Math Problems Vault

Problem 34.1

Construct a stationary sequence whose covariance function $b(n)$ equals 1 for even $n$ and 0 for odd $n$ .

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Remark.

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Let $\{ \xi_{n}\}_{n=-\infty }^{\infty }$ be a stationary sequence with covariance function $b(n)$ . Then there exists a nonnegative finite measure $F(t)$ such that for every $n$ the equality $b(n) = \int_{[-\pi ,\pi )} e^{int} F(dt)$ holds. The function $F(x) = F(-\infty , x)$ is called the spectral function of the sequence $\{ \xi_{n}\}$ . If the measure $F$ is absolutely continuous, its derivative is called the spectral density of the sequence $\{ \xi_{n}\}$ . Moreover, there exists a process $\mu (t)$ with zero mean, orthogonal increments, and structure function $F(t)$ , such that for every $n$ the spectral representation $\xi_{n} = \int_{[-\pi ,\pi )} e^{int}\mu (dt)$ holds.

Problem 34.2

Let $\xi$ be a random variable with the standard normal distribution. Set $\xi_{n} = \xi$ for every integer $n$ . Is the random sequence $\{ \xi_{n}\}$ stationary? If so, find its spectral measure and spectral representation.

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Problem 34.3

Let $\xi$ and $\eta$ be independent random variables, each taking the values $-1$ and $1$ with probabilities $1/2$ . Set $\xi_{2n+1} = \xi$ and $\xi_{2n} = \eta$ for every integer $n$ . Is the random sequence $\{ \xi_{n}\}$ stationary? If so, find its spectral measure and spectral representation.

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Problem 34.4

Let $\xi$ and $\eta$ be uncorrelated random variables, each having the standard normal distribution. Set $\xi_{2n+1} = \xi$ and $\xi_{2n} = \eta$ for every integer $n$ . Is the random sequence $\{ \xi_{n}\}$ stationary? If so, find its spectral measure and spectral representation.

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Problem 34.5

Let $\ldots , \eta_{-1}, \eta_{0}, \eta_{1}, \ldots$ be independent random variables uniformly distributed on the interval $[-2, 2]$ . Set $\xi_{n} = \eta_{n-1}\eta_{n}$ for every integer $n$ . Is the random sequence $\{ \xi_{n}\}$ stationary? If so, find the decomposition of this sequence into deterministic and purely nondeterministic components, and construct the best linear predictor of $\xi_{1}$ from the observations $\{ \xi_{n}, n \leq 0\}$ .

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Problem 34.6

Let $\ldots , \eta_{-1}, \eta_{0}, \eta_{1}, \ldots$ be uncorrelated random variables with zero means and unit variances. Set $\xi_{n} = \eta_{n-3}+\eta_{n-2}+\eta_{n-1}+\eta_{n}$ for every integer $n$ . Is the random sequence $\{ \xi_{n}\}$ stationary? If so, find the decomposition of this sequence into deterministic and purely nondeterministic components, and construct the best linear predictor of $\xi_{2}$ from the observations $\{ \xi (n), n \leq 0\}$ .

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Problem 34.7

Let $\ldots , \eta_{-1}, \eta_{0}, \eta_{1}, \ldots$ be independent random variables with the standard normal distribution. Set $\xi_{n} = \eta_{n-2}+\eta_{n-1}+\eta_{n}$ . Is the random sequence $\{ \xi_{n}\}$ stationary? If so, find the decomposition of this sequence into deterministic and purely nondeterministic components, and construct the best linear predictor of $\xi_{1}$ from the observations $\{ \xi_{n}, n \leq 0\}$ .

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Problem 34.8

Let $\ldots , \eta_{-1}, \eta_{0}, \eta_{1}, \ldots$ and $\xi$ be independent random variables uniformly distributed on the interval $[-1, 1]$ . Set $\xi_{n} = \eta_{n-1}+\eta_{n}+\xi$ for every integer $n$ . Is the random sequence $\{ \xi_{n}\}$ stationary? If so, find the decomposition of this sequence into deterministic and purely nondeterministic components, and construct the best linear predictor of $\xi_{1}$ from the observations $\{ \xi_{n}, n \leq 0\}$ .

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Problem 34.9

(a)(b)

Let $w(u)$ be a standard Wiener process. Find the distributions of the following random sequences:

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(a)

$\xi_{n} = \displaystyle \int_{-\pi }^{\pi } e^{inu}\, dw(\pi +u)$ ;

(b)

$\xi_{n} = \displaystyle \int_{-\pi }^{\pi } e^{2inu}\, dw(\pi +u)$ .