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Chapter 15: Problem 2

(Cholesky factorization) Let \(C\) be a positive definite symmetric real \(d \times d\) matrix. Then there exists a real \(d \times d\) matrix \(A=\left(a_{k l}\right)\) with \(A \cdot A^{T}=C\). The matrix \(A\) can be chosen to be lower triangular. Let \(W:=\left(W_{1}, \ldots, W_{d}\right)^{T}\), where \(W_{1}, \ldots, W_{d}\) are independent and \(\mathcal{N}_{0,1}\)-distributed. Define \(X:=A W+\mu\). Show that \(X \sim \mathcal{N}_{\mu, C}\).

Short Answer

Expert verified
Yes, \(X\) is normally distributed with mean \(\mu\) and covariance \(C\).

Step by step solution

01

Understand the Problem Statement

We need to prove that if \(C\) is a positive definite symmetric matrix, and \(A\) is its Cholesky factor such that \(A \cdot A^T = C\), then \(X = AW + \mu\) is normally distributed with mean \(\mu\) and covariance matrix \(C\). Also, we know \(W\) components are independent and standard normally distributed.
02

Review Basic Properties of Cholesky Decomposition

Recall that a positive definite symmetric matrix \(C\) can be decomposed into the form \(C = A \cdot A^T\), where \(A\) is a lower triangular matrix in Cholesky factorization. This ensures \(C\) is positive definite and symmetric.
03

Define the Distribution of Vector W

Given that \(W\) is a vector of independent \(\mathcal{N}(0,1)\) random variables, its covariance matrix is the identity matrix \(I\), and its mean vector is \(0\).
04

Express X in Terms of W and A

The vector \(X = AW + \mu\). This means \(X\) is a linear transformation of the standard normal vector \(W\), shifted by a constant vector \(\mu\). We know linear transformations of normals remain normal.
05

Calculate the Mean of X

The expected value of \(X\) is \(\mathbb{E}(X) = \mathbb{E}(AW + \mu) = A\mathbb{E}(W) + \mu = 0 + \mu = \mu\), since \(\mathbb{E}(W) = 0\).
06

Calculate the Covariance of X

The covariance of \(X\) is \(\text{Cov}(X) = \text{Cov}(AW) = A\text{Cov}(W)A^T = AI A^T = AA^T = C\). Since \(W\) is standard normal, \(\text{Cov}(W) = I\).
07

Conclusion

Since \(X\) has mean \(\mu\) and covariance matrix \(C\), by definition, \(X \sim \mathcal{N}(\mu, C)\). Therefore, the statement is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Positive Definite Matrix
A positive definite matrix is an essential concept in linear algebra and statistics. It is a symmetric matrix with attributes that make it quite special and useful. To understand a positive definite matrix, remember:
  • All its eigenvalues are greater than zero. This ensures the matrix is invertible and non-singular.
  • For any non-zero vector \( x \), the quadratic form \( x^T C x \) is always positive. This property is crucial in optimization and stability analysis.
  • A positive definite matrix is symmetric, which means \( C = C^T \).
These properties are used in various applications such as optimization, statistics, and the Cholesky factorization. Cholesky factorization particularly relies on the matrix being positive definite, allowing it to be decomposed into the product of a lower triangular matrix and its transpose. This decomposition is pivotal in solving matrix equations and simplifying complex problems.
Normal Distribution
The normal distribution, also known as Gaussian distribution, is a continuous probability distribution characterized by a symmetrical bell-shaped curve. It鈥檚 foundational in statistics and many fields due to its natural occurrence in numerous phenomena. Key features include:
  • Defined by its mean \( \mu \) and variance \( \sigma^2 \), where the mean indicates the central tendency, and the variance measures the spread.
  • In the context of standard normal distribution, the mean is 0 and the variance is 1, represented as \( \mathcal{N}(0, 1) \).
  • Properties like symmetry about the mean, and the total area under the curve equating to 1, make it a powerful tool for statistical inference.
In the exercise, the vector \( W \) composed of independent standard normal variables is crucial because transformations of normal distributions remain normal. Understanding how such distributions transform is vital for synthetic data generation, hypothesis testing, and modeling.
Covariance Matrix
A covariance matrix provides a detailed summary of the relationships between multiple variables. Particularly in multivariate analysis, it is invaluable. Here's what you should know:
  • Each element \( \text{Cov}(X_i, X_j) \) in the matrix represents the covariance between a pair of random variables.
  • The diagonal entries represent the variances of each individual variable, while the off-diagonal entries represent the covariances.
  • A covariance matrix operates as a generalization of variance to higher dimensions for multivariate data, and it is symmetric.
In our exercise, the covariance matrix \( C \) describes the spread and interdependence of the components of vector \( X \). This matrix is decomposed using Cholesky factorization, facilitating the understanding and manipulation of multidimensional data sets. Recognizing how covariance impacts data is critical in various statistical tasks, including principal component analysis and portfolio optimization.

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Most popular questions from this chapter

Let \(X\) be a real random variable with characteristic function \(\varphi\). Assume that there is a sequence \(\left(t_{n}\right)_{n \in \mathbb{N}}\) of real numbers such that \(\left|t_{n}\right| \downarrow 0\) and \(\left|\varphi\left(t_{n}\right)\right|=1\) for any \(n\). Show that there exists a \(b \in \mathbb{R}\) such that \(X=b\) almost surely. If in addition, \(\varphi\left(t_{n}\right)=1\) for all \(n\), then \(X=0\) almost surely.

\( Show that, for \)\alpha>2\(, the function \)\phi_{\alpha}(t)=e^{-|t|^{\alpha}}$ is not a characteristic function. Hint: Assume the contrary and show that the corresponding random variable would have variance zero.

Let \(\varphi\) be the characteristic function of the \(d\)-dimensional random variable \(X\). Assume that \(\varphi(t)=1\) for some \(t \neq 0\). Show that \(\mathbf{P}\left[X \in H_{t}\right]=1\), where $$ \begin{aligned} H_{t} &=\left\\{x \in \mathbb{R}^{d}:\langle x, t\rangle \in 2 \pi \mathbb{Z}\right\\} \\ &=\left\\{y+z \cdot\left(2 \pi t /\|t\|_{2}^{2}\right): z \in \mathbb{Z}, y \in \mathbb{R}^{d} \text { with }\langle y, t\rangle=0\right\\} \end{aligned} $$ Infer that \(\varphi(t+s)=\varphi(s)\) for all \(s \in \mathbb{R}^{d}\).

Let \(X\) be a real random variable with characteristic function \(\varphi\). \(X\) is called lattice distributed if there are \(a, d \in \mathbb{R}\) such that \(\mathbf{P}[X \in a+d \mathbb{Z}]=1\). Show that \(X\) is lattice distributed if and only if there exists a \(u \neq 0\) such that \(|\varphi(u)|=1\)

Let \((\Omega, \tau)\) be a separable topological space that satisfies the \(T_{3 \frac{1}{2}}\) separation axiom: For any closed set \(A \subset \Omega\) and any point \(x \in \Omega \backslash A\), there exists a continuous function \(f: \Omega \rightarrow[0,1]\) with \(f(x)=0\) and \(f(y)=1\) for all \(y \in A\). (Note in particular that every metric space is a \(T_{3 \frac{1}{2}}\)-space.) Show that \(\sigma\left(C_{b}(\Omega)\right)=\mathcal{B}(\Omega)\); that is, the Borel \(\sigma\)-algebra is generated by the bounded continuous functions \(\Omega \rightarrow \mathbb{R}\).
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