本章导读 本章把布朗运动刻画为高斯过程。§2.1 多元正态分布：2.1.1 定义（标准正态随机向量 Def 2.1；多元正态的两个等价定义 Def 2.2——定义 1：\(\mathbf X=A\mathbf Z+\mu\)；定义 2：任意线性组合为一元正态）；2.1.2 两定义的等价性（卷积 Def 2.4、引理 2.1：独立正态的线性组合仍正态 [Ch 1 构造中用到]）；2.1.3 多元正态的密度。§2.2 高斯过程（Def 2.5；命题 2.1：标准 BM 是高斯过程，均值 \(0\)、协方差 \(\Gamma_{st}=s\wedge t\)；注 2.2：可据此把 BM 等价重定义为零均值、协方差 \(s\wedge t\)、路径以概率 1 连续的高斯过程）。无图。

定义 2.1、2.2（多元正态的两个等价定义）/ Definitions 2.1, 2.2 定义 2.1（标准正态随机向量）：实随机向量 \(\mathbf Z=(Z_1,Z_2,\dots,Z_l)'\) 称为标准正态随机向量，若其所有分量 \(Z_n\)（\(n=1,\dots,l\)）都是 i.i.d. 一元标准正态。定义 2.2（多元正态分布·定义 1）：实随机向量 \(\mathbf X=(X_1,\dots,X_k)'\) 称为正态随机向量、其分量多元正态分布，当且仅当存在标准正态随机向量 \(\mathbf Z_{l\times1}\) 与矩阵 \(A_{k\times l}\) 使得 \(\mathbf X=A\mathbf Z+\mu\)，其中 \(\mu\) 是 \(\mathbf X\) 的均值向量。换言之，多元正态向量的任一分量都可表为若干 i.i.d. 标准正态变量的线性组合加其均值，即 \(\forall i\)，\(X_i=\mu_i+a_1 Z_1+a_2 Z_2+\dots+a_l Z_l\)（某 \(a_1,\dots,a_l\in\mathbb R\)）。定义 2.2（多元正态分布·定义 2）：\(\mathbf X=(X_1,\dots,X_k)'\) 称为正态随机向量、分量多元正态分布，当且仅当其分量的任意线性组合都是一元正态，即 \(\forall\mathbf b=(b_1,\dots,b_k)'\in\mathbb R^k\)，\(Y=\mathbf b'\mathbf X=b_1 X_1+\dots+b_k X_k\sim\mathcal N(\mu_Y,\sigma_Y^2)\)。Definition 2.1 (Standard normal random vector): a real random vector \(\mathbf Z=(Z_1,Z_2,\dots,Z_l)'\) is called a standard normal random vector if all of its components \(Z_n\) (\(n=1,\dots,l\)) are i.i.d. univariate standard normal. Definition 2.2 (Multivariate normal distribution-1): a real random vector \(\mathbf X=(X_1,\dots,X_k)'\) is called a normal random vector and its components are multivariate normally distributed if and only if there exist a standard normal random vector \(\mathbf Z_{l\times1}\) and a matrix \(A_{k\times l}\) such that \(\mathbf X=A\mathbf Z+\mu\), where \(\mu\) is the mean vector of \(\mathbf X\). In other words, any component of a multivariate normally distributed random vector can be represented by a linear combination of some i.i.d. standard normal variables plus its mean, i.e. for all \(i\), \(X_i=\mu_i+a_1 Z_1+a_2 Z_2+\dots+a_l Z_l\) for some \(a_1,\dots,a_l\in\mathbb R\). Definition 2.2 (Multivariate normal distribution-2): \(\mathbf X=(X_1,\dots,X_k)'\) is called a normal random vector with multivariate normally distributed components if and only if any linear combination of its components is (univariate) normally distributed, i.e. for all \(\mathbf b=(b_1,\dots,b_k)'\in\mathbb R^k\), \(Y=\mathbf b'\mathbf X=b_1 X_1+\dots+b_k X_k\sim\mathcal N(\mu_Y,\sigma_Y^2)\).

定义 2.4 与引理 2.1 / Definition 2.4 and Lemma 2.1 定义 2.4（卷积）：卷积是定义在 \(\mathbb R\) 上两函数 \(f,g\) 的算子 $$：\((f*g)(x)\equiv\int_{-\infty}^\infty f(t)g(x-t)\,dt\)。引理 2.1：一组独立正态分布随机变量 \((X_1,X_2,\dots,X_s)\) 的任意线性组合都是正态分布。注 2.1：卷积的直觉是——\(Y=y\) 的概率密度应等于 \(X_i,X_j\) 取值的所有可能组合之"和"。Definition 2.4 (Convolution): a convolution is the operator $$ for two functions \(f,g\) on \(\mathbb R\): \((f*g)(x)\equiv\int_{-\infty}^\infty f(t)g(x-t)\,dt\). Lemma 2.1: any linear combination of a set of independent normally distributed random variables \((X_1,X_2,\dots,X_s)\) is normally distributed. Remark 2.1: the intuition for the convolution is that the probability density of \(Y=y\) should be the sum of the possibilities of all combinations of values of \(X_i,X_j\).

引理 2.1 与两定义等价性的证明 / Proof of Lemma 2.1 and the equivalence 引理 2.1（归纳）：先看两个独立正态 \(X_i\sim\mathcal N(\mu_i,\sigma_i^2)\)、\(X_j\sim\mathcal N(\mu_j,\sigma_j^2)\) 的简单和 \(Y=X_i+X_j\)。其密度为卷积 \(g(y)=(f_i*f_j)(y)=\int_{-\infty}^\infty f_i(t)f_j(y-t)\,dt\)，经配方（Gaussian 卷积的标准计算）化简后正是 \(\mathcal N(\mu_i+\mu_j,\sigma_i^2+\sigma_j^2)\) 的密度。再看任意线性组合 \(V=aX_i+bX_j\)：记 \(W_i=aX_i\sim\mathcal N(a\mu_i,a^2\sigma_i^2)\)、\(W_j=bX_j\sim\mathcal N(b\mu_j,b^2\sigma_j^2)\)，二者独立，故 \(V=W_i+W_j\) 一元正态。最后对 \(V\) 与另一 \(cX_k\)（\(c\in\mathbb R\)）重复此过程，即证任意线性组合一元正态。\(\blacksquare\) 两定义等价：由定义 2（定义 2.2-2）每个 \(\mathbf X\) 元素是 \(\mathbf Z\) 元素的线性组合，故 \(\mathbf X\) 元素的任意线性组合仍是 \(\mathbf Z\) 元素的线性组合，由引理 2.1 一元正态，给出定义 2.2-2。反向：引理 2.1 的证明重度依赖 \(X_i,X_j\) 的独立性与正态性，故要使定义 2 的性质成立，需每个 \(\mathbf X\) 元素是若干独立正态变量的线性组合——它可化为 i.i.d. 标准正态变量的线性组合，给出定义 2.2-1。\(\blacksquare\)Lemma 2.1 (by induction): first consider the simple sum \(Y=X_i+X_j\) of two independent normals \(X_i\sim\mathcal N(\mu_i,\sigma_i^2)\), \(X_j\sim\mathcal N(\mu_j,\sigma_j^2)\). Its density is the convolution \(g(y)=(f_i*f_j)(y)=\int_{-\infty}^\infty f_i(t)f_j(y-t)\,dt\), which after completing the square (the standard Gaussian-convolution computation) is exactly the density of \(\mathcal N(\mu_i+\mu_j,\sigma_i^2+\sigma_j^2)\). Then for an arbitrary linear combination \(V=aX_i+bX_j\): denote \(W_i=aX_i\sim\mathcal N(a\mu_i,a^2\sigma_i^2)\), \(W_j=bX_j\sim\mathcal N(b\mu_j,b^2\sigma_j^2)\), which are independent, so \(V=W_i+W_j\) is univariate normal. Finally, repeating with \(V\) and another \(cX_k\) (\(c\in\mathbb R\)) proves any linear combination is univariate normal. \(\blacksquare\) Equivalence: by Definition 2 (Definition 2.2-2) each element of \(\mathbf X\) is a linear combination of elements in \(\mathbf Z\), so any linear combination of elements in \(\mathbf X\) is again a linear combination of elements in \(\mathbf Z\), which is univariate normal by Lemma 2.1, giving Definition 2.2-2. Conversely, the proof of Lemma 2.1 heavily depends on the independence and normality of \(X_i,X_j\), so for the property in Definition 2 to hold we need each element in \(\mathbf X\) to be a linear combination of some independent normal random variables — which can be transformed to a linear combination of i.i.d. standard normal random variables, giving Definition 2.2-1. \(\blacksquare\)

多元正态密度 / Density of the multivariate normal 多元正态向量 \(\mathbf X=(X_1,\dots,X_k)'\sim\mathcal N(\mu,\Omega)\) 的联合密度为The joint density of a multivariate normally distributed vector \(\mathbf X=(X_1,\dots,X_k)'\sim\mathcal N(\mu,\Omega)\) is

$$f_{\mathbf X}(x_1,x_2,\dots,x_k)=\frac1{\sqrt{(2\pi)^k|\Omega|}}\,e^{-\frac12(\mathbf x-\mu)'\Omega^{-1}(\mathbf x-\mu)}$$

这很直观，因为 \((\mathbf x-\mu)'\Omega^{-1}(\mathbf x-\mu)\sim\chi_k^2=z_1^2+z_2^2+\dots+z_k^2\)，其中 \(z_i\) 是独立标准正态。which is very intuitive since \((\mathbf x-\mu)'\Omega^{-1}(\mathbf x-\mu)\sim\chi_k^2=z_1^2+z_2^2+\dots+z_k^2\), where the \(z_i\) are independent standard normal.

定义 2.5 与命题 2.1 / Definition 2.5 and Proposition 2.1 定义 2.5（高斯过程）：过程 \(\{X_t\}\) 是高斯过程，若其任意有限子集 \(\{X_{t_1},\dots,X_{t_n}\}\) 都服从联合正态分布。高斯过程由均值 \(m_t=\mathbb E[X_t]\) 与协方差 \(\Gamma_{st}=\mathrm{Cov}(X_s,X_t)\) 刻画。命题 2.1：标准布朗运动 \(\{B_t\}\) 是高斯过程。其均值 \(m_t=\mathbb E[B_t]=0\)，协方差（设 \(sDefinition 2.5 (Gaussian Process): a process \(\{X_t\}\) is a Gaussian process if any of its finite subcollections \(\{X_{t_1},\dots,X_{t_n}\}\) has a joint normal distribution. A Gaussian process is characterized by its mean \(m_t=\mathbb E[X_t]\) and covariance \(\Gamma_{st}=\mathrm{Cov}(X_s,X_t)\). Proposition 2.1: a standard Brownian motion \(\{B_t\}\) is a Gaussian process. Its mean is \(m_t=\mathbb E[B_t]=0\), and its covariance (for \(s

命题 2.1 证明 / Proof of Proposition 2.1 任取子集 \(\{B_{t_1},\dots,B_{t_n}\}\)，只需证其联合正态。定义 \(Z_i=\frac{B_{t_i}-B_{t_{i-1}}}{\sqrt{t_i-t_{i-1}}}\)，由标准 BM 性质 \(Z_i\) 独立标准正态；而每个 \(B_{t_j}\) 都是 \(\{Z_1,\dots,Z_n\}\) 的线性组合，故由定义 2.2，\(\{B_{t_1},\dots,B_{t_n}\}\) 联合正态。\(\blacksquare\)Take any subcollection \(\{B_{t_1},\dots,B_{t_n}\}\); we only need to show it has a joint normal distribution. Define \(Z_i=\frac{B_{t_i}-B_{t_{i-1}}}{\sqrt{t_i-t_{i-1}}}\); by the standard-BM property the \(Z_i\) are independent standard normal, and each \(B_{t_j}\) is a linear combination of \(\{Z_1,\dots,Z_n\}\), so by Definition 2.2, \(\{B_{t_1},\dots,B_{t_n}\}\) is jointly normal. \(\blacksquare\)

注 2.2（BM 的等价重定义 / Remark 2.2） 由命题 2.1，可把布朗运动等价地重定义为一个零均值（中心化）高斯过程，其协方差 \(\Gamma_{st}=s\wedge t\)（即 \(s\) 与 \(t\) 中较小者），且路径以概率 1 连续。By Proposition 2.1, Brownian motion can be equivalently redefined as a centered (mean-zero) Gaussian process with covariance \(\Gamma_{st}=s\wedge t\) (the smaller of \(s\) and \(t\)) and paths continuous with probability one.