STAT151A Homework 3: Due February 23rd

\(\,\)

1 Normal intervals

For these problems, assume I give you a computer program that can compute the function \(\Phi(z) = \mathbb{P}\left(\tilde{z} \le z\right)\) where \(\tilde{z}\) is a standard scalar-valued random variable.

Let \(\tilde{x}\) denote a scalar-valued \(N(\mu, \sigma^2)\) random variable. Using only \(\Phi(z)\) and elementary arithmetic, construct functions that evaluate the following:

(a)

\(a \mapsto \mathbb{P}\left(\tilde{x} \le a\right)\)

(b)

\(b \mapsto \mathbb{P}\left(\tilde{x} \ge b\right)\)

(c)

\(a, b \mapsto \mathbb{P}\left(b \le \tilde{x} \le a\right)\)

(d)

\(a \mapsto \mathbb{P}\left(\left|\tilde{x}\right| \le a\right)\)

(e)

\(a \mapsto \mathbb{P}\left(\left|\tilde{x}\right| \ge a\right)\)

(f)

\(a \mapsto \mathbb{P}\left(\left|\tilde{x}\right| > a\right)\)

(g)

\(a \mapsto \mathbb{P}\left(\left|\tilde{x}\right| = a\right)\)

2 Multivariate CLT

Let \(\tilde{\boldsymbol{x}}_n\) denote an IID sequence of random variables in \(\mathbb{R}^{P}\) (not necessarily normal), each with zero mean and finite covariance matrix \(\boldsymbol{\Sigma}\). Let \(\boldsymbol{a}\in \mathbb{R}^{P}\) denote a fixed vector.

(a)

Using the univariate CLT, find the limiting distribution of

\[ \frac{1}{\sqrt{N}} \sum_{n=1}^N\boldsymbol{a}^\intercal\tilde{\boldsymbol{x}_n}. \]

(b)

Using the multivariate CLT and the continuous mapping theorem, find the limiting distribution of

\[ \boldsymbol{a}^\intercal\left( \frac{1}{\sqrt{N}} \sum_{n=1}^N\tilde{\boldsymbol{x}_n}\right). \]

(c)

Now, suppose that \(P = 2\) and

\[\boldsymbol{\Sigma}= \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}. \]

Note that we can write \[ \tilde{\boldsymbol{x}}_n = \begin{pmatrix} \tilde{\boldsymbol{x}}_{n1} \\ \tilde{\boldsymbol{x}}_{n2} \end{pmatrix}, \] where \(\tilde{\boldsymbol{x}}_{n1}\) and \(\tilde{\boldsymbol{x}}_{n2}\) are scalars. Find the limiting distributions of each of the following expressions:

\[ \begin{aligned} \frac{1}{\sqrt{N}} \sum_{n=1}^N\tilde{\boldsymbol{x}}_{n1} \rightarrow& \textrm{?}\\ \frac{1}{\sqrt{N}} \sum_{n=1}^N\tilde{\boldsymbol{x}}_{n2} \rightarrow& \textrm{?} \\ \frac{1}{\sqrt{N}} \sum_{n=1}^N(\tilde{\boldsymbol{x}}_{n1} + \tilde{\boldsymbol{x}}_{n2}) \rightarrow& \textrm{?} \end{aligned} \]

(This result demonstrates why it’s not enough to only look at the marginal distribution of the vector components when using a multivariate CLT.)

3 Valid covariance matrices

Suppose I were to tell you that the vector-valued random variable \(\boldsymbol{x}\) has a covariance matrix \(\mathrm{Cov}\left(\boldsymbol{x}\right) = \boldsymbol{\Sigma}\) where \(\boldsymbol{\Sigma}\) is not positive semi-definite (i.e., \(\boldsymbol{\Sigma}\) has at least one negative eigenvalue). Show that, if this were true, you could construct a scalar-valued random variable with negative variance, which is impossible.

(It follows from this argument every covariance matrix must be postive semi-definite.)