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The Ornstein-Uhlenbeck process $X(t)$ is a centered, Gaussian process with covariance function $$B(s,t) = e^{-\vert t-s \vert /2}$$

The spectral measure is abs. cont. w.r.t. the Lebesgue measure with density

$$f(\lambda) = \frac{1}{2\pi} \frac{1}{\lambda^2+1/4}$$, i.e. Cauchy distribution with parameter $1/2$.

How can one find an explicit representation of the spectral process $Z(\lambda)$?


EDIT: Is it an easier task to find the finite dimensional distributions of the spectral process $(Z(\lambda))_{\lambda \in\mathbb R}$ without explicitly constructing the process?


EDIT2: Yes, it is easier to determine the distribution than to construct an explicit representation of the spectral process.

I first repeat the construction of an isometry that seems to appear in the standard proof of the spectral representation:

Let $H_X$ be the closure in $L^2$ norm of the set of all linear combinations of the form $\sum_{j=1}^n c_j X_{t_j}$, where $H_X$ is a dense linear subspace of $L^2(\mu)$. Then define the isometry

$$I : L^2(\mu) \rightarrow H_X, \quad\sum_{j=1}^n c_j e^{i\lambda t_j} \mapsto \sum_{j=1}^n c_j X_{t_j}$$

Now, one has to consider two important cases ($Z$ is the spectral process to $X$):

$$I(\mathbb I_{(-\infty,\lambda]}) = Z(\lambda)\qquad I(e^{it\lambda}) = X(t)$$

Therefore, in order to say something about $Z$, one wants say something about the isometry $I$ applied to $\mathbb I_{(-\infty,\lambda]}$.

The isometry gives information about the first and second moment of $Z(\lambda)$. Since $X(t)$ is centered, so is $Z(\lambda)$, since $\forall f \in L^2(\mu), \mathbb E[I(f)] = \sum_{k=1}^n c_k \mathbb E[X_{t_k}]=0$. Moreover, by the isometry property:

$$\mathbb E[Z(\lambda_1)Z(\lambda_2)] = \int_{-\infty}^{\infty} \mathbb I_{(-\infty,\lambda_1]}(x) \mathbb I_{(-\infty,\lambda_2]}(x) \mu(dx) \\= \int_{-\infty}^{\min\{\lambda_1,\lambda_2\}} \frac{2}{\pi (1+4x^2)}dx = \frac{1}{\pi} \arctan(2(\min\{\lambda_1,\lambda_2\})) + \frac{1}{2}.$$

Now, $(Z_{\lambda_1},\dots,Z_{\lambda_N})$ is multivariate Gaussian, since sums and limits of Gaussian random variables are again Gaussian, and thus completely determined by the first two moments.

The above question remains though:

How can one find an explicit representation of the spectral process $Z(\lambda)$?

share|improve this question
    
I've added the distribution of the spectral process. –  madison54 Apr 5 '13 at 14:10

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