf{P} \) . To show it is the maximum eigenvalue, denote by \( \mathbf{u} \) the left eigenvector of \( \mathbf{P} \) with eigenvalue \( \lambda \) . Then\n\n\[ \lambda {u}_{i} = \mathop{\sum }\limits_{{j \in S}}{u}_{j}{p}_{ji} \]\n\nwhich implies that\n\n\[ \lef... | Yes |
Lemma 3.7. Irreducibility is equivalent to the property that every pair of nodes in the state space communicates with each other. | The proof is left as an exercise for the readers. | No |
Theorem 3.8 (Perron-Frobenius theorem). Let \( \mathbf{A} \) be an irreducible nonnegative matrix, and let \( \rho \left( \mathbf{A}\right) \) be its spectral radius: \( \rho \left( \mathbf{A}\right) = \mathop{\max }\limits_{\lambda }\left| {\lambda \left( \mathbf{A}\right) }\right| \) . Then:\n\n(1) There exist positi... | We refer to [HJ85] for the proof. | No |
Consider the Markov chain with the transition probability matrix\n\n\[ \mathbf{P} = \left\lbrack \begin{matrix} 0 & 1 & 0 & 0 & 0 \\ {0.3} & {0.4} & {0.3} & 0 & 0 \\ {0.3} & 0 & {0.4} & {0.3} & 0 \\ 0 & 0 & 0 & {0.5} & {0.5} \\ 0 & 0 & 0 & {0.5} & {0.5} \end{matrix}\right\rbrack \] | This is reducible. But one can verify that the chain has a unique invariant distribution\n\n\[ \mathbf{\pi } = \left( {0,0,0,{0.5},{0.5}}\right) . \] | No |
Theorem 3.14. For any fixed time \( t \), the distribution of \( {N}_{t} \) is Poisson with parameter \( {\lambda t} \) . | Proof. Let \( {p}_{m}\left( t\right) = \mathbb{P}\left( {{N}_{t} = m}\right) \) . Taking \( h \ll 1 \) we have\n\n\[ \n{p}_{0}\left( {t + h}\right) = {p}_{0}\left( t\right) \left( {1 - {\lambda h}}\right) + o\left( h\right) , \n\]\n\nor equivalently\n\n\[ \n\frac{{p}_{0}\left( {t + h}\right) - {p}_{0}\left( t\right) }{... | Yes |
Theorem 3.15. The jump chain \( \left\{ {Y}_{n}\right\} \) is a Markov chain with \( \widetilde{Q} \) as the transition probability matrix, and the holding times \( {H}_{1},{H}_{2},\ldots \) are independent exponential random variables with parameters \( {q}_{{Y}_{0}},{q}_{{Y}_{1}},\ldots \), respectively. | The proof relies on the strong Markov property of the \( Q \) -process. See Section E of the appendix and [Nor97]. | No |
Theorem 3.16. The Q-process \( X \) is irreducible if and only if the embedded chain \( Y \) is irreducible. | Proof. First note that for any \( i \neq j \), if \( {q}_{ij} > 0 \), we have\n\n\[ \n{p}_{ij}\left( t\right) \geq {\mathbb{P}}^{i}\left( {{J}_{1} \leq t,{Y}_{1} = j,{H}_{2} > t}\right) = {\int }_{0}^{t}\exp \left( {-{q}_{i}u}\right) {q}_{ij}{du}\exp \left( {-{q}_{j}t}\right) \n\]\n\n\[ \n= \left( {1 - {e}^{-{q}_{i}t}}... | Yes |
Theorem 3.17 (Convergence to equilibrium). Assume that \( \mathbf{Q} \) is irreducible. Then for any initial distribution \( {\mathbf{\mu }}_{0} \)\n\n\[ \mathbf{\mu }\left( t\right) = {\mathbf{\mu }}_{0}\mathbf{P}\left( t\right) \rightarrow \mathbf{\pi }\;\text{ exponentially fast as }t \rightarrow \infty ,\]\n\nwhere... | Proof. From irreducibility and Theorem 3.16, we know that \( {p}_{ij}\left( t\right) > 0 \) for any \( i \neq j \) and \( t > 0 \) . It is straightforward to see that\n\n\[ {p}_{ii}\left( t\right) \geq {\mathbb{P}}^{i}\left( {{J}_{1} > t}\right) = {e}^{-{q}_{i}t} > 0 \]\n\nfor any \( i \in S \) and \( t > 0 \), where \... | Yes |
Theorem 3.18 (Ergodic theorem). Assume that \( \mathbf{Q} \) is irreducible. Then for any bounded function \( f \) we have\n\n\[ \frac{1}{T}{\int }_{0}^{T}f\left( {X}_{s}\right) {ds} \rightarrow \langle f{\rangle }_{\pi },\;\text{ a.s. } \]\n\nas \( T \rightarrow \infty \), where \( \pi \) is the unique invariant distr... | The proof can be found in [Dur10, Nor97]. | No |
Theorem 4.1. The period of an LCG is \( m \) if and only if\n\n(i) \( b \) and \( m \) are relatively prime;\n\n(ii) every prime factor of \( m \) divides \( a - 1 \) ;\n\n(iii) if \( 4 \mid m \), then \( 4 \mid \left( {a - 1}\right) \) . | If one chooses \( m = {2}^{k}, a = {4c} + 1 \), and \( b \) odd, then these conditions are satisfied. | Yes |
Proposition 4.3 (Inverse transformation method). Let \( F \) be the distribution function of \( X \) ; i.e., \( F\left( x\right) = \mathbb{P}\left( {X \leq x}\right) \) . Let \( U \) be a \( \mathcal{U}\left\lbrack {0,1}\right\rbrack \) random variable. Define the generalized inverse of \( F \) by\n\n\[ \n{F}^{ - }\lef... | Proof. If \( F \) is continuous and strictly increasing, we have \( {F}^{ - }\left( u\right) = {F}^{-1}\left( u\right) \) and\n\n\[ \n\mathbb{P}\left( {X \leq x}\right) = \mathbb{P}\left( {{F}^{-1}\left( U\right) \leq x}\right) = \mathbb{P}\left( {U \leq F\left( x\right) }\right) = F\left( x\right) .\n\]\n\nIn the gene... | Yes |
Consider the integral\n\n\[ \nI\left( f\right) = {\int }_{-\infty }^{+\infty }\frac{1}{\sqrt{2\pi }}{\left( 1 + r\right) }^{-1}{e}^{-\frac{{x}^{2}}{2}}{dx} \n\]\n\nwhere \( r = {e}^{\sigma x},\sigma \gg 1 \) . | Notice that\n\n\[ \n{\left( 1 + r\right) }^{-1} \approx h\left( x\right) = \left\{ \begin{array}{ll} 1, & x \leq 0 \\ 0, & x > 0 \end{array}\right. \n\]\n\nWe have\n\n\[ \nI\left( f\right) = \frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{+\infty }\left( {{\left( 1 + r\right) }^{-1} - h\left( x\right) }\right) {e}^{-\frac{{x... | Yes |
Example 4.5. Shown in Figure 4.4 is a one-dimensional model for the magnetization of a ferromagnet. The lattice has \( M \) sites, the state space \( S = \{ \mathbf{x}\} = \{ + 1, - 1{\} }^{M} \) . It has \( {2}^{M} \) states in total. The states at the sites are called spins. The microscopic configurations are describ... | In equilibrium statistical mechanics, we are interested in the thermodynamic average of some function \( f\left( \mathbf{x}\right) \) given by \[ \langle f\rangle = \mathop{\sum }\limits_{{\mathbf{x} \in S}}f\left( \mathbf{x}\right) \pi \left( \mathbf{x}\right) \;\text{ or }\;{\int }_{S}f\left( \mathbf{x}\right) \pi \l... | Yes |
Theorem 4.6. The Gibbs distribution has the limit\n\n\\[ \n\\mathop{\\lim }\\limits_{{\\beta \\rightarrow + \\infty }}{\\pi }_{\\beta }\\left( x\\right) = \\left\\{ \\begin{array}{ll} \\frac{1}{\\left| \\mathcal{M}\\right| }, & \\text{ if }x \\in \\mathcal{M}, \\\\ 0, & \\text{ othewise,} \\end{array}\\right.\n\\]\n\nw... | Proof. Define \\( m = \\mathop{\\min }\\limits_{x}H\\left( x\\right) \\) . Then\n\n\\[ \n{\\pi }_{\\beta }\\left( x\\right) \\; = \\;\\frac{{e}^{-\\beta \\left( {H\\left( x\\right) - m}\\right) }}{\\mathop{\\sum }\\limits_{{z \\in \\mathcal{M}}}{e}^{-\\beta \\left( {H\\left( z\\right) - m}\\right) } + \\mathop{\\sum }\... | Yes |
Theorem 4.7 (Convergence of simulated annealing). Assume that \( H \) is defined over a finite set \( \mathcal{X} \) and \( Q \) is a symmetric irreducible proposal matrix. If the annealing procedure is chosen such that \( \beta \left( n\right) \leq C\log n \), where \( C \) only depends on the structure of \( Q \) and... | The proof of this theorem can be found in [Win03]. | No |
Consider the \( Q \) -process \( {X}_{t} \) on \( S = \{ 1,2,\ldots, I\} \) with generator \( \mathbf{Q} \) defined as in (3.16). We have | \[\n\left( {\mathcal{A}f}\right) \left( i\right) = \mathop{\lim }\limits_{{t \rightarrow 0 + }}\frac{{\mathbb{E}}^{i}f\left( {X}_{t}\right) - f\left( i\right) }{t} = \mathop{\lim }\limits_{{t \rightarrow 0 + }}\frac{1}{t}\left( {\mathop{\sum }\limits_{{j \in S}}\left( {{P}_{ij}\left( t\right) - {\delta }_{ij}}\right) f... | Yes |
Example 5.8 (Poisson process). Consider the Poisson process \( {X}_{t} \) on \( \mathbb{N} \) with rate \( \lambda \) . We have\n\n\[ \left( {\mathcal{A}f}\right) \left( n\right) = \mathop{\lim }\limits_{{t \rightarrow 0 + }}\frac{{\mathbb{E}}^{n}f\left( {X}_{t}\right) - f\left( n\right) }{t} = \mathop{\lim }\limits_{{... | \[ = \mathop{\lim }\limits_{{t \rightarrow 0 + }}\frac{1}{t}\left( {f\left( n\right) \left( {{e}^{-{\lambda t}} - 1}\right) + f\left( {n + 1}\right) {\lambda t}{e}^{-{\lambda t}} + \mathop{\sum }\limits_{{k = n + 2}}^{\infty }f\left( k\right) \frac{{\left( \lambda t\right) }^{k - n}}{\left( {k - n}\right) !}{e}^{-{\lam... | Yes |
Theorem 5.10. Assume the stochastic process \( {\left\{ {X}_{t}\right\} }_{t \in \left\lbrack {0, T}\right\rbrack } \) satisfies the condition\n\n\[ \mathbb{E}{\int }_{0}^{T}{X}_{t}^{2}{dt} < \infty \]\n\nThen \( m \in {L}_{t}^{2} \) in the sense that\n\n\[ {\int }_{0}^{T}{m}^{2}\left( t\right) {dt} < \infty \]\n\nFurt... | Proof. First, we have\n\n\[ {\int }_{0}^{T}{m}^{2}\left( t\right) {dt} = {\int }_{0}^{T}{\left( \mathbb{E}{X}_{t}\right) }^{2}{dt} \leq {\int }_{0}^{T}\mathbb{E}{X}_{t}^{2}{dt} < \infty .\n\nIn addition, we have\n\n\[ {\int }_{0}^{T}{\int }_{0}^{T}{K}^{2}\left( {s, t}\right) {dsdt} = {\int }_{0}^{T}{\int }_{0}^{T}{\lef... | Yes |
Theorem 5.11. Assume that \( {X}_{1},{X}_{2},\ldots \) is a sequence of Gaussian random variables that converges to \( X \) in probability. Then \( X \) is also Gaussian. | Proof. Let us denote\n\n\[ \n{m}_{k} = \mathbb{E}{X}_{k},\;{\sigma }_{k}^{2} = \operatorname{Var}{X}_{k}.\n\]\n\nThen from part \( \left( v\right) \) of the Theorem 1.32 and Lévy’s continuity theorem, we have\n\n\[ \n{e}^{{i\xi }{m}_{k} - \frac{1}{2}{\sigma }_{k}^{2}{\xi }^{2}} = \mathbb{E}{e}^{{i\xi }{X}_{k}} \rightar... | Yes |
Theorem 5.13 (Karhunen-Loève expansion). Let \( {\left( {X}_{t}\right) }_{t \in \left\lbrack {0,1}\right\rbrack } \) be a Gaussian process with mean 0 and covariance function \( K\left( {s, t}\right) \) . Assume that \( K \) is continuous. Let \( \left\{ {\lambda }_{k}\right\} ,\left\{ {\phi }_{k}\right\} \) be the seq... | Proof. We need to show that the random series is well-defined and that it is a Gaussian process with the desired mean and covariance function.\n\nFirst consider the operator \( \mathcal{K} : {L}^{2}\left( \left\lbrack {0,1}\right\rbrack \right) \rightarrow {L}^{2}\left( \left\lbrack {0,1}\right\rbrack \right) \) define... | Yes |
Example 6.1 (Random walk). Let \( \\left\\{ {\\xi }_{i}\\right\\} \) be i.i.d. random variables such that \( {\\xi }_{i} = \\pm 1 \) with probability \( 1/2 \), and let\n\n\[ \n{X}_{n} = \\mathop{\\sum }\\limits_{{k = 1}}^{n}{\\xi }_{k},\\;\\text{ i.e.,}\\{X}_{0} = 0.\n\]\n\nHere \( \\left\\{ {X}_{n}\\right\\} \) is ca... | The mean position and mean squared deviation are\n\n\[ \n\\mathbb{E}{X}_{N} = 0,\\;\\mathbb{E}{X}_{N}^{2} = N.\n\]\n\nThe root mean squared displacement is \( \\sqrt{N} \) . | Yes |
Theorem 6.5. Let \( \\left\\{ {\\alpha }_{k}^{\\left( n\\right) }\\right\\} \) be a sequence of i.i.d. Gaussian random variables with distribution \( N\\left( {0,1}\\right) \) . Then, almost surely,\n\n\[ \n{W}_{t}^{N} = \\mathop{\\sum }\\limits_{{n = 0}}^{N}\\mathop{\\sum }\\limits_{{k \\in {I}_{n}}}{\\alpha }_{k}^{\\... | Proof. First we show that almost surely, \( {W}_{t}^{N} \) converges uniformly to some continuous function. For this purpose, we note that for the Gaussian random variable \( \\xi \\sim N\\left( {0,1}\\right) \),\n\n\[ \n\\mathbb{P}\\left( {\\left| \\xi \\right| > x}\\right) = \\sqrt{\\frac{2}{\\pi }}{\\int }_{x}^{\\in... | Yes |
Compute the expectation\n\n\\[ \n\\mathbb{E}\\exp \\left( {-\\frac{1}{2}\\int }_{0}^{1}{W}_{t}^{2}{dt}\\right) \n\\] | Solution. This is an example of a Wiener functional. Using the Karhunen-Loève expansion, we get\n\n\\[ \n\\int }_{0}^{1}{W}_{t}^{2}{dt} = \\int }_{0}^{1}\\mathop{\\sum }\\limits_{{k, l}}\\sqrt{{\\lambda }_{k}{\\lambda }_{l}}{\\alpha }_{k}{\\alpha }_{l}{\\phi }_{k}\\left( t\\right) {\\phi }_{l}\\left( t\\right) {dt}\n\\... | Yes |
Theorem 6.7. Wiener process has the following symmetry properties.\n\n(1) Time-homogeneity: For any \( s > 0,{W}_{t + s} - {W}_{s}, t \geq 0 \), is a Wiener process.\n\n(2) Symmetry: The process \( - {W}_{t}, t \geq 0 \), is a Wiener process.\n\n(3) Scaling: For every \( c > 0 \), the process \( c{W}_{t/{c}^{2}}, t \ge... | The proof is straightforward and is left as an exercise. | No |
Theorem 6.9 (Unbounded variation of the Wiener path). On any finite interval, the total variation of a Wiener path is almost surely infinite. | Proof. Because of (6.18), there is a subset \( {\Omega }_{0} \subset \Omega \) such that \( \mathbb{P}\left( {\Omega }_{0}\right) = 1 \) , and a subsequence of subdivisions, still denoted as \( \left\{ {\Delta }_{n}\right\} \), such that for any pair of rational numbers \( \left( {p, q}\right), p < q \) ,\n\n\[ \n{Q}_{... | Yes |
Theorem 6.10 (Smoothness of the Wiener path). Let \( {\Omega }_{\alpha } \) be the set of functions that are Hölder continuous with exponent \( \alpha \left( {0 < \alpha < 1}\right) \) :\n\n\[ \n{\Omega }_{\alpha } = \left\{ {f \in C\left\lbrack {0,1}\right\rbrack ,\mathop{\sup }\limits_{{0 \leq s, t \leq 1}}\frac{\lef... | The proof of Theorem 6.10 relies on the modification concept and the following Kolmogorov continuity theorem, which can be found in [RY05]. | No |
Theorem 6.14 (Wiener chaos expansion). Let \( W \) be a standard Wiener process on \( \left( {\Omega ,\mathcal{F},\mathbb{P}}\right) \). Let \( F \) be a Wiener functional in \( {L}^{2}\left( \Omega \right) \). Then \( F \) can be represented as\n\n\[ F\left\lbrack W\right\rbrack = \mathop{\sum }\limits_{{\alpha \in \m... | This is an infinite-dimensional analog of the Fourier expansion with weight function \( {\left( \sqrt{2\pi }\right) }^{-1}{e}^{-{x}^{2}/2} \) in each dimension. Its proof can be found in CM47. | No |
Lemma 7.1. Assume that \( 0 \leq S \leq T \) . The stochastic integral for the simple functions satisfies\n\n(7.7)\n\n\[ \mathbb{E}\left( {{\int }_{S}^{T}f\left( {\omega, t}\right) d{W}_{t}}\right) = 0 \]\n\n(7.8)\n\n\[ \text{ (Itô isometry) }\;\mathbb{E}{\left( {\int }_{S}^{T}f\left( t,\omega \right) d{W}_{t}\right) }... | Proof. The first property is straightforward from the independence between \( \delta {W}_{j} \mathrel{\text{:=}} {W}_{{t}_{j + 1}} - {W}_{{t}_{j}},{e}_{j}\left( \omega \right) \), and \( \delta {W}_{j} \sim N\left( {0,{t}_{j + 1} - {t}_{j}}\right) \) . For the second property we have\n\n\[ \mathbb{E}{\left( {\int }_{S}... | Yes |
For \( f \in \mathcal{V}\left\lbrack {S, T}\right\rbrack \), the Itô integral satisfies\n\n\[ \mathbb{E}\left( {{\int }_{S}^{T}f\left( {\omega, t}\right) d{W}_{t}}\right) = 0 \] | Proof. Based on Lemma 7.1, we have\n\n\[ \left| {\mathbb{E}\left( {{\int }_{S}^{T}f\left( {\omega, t}\right) d{W}_{t}}\right) }\right| = \left| {\mathbb{E}\left( {{\int }_{S}^{T}f\left( {\omega, t}\right) d{W}_{t} - {\int }_{S}^{T}{\phi }_{n}\left( {\omega, t}\right) d{W}_{t}}\right) }\right| \]\n\n\[ \leq {\left\lbrac... | Yes |
For the Itô integral we have\n\n\[ {\int }_{0}^{t}{W}_{s}d{W}_{s} = \frac{{W}_{t}^{2}}{2} - \frac{t}{2}\;\text{ a.s. } \] | Proof. Take the dyadic subdivision with mesh size \( {2}^{-n} \) on \( \left\lbrack {0, T}\right\rbrack \) . From the definition of Itô integral\n\n\[ {\int }_{0}^{t}{W}_{s}d{W}_{s} \approx \mathop{\sum }\limits_{j}{W}_{{t}_{j}}\left( {{W}_{{t}_{j + 1}} - {W}_{{t}_{j}}}\right) = \mathop{\sum }\limits_{j}\frac{2{W}_{{t}... | Yes |
Proposition 7.6. Assume that \( f \) is bounded and continuous in \( t \) for \( t \in \) \( \left\lbrack {0, T}\right\rbrack \) almost surely. Then\n\n\[ \mathop{\sum }\limits_{j}f\left( {\omega ,{t}_{j}^{ * }}\right) {\left( {W}_{{t}_{j + 1}} - {W}_{{t}_{j}}\right) }^{2} \rightarrow {\int }_{0}^{T}f\left( {\omega, s}... | Proof. A straightforward calculation gives\n\n\[ \mathbb{E}{\left( \mathop{\sum }\limits_{j}f\left( {t}_{j}\right) \delta {W}_{{t}_{j}}^{2} - \mathop{\sum }\limits_{j}f\left( {t}_{j}\right) \delta {t}_{j}\right) }^{2} \]\n\n\[ = \mathbb{E}\left( {\mathop{\sum }\limits_{{j, k}}f\left( {t}_{j}\right) f\left( {t}_{k}\righ... | Yes |
Theorem 7.7 (One-dimensional Itô formula). Let \( f \) be a twice differentiable function, and let \( {Y}_{t} = f\left( {X}_{t}\right) \) where \( {X}_{t} \) is an Itô process defined in (7.16). Then \( {Y}_{t} \) is also an Itô process and\n\n\[ d{Y}_{t} = \left( {b\left( {t,\omega }\right) {f}^{\prime }\left( {X}_{t}... | Remark 7.8. Equation (7.17) can be derived formally using Taylor expansion and the calculation rules:\n\n\[ d{t}^{2} = 0,\;{dtd}{W}_{t} = d{W}_{t}{dt} = 0,\;{\left( d{W}_{t}\right) }^{2} = {dt}. \]\n\nFirst, we have\n\n\[ d{Y}_{t} = {f}^{\prime }\left( {X}_{t}\right) d{X}_{t} + \frac{1}{2}{f}^{\prime \prime }\left( {X}... | Yes |
Theorem 7.9 (Multidimensional Itô formula). Let \( {\mathbf{X}}_{t} \) be an Itô process defined by \( d{\mathbf{X}}_{t} = \mathbf{b}\left( {\omega, t}\right) {dt} + \mathbf{\sigma }\left( {\omega, t}\right) d{\mathbf{W}}_{t} \), where \( {\mathbf{X}}_{t} \in {\mathbb{R}}^{n},\mathbf{\sigma } \in {\mathbb{R}}^{n \times... | Remark 7.10. Equation (7.20) can be derived formally using the calculation rules\n\n(7.21)\n\n\[ d{t}^{2} = 0,\;{dtd}{W}_{t}^{i} = d{W}_{t}^{i}{dt} = d{W}_{t}^{i}d{W}_{t}^{j} = 0\;\left( {i \neq j}\right) ,\;{\left( d{W}_{t}^{i}\right) }^{2} = {dt}. \]\n\nUsing Taylor expansion, we have\n\n(7.22)\n\n\[ d{Y}_{t} = \nabl... | Yes |
An example of integration by parts is\n\n\[ \n{\int }_{0}^{t}{sd}{W}_{s} = t{W}_{t} - {\int }_{0}^{t}{W}_{s}{ds} \n\] | Proof. Define \( f\left( {x, y}\right) = {xy},{X}_{t} = t,{Y}_{t} = {W}_{t} \) . Then from the multidimensional Itô formula\n\n\[ \n{df}\left( {{X}_{t},{Y}_{t}}\right) = {X}_{t}d{Y}_{t} + {Y}_{t}d{X}_{t} + d{X}_{t}d{Y}_{t} \n\]\n\nSince \( {dtd}{W}_{t} = 0 \), we obtain \( d\left( {t{W}_{t}}\right) = {td}{W}_{t} + {W}_... | Yes |
\[ {\int }_{0}^{t}d{W}_{{t}_{1}}{\int }_{0}^{{t}_{1}}d{W}_{{t}_{2}}\ldots {\int }_{0}^{{t}_{n - 1}}d{W}_{{t}_{n}} = \frac{1}{n!}{t}^{\frac{n}{2}}{H}_{n}\left( \frac{{W}_{t}}{\sqrt{t}}\right) ,\] | Proof. It is easy to verify that \[ {\int }_{0}^{t}{W}_{s}d{W}_{s} = \frac{t}{2!}{H}_{2}\left( \frac{{W}_{t}}{\sqrt{t}}\right) \] where \( {H}_{2}\left( x\right) = {x}^{2} - 1 \) is the second-order Hermite polynomial. In the same fashion, we have \[ {\int }_{0}^{t}\left( {{\int }_{0}^{s}{W}_{u}d{W}_{u}}\right) d{W}_{s... | No |
Theorem 7.13 (Burkholder-Davis-Gundy inequality). For any \( m > 0 \) , there exist constants \( {k}_{m},{K}_{m} > 0 \) such that\n\n\[ \n{k}_{m}\mathbb{E}\left( {Q}_{T}^{m}\right) \leq \mathbb{E}\left( {\mathop{\sup }\limits_{{0 \leq t \leq T}}{\left| {\mathbf{M}}_{t}\right| }^{2m}}\right) \leq {K}_{m}\mathbb{E}\left(... | See [KS91] for detailed proof. | No |
Theorem 7.14. Assume that the coefficients \( \mathbf{b} \in {\mathbb{R}}^{n},\mathbf{\sigma } \in {\mathbb{R}}^{n \times m} \) satisfy the global Lipschitz and linear growth conditions:\n\n(7.28)\n\n\[ \left| {\mathbf{b}\left( {\mathbf{x}, t}\right) - \mathbf{b}\left( {\mathbf{y}, t}\right) }\right| + \left| {\mathbf{... | Proof. We will only consider the one-dimensional case. The high-dimensional case is similar. First we prove uniqueness. Let \( {X}_{t},{\widehat{X}}_{t} \in \mathcal{V}\left\lbrack {0, T}\right\rbrack \) be solutions of the SDEs (7.27) with the same initial value \( {X}_{0} \) . Then\n\n\[ {X}_{t} - {\widehat{X}}_{t} =... | Yes |
The Ornstein-Uhlenbeck (OU) process is the solution of the following linear SDE with the additive noise:\n\n\[ d{X}_{t} = - \gamma {X}_{t}{dt} + {\sigma d}{W}_{t},{\left. \;{X}_{t}\right| }_{t = 0} = {X}_{0}. \] | Solution. Multiplying both sides of (7.38) by \( {e}^{\gamma t} \) and integrating, we get\n\n\[ {e}^{\gamma t}{X}_{t} - {X}_{0} = {\int }_{0}^{t}\sigma {e}^{\gamma s}d{W}_{s} \]\n\nThus\n\n\[ {X}_{t} = {e}^{-{\gamma t}}{X}_{0} + \sigma {\int }_{0}^{t}{e}^{-\gamma \left( {t - s}\right) }d{W}_{s} \]\n\nis the solution. ... | Yes |
The geometric Brownian motion is the solution of a linear SDE with multiplicative noise\n\n\[ d{N}_{t} = r{N}_{t}{dt} + \alpha {N}_{t}d{W}_{t},{\left. \;{N}_{t}\right| }_{t = 0} = {N}_{0}. \] | Solution. Dividing both sides by \( {N}_{t} \), we have \( d{N}_{t}/{N}_{t} = {rdt} + {\alpha d}{W}_{t} \) . Applying Itô’s formula to \( \log {N}_{t} \), we get\n\n\[ d\left( {\log {N}_{t}}\right) = \frac{1}{{N}_{t}}d{N}_{t} - \frac{1}{2{N}_{t}^{2}}{\left( d{N}_{t}\right) }^{2} \]\n\n\[ = \frac{1}{{N}_{t}}d{N}_{t} - \... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.