Math Problems Vault

Problem 9.6.1

Show that the function $c(s, t) = \min \left\{ s, t\right\}$ is positive definite. That is, show that

\sum _{j, k = 1}^{n} c\left(t_{k}, t_{j}\right) z_{j} \bar{z}_{k} > 0

for all $0 \leq t_{1} < t_{2} < \cdots < t_{n}$ and all complex numbers $z_{1}, z_{2}, \ldots , z_{n}$ at least one of which is non-zero.

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Problem 9.6.2

Let $X_{1}, X_{2}, \ldots$ be a stationary Gaussian sequence with zero means and unit variances which satisfies the Markov property. Find the spectral density function of the sequence in terms of the constant $\rho = \operatorname {Cov}\left[X_{1}, X_{2}\right]$ .

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Problem 9.6.3

Show that a Gaussian process is strongly stationary if and only if it is weakly stationary.

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Problem 9.6.4

(a)(b)

Let $X$ be a stationary Gaussian process with zero mean, unit variance, and autocovariance function $c(t)$ . Find the autocovariance functions of the processes

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(a)

$X^{2} = \left\{ X(t)^{2}:-\infty < t < \infty \right\}$ ,

(b)

$X^{3} = \left\{ X(t)^{3}:-\infty < t < \infty \right\}$ .

Problem 9.6.5

(a)(b)

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(a)

Let $W$ be a standard Wiener process, and $T:[0, \infty ) \rightarrow [0, \infty )$ a non-random, rightcontinuous, non-decreasing function with $T(0) = 0$ . Show that $X(t): = W(T(t))$ defines a Gaussian process. Find the characteristic function of $X(t)$ and the covariance function $c(s, t) = \operatorname {Cov}\left[X(s), X(t)\right]$ .

(b)

Let $W$ be a Wiener process with $\operatorname {Var}\left[W(t)\right] = 2 t$ , and let $T$ be an independent $\alpha$ -stable subordinator, where $0 < \alpha < 1$ . Show that $X(t) = W(T(t))$ is a symmetric ( $2 \alpha )$ -stable Lévy process.

Problem 9.6.6

Let $X(s, t)$ be a two-parameter, zero-mean, Gaussian process on the positive quadrant $[0, \infty )^{2}$ of $\mathbb {R}^{2}$ , with covariance function $\operatorname {Cov}\left[X(s, t), X(u, v)\right] = (s \wedge u)(t \wedge v)$ , where $x \wedge y = \min \left\{ x, y\right\}$ . For any rectangle $R$ with corners $(s, t),(u, t),(s, v),(u, v)$ where $0 \leq s < u < \infty$ and $0 \leq t < v < \infty$ , define

Y(R) = X(s, t)+X(u, v)-X(s, v)-X(u, t)

Find $\operatorname {Var}\left[Y(R)\right]$ , and show that $Y$ has independent increments.

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