Math Problems Vault

Problem 9.3.1

Let $X_{n} = A \cos (n \lambda )+B \sin (n \lambda )$ where $A$ and $B$ are uncorrelated random variables with zero means and unit variances. Show that $X$ is stationary with a spectrum containing exactly one point.

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Problem 9.3.2

Let $U$ be uniformly distributed on $(-\pi , \pi )$ , and let $V$ be independent of $U$ with distribution function $F$ . Show that $X_{n} = e^{i(U-V n)}$ defines a stationary (complex) sequence with spectral distribution function $F$ .

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Problem 9.3.3

Find the autocorrelation function of the stationary process $\left\{ X(t):-\infty < t < \infty \right\}$ whose spectral density function is:

(i) $N(0,1)$ ,

(ii) $f(x) = \frac{1}{2} e^{-\left|x\right|},-\infty < x < \infty$ .

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Problem 9.3.4

Let $X_{1}, X_{2}, \ldots$ be a real-valued stationary sequence with zero means and autocovariance function $c(m)$ . Show that

\operatorname {Var}\left[\frac{1}{n} \sum _{j = 1}^{n} X_{j}\right] = c(0) \int _{(-\pi , \pi ]}\left(\frac{\sin (n \lambda / 2)}{n \sin (\lambda / 2)}\right)^{2} d F(\lambda )

where $F$ is the spectral distribution function. Deduce that $n^{-1} \sum_{j = 1}^{n} X_{j} \xrightarrow {\text{ m.s. }} 0$ if and only if $F(0)-F(0-) = 0$ , and show that

c(0)\left\{ F(0)-F(0-)\right\} = \lim _{n \rightarrow \infty } \frac{1}{n} \sum _{j = 0}^{n-1} c(j)

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Problem 9.3.5

(a)(b)(c)

Let $Z = \left\{ Z_{n}: n \in \mathbb {Z}\right\}$ be a sequence of uncorrelated random variables with means 0 and variances 1, and suppose the sequence $X = \left\{ X_{n}: n \in \mathbb {Z}\right\}$ satisfies $X_{n} = \phi X_{n-1}+Z_{n}+\theta Z_{n-1}$ .

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(a)

For what values of $\phi , \theta$ is $X$ stationary?

(b)

For what values of $\phi , \theta$ does $Z_{n}$ have a series representation in terms of the $X_{m}$ ?

(c)

Show that the autocorrelation function $\rho$ of $X$ satisfies

\rho (1) = \frac{(\phi +\theta )(1+\phi \theta )}{1+2 \phi \theta +\theta ^{2}}

and find $\rho (k)$ for $k > 1$ . Hence evaluate the spectral density in a finite form.

Problem 9.3.6

Let $Z = \left\{ Z_{n}: n \in \mathbb {Z}\right\}$ be a sequence of uncorrelated random variables with means 0 and variances $1$ , and suppose $X_{n} = \phi X_{n-1}+\theta X_{n-2}+Z_{n}$ for $n \in \mathbb {Z}$ , where $\phi^{2}+4 \theta > 0$ . Suppose $d_{k}$ is a real sequence such that

X_{n} = \sum _{k = 0}^{\infty } d_{k} Z_{n-k}

Show that $d_{k} = a_{1} r_{1}^{k}+a_{2} r_{2}^{k}$ , where $a_{1}+a_{2} = 1$ and $a_{1} r_{1}+a_{2} r_{2} = \phi$ . Show further that

a_{1} = \frac{r_{1}}{r_{1}-r_{2}}, \quad a_{2} = \frac{-r_{2}}{r_{1}-r_{2}}, \quad r_{1} r_{2} = -\theta , \quad r_{1}^{2}+r_{2}^{2} = \phi ^{2}+2 \theta

Hence find $\mathbb {E}\left[X_{n}^{2}\right]$ in terms of $\phi$ and $\theta$ when $\left|r_{1}\right|,\left|r_{2}\right| < 1$ .

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Problem 9.3.7

Let $X = \left\{ X_{n}: n \in \mathbb {Z}\right\}$ be a stationary process with zero means and variances $\sigma^{2}$ , and let $\widehat{X}_{n+1}$ be the best linear predictor of $X_{n+1}$ given $\left\{ X_{r}: r \leq n\right\}$ . (See Section 9.2.) The Kolmogorov-Szegő formula states that

\operatorname {Var}\left[X_{n+1}-\widehat{X}_{n+1}\right] = \exp \left\{ \frac{1}{2 \pi } \int _{-\pi }^{\pi } \log \left(2 \pi \sigma ^{2} f(\lambda )\right) d \lambda \right\}

where $f$ is the spectral density of $X$ . Verify this for the autoregressive scheme of Example (9.3.23).

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