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A Hilbert Space for Stochastic Processes

Module by: Don Johnson. E-mail the author

The result of primary concern here is the construction of a Hilbert space for stochastic processes. The space consisting of random variables XX having a finite mean-square value is (almost) a Hilbert space with inner product EXY X Y . Consequently, the distance between two random variables XX and YY is dXY=EXY21/2 d X Y X Y 2 1/2 Now (dXY=0)(EXY2=0) d X Y 0 X Y 2 0 . However, this does not imply that X=Y X Y . Those sets with probability zero appear again. Consequently, we do not have a Hilbert space unless we agree X=Y X Y means PrX=Y=1 X Y 1 .

Let Xt X t be a process with EX2t< X t 2 . For each tt, Xt X t is an element of the Hilbert space just defined. Parametrically, Xt X t is therefore regarded as a "curve" in a Hilbert space. This curve is continuous if limit  tuEXtXu2=0 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 L 2 T i T f . Consider the collection of real-valued stochastic processes Xt X t for which T i T f EX2tdt< 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: EXt,Yt=E T i T f XtYtdt 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 Xt,Yt 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 φ i t , i=1 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 :Xt=i=1 X i φ i t t T i t T f X t i 1 X i φ i t
(1)
where X i X i , the representation of Xt X t , is a sequence of random variables given by X i =Xt, φ i t X i X t φ i t or X i = T i T f Xt φ i tdt 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 0T 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 Xt X t and Yt Y t are equal almost everywhere if the distance between XtYt 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."

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