1.13 A hilbert space for stochastic processes

The result of primary concern here is the construction of a Hilbert space for stochastic processes. The space consisting ofrandom variables $X$ having a finite mean-square value is (almost) a Hilbert space with inner product $(XY)$ . Consequently, the distance between two random variables $X$ and $Y$ is $$d(X, Y)=((X-Y)^2)^{1/2}$$ Now $(d(X, Y)=0)\implies (((X-Y)^2)=0)$ . 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 $$\lim_{t\to u}((X(t)-X(u))^2)=0$$ 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 $$\int_{T_i}^{T_f} (X(t)^2)\,d t$$∞ Stochastic processes in this collection are easily verified to constitute a linear vector space. Define an inner product forthis space as: $$(X(t)\cdot Y(t))=(\int_{T_i}^{T_f} X(t)Y(t)\,d t)$$ While this equation is a valid inner product, the left-hand side will be used to denote the inner product instead of the notationpreviously defined. We take $X(t)\cdot Y(t)$ to be the time-domain inner product as shown here . In this way, the deterministic portion of the inner product and theexpected value portion are explicitly indicated. This convention allows certain theoretical manipulations to beperformed more easily.

One of the more interesting results of the theory of stochastic processes is that the normed vector space for processespreviously defined is separable. Consequently, there exists a complete (and, by assumption, orthonormal) set $\{{}_i(t)\}$ , $i=\{1, \}$ of deterministic (nonrandom) functions which constitutes a basis. A process in the space of stochasticprocesses can be represented as