A Hilbert Space for Stochastic Processes 1.3 2003/05/22 2003/08/08 17:24:02.541 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Eileen Krause erkrause@rice.edu Kyle Clarkson kclarks@rice.edu Elizabeth Gregory lizzardg@rice.edu Kevin Duh kevinduh@rice.edu Mariyah Poonawala mariyah@rice.edu Matthew Jeanes mjeanes@rice.edu Jeffrey Silverman jsilv@rice.edu The result of primary concern here is the construction of a Hilbert space for stochastic processes. The space consisting of random variables X having a finite mean-square value is (almost) a Hilbert space with inner product X Y . Consequently, the distance between two random variables X and Y is d X Y X Y 2 1/2 Now d X Y 0 X Y 2 0 . However, this does not imply that X Y . Those sets with probability zero appear again. Consequently, we do not have a Hilbert space unless we agree X Y means X Y 1 . Let X t be a process with X t 2 . For each t, X t is an element of the Hilbert space just defined. Parametrically, X t is therefore regarded as a "curve" in a Hilbert space. This curve is continuous if t u X t X u 2 0 Processes satisfying this condition are said to be continuous in the quadratic mean. The vector space of greatest importance is analogous to L 2 T i T f . Consider the collection of real-valued stochastic processes X t for which t T i T f X t 2 Stochastic processes in this collection are easily verified to constitute a linear vector space. Define an inner product for this space as: X t Y t t T i T f X t Y t While this equation is a valid inner product, the left-hand side will be used to denote the inner product instead of the notation previously defined. We take X t Y t to be the time-domain inner product as shown here. In this way, the deterministic portion of the inner product and the expected value portion are explicitly indicated. This convention allows certain theoretical manipulations to be performed more easily. One of the more interesting results of the theory of stochastic processes is that the normed vector space for processes previously defined is separable. Consequently, there exists a complete (and, by assumption, orthonormal) set φ i t , i 1 … of deterministic (nonrandom) functions which constitutes a basis. A process in the space of stochastic processes can be represented as t T i t T f X t i 1 X i φ i t where X i , the representation of X t , is a sequence of random variables given by X i X t φ i t or X i t T i T f X t φ i t Strict equality between a process and its representation cannot be assured. Not only does the analogous issue in L 2 0 T occur with respect to representing individual sample functions, but also sample functions assigned a zero probability of occurrence can be troublesome. In fact, the ensemble of any stochastic process can be augmented by a set of sample functions that are not well-behaved (e.g., a sequence of pulses) but have probability zero. In a practical sense, this augmentation is trivial: such members of the process cannot occur. Therefore, one says that two processes X t and Y t are equal almost everywhere if the distance between X t Y t is zero. The implication is that any lack of strict equality between the processes (strict equality means the processes match on a sample-function-by-sample-function basis) is "trivial."